Research ArticleMultistage Effort and the Equity Structure ofVenture Investment Based on Reciprocity Motivation

Chuan Ding1 Jiacheng Chen2 Xin Liu3 and Junjun Zheng4

1 Institute of Financial Mathematics Southwestern University of Finance and Economics Chengdu 610074 China2School of Finance Southwestern University of Finance and Economics Chengdu 610074 China3Department of Accounting Tianfu College of Southwest University of Finance and Economics Mianyang 621000 China4School of Economics and Management Wuhan University Wuhan 430072 China

Correspondence should be addressed to Chuan Ding dingchuanswufeeducn

Received 27 April 2015 Revised 17 August 2015 Accepted 26 August 2015

Academic Editor Alicia Cordero

Copyright copy 2015 Chuan Ding et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For venture capitals it is a long process from an entry to its exit In this paper the activity of venture investment will be divided intomultistages And according to the effort level entrepreneurs will choose the venture capitalists will provide an equity structure atthe very beginning As a benchmark for comparison we will establish two game models on multistage investment under perfectrationality a cooperative gamemodel and a noncooperative one Further as a cause of pervasive psychological preference behaviorreciprocitymotivationwill influence the behavior of the decision-makers Given this situation Rabinrsquos reciprocitymotivation theorywill be applied to the multistage game model of the venture investment and multistage behavior game model will be established aswell based on the reciprocity motivation By looking into the theoretical derivations and simulation studies we find that if venturecapitalists and entrepreneurs both have reciprocity preferences their utility would have been Pareto improvement compared withthose under perfect rationality

1 Introduction

Venture capitals have been playing a crucial role in newunlisted SMEs for a long time especially in new high-techenterprises For example in America venture capital insti-tutions invest 70 of their capitals into high-tech industriesdominated by ITMany successful high-tech enterprises suchas Microsoft and Netscape all once received support duringtheir development by venture capitals According to our datain China 2927 of venture capital institutions focus onhigh-growth fields and 1675 of them focus on high-techareas Venture capital is risky simultaneously even thoughit is vital to the economic development Therefore how tomaintain and increase the value of venture capital has been asignificant study field for scholars Scholars target designinga certain kind of financial contracts to motivate ventureentrepreneurs (later denoted as EN) to pay more effortthat will improve the venture projectsrsquo success probabilityHowever several problems arise Firstly the efforts andactions of EN are implicit thus there is no contracts to

motivate all the efforts of EN so how to make up for thisdefect The actions that financial contracts cannot includecan be resolved through mutual reciprocal affection Suchresearch is inspired by practice For example CEO quietlyprepared for an employee a birthday party (the cost of whichis not included in the wage contract) thus leading to thestaff rsquos willing to pay more efforts to the CEO Employeesrsquopaying more effort will thus naturally improve the outputand therefore both the CEO and employees will benefitfrom it This reciprocal behavior in behavioral economicsis called the theory of reciprocal fairness preference Thereare many other examples in which reciprocity theory hasbeen applied to improve the performance of enterprisesexamples such cases are outstanding in South Korea Japanand China There also exists such a situation in a venturecapital venture capitalist (later denoted as VC) paid the fixedincome to EN and EN will increase efforts to repay VCtherefore the mutual reciprocal of both VC and EN improvesthe utility of both sides Secondly in general VC prefersmultistage investment to venture projectsThis is because VC

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 689362 13 pageshttpdxdoiorg1011552015689362

2 Discrete Dynamics in Nature and Society

will continue investment if the venture project is of goodquality and will withdraw from the project if the projectfails Thirdly VC and ENrsquos reciprocity need to be extendedinto the long term Therefore in this paper we will study theventure investmentrsquos decision-making from two perspectivesmultistages and the reciprocal fairness preference as well

2 Literature Review

Maintaining and increasing the value of venture capital wasan important task in the management of venture capital VCshoped to choose venture projects which had good prospectsin the future and also hoped to select the qualified enterpriseand the manager operated the venture capital to make itappreciated However at the beginning of venture capitalthe VCs usually did not clearly know the ENsrsquo managementability the effort level and the return and risk condition of theproject As a result they often made decisions according tothe principle of maximizing their own benefit such as capitalabuse and over investment which result in moral hazardproblem [1ndash8] Given this situation designing an appropriateequity contract was necessary for the VC to control moralhazard Aiming to ensure that the ENrsquos effort benefited theVC the equity contract realized benefit and risk sharingbetween both sides through coordination mechanism [1ndash3 9ndash11] Sahlman [10] believed that the VC should offerincentive restrictions to the EN based on the observedinformation so that the expected benefit of both sides couldconverge Shleifer and Vishny [11] argued that if controlrights of venture capital institutions were mainly assigned tominor shareholders theVCrsquos benefit to some extent could beprotected against potential losses but it simultaneouslymightcause the issue of insider control

According to current literatures on venture capital stagedfinancing could mitigate moral hazard [12] Dahiya and Ray[12] thought that the VC regarded staging as a mechanismStaged financing provided the VC with the option of endingprojects with low early returns This sorted ventures into twogroups stay or quit It was efficient to quit if the early returnswere weak and to stay otherwise If the ENrsquos early output waslow the VC and the EN might not work hard instead theycollected their respective outside options Venture capitalusually experienced several rounds of investment from seedcapital to final exit whichwas a long-term process Amit et al[13] and Gompers and Lerner [14] researched the multistageissue on venture capital Neher [15] believed that upfrontfinancing was feasible in early time but once the investmentwas sunk the EN was insolvent to deal with the VCrsquosclaim However staged financing could solve these problemsbecause early rounds of investment created collateral forlater financing Ramy and Arieh [9] studied the relationshipbetween the VC and the EN and offered a multistage gamemodel based onmoral hazardThemodel considered the ENrsquoseffort in different stages and designed a multistage incentivecontract to mitigate moral hazard Research showed that theVC should postpone the ENrsquos incentive and the optimalform of contract was debt financing Since Ramy and Arieh[9] only researched the single moral hazard of the EN andignored the moral hazard of the VC Zhang and Wei [16]

made an improvement considering doublemoral hazard andthe effort of both sides in different stages They deduced theoptimal incentive contract analyzed several factors affectingthe contract design andworked out the optimal exit point fortheVC Jin et al [8] supposed that the output functionwas thefunction of the effort in each stage and they studied the VCrsquosfinancing problem in two stages

Early researches only considered the efforts of EN andestablished the principal-agent model called a unilateralmoral hazard model to motivate EN to pay more effortsHowever the success of the project required both ENrsquosexpertise and the need for VCrsquos rich experience in marketingTherefore both sides needed to pay more efforts As withthe previous analysis EN and VC might hide their effortswhich might lead to the double moral hazard and somescholars had established a double moral hazard model Basedon Ramy and Ariehrsquos research Zhang and Wei [16] took thedouble moral hazard into consideration and next were ableto get the design of the contract in multistage investmentWu et al [17] thought personal bounded rationality and theuncertainty of the future benefits of the investment projectmade VCs and ENs face the risk of double moral hazardthey divided the venture capital project into early days andthe development product market stage De Bettignies andBrander [18] examined the ENrsquos choice between bank financeand venture capital With venture capital finance there wasa two-sided moral hazard problem as both EN and VCprovided unverifiable effort The EN benefited from the VCrsquosmanagerial input butmust surrender partial ownership of theventure thus diluting the ENrsquos incentive to provide effort Inincomplete contracts framework De Bettignies [19] thoughtthe EN could design contracts contingent on three possiblecontrol right allocations entrepreneur control investor con-trol and joint control with each allocation inducing differenteffort levels by both the EN and the investorThere were otherrich results see [1 20]

From above we might see that the efforts of VC and ENcould be divided into the single moral hazard and the doublemoral hazard in the management of venture capital

Then these research literatures on multistage venturecapital were all involved with a hypothesis the VC andthe EN were both perfectly rational In recent years manyresearchers had been challenging this hypothesis of tradi-tional economics They argued that not all the behaviorscould be explained by utility maximization of neoclassicaleconomics that is decision-makers were bounded rational-ityThebehavior of decision-makerswith bounded rationalitycould be presented in many aspects and one of them wasreciprocity motivation Ultimatum game dictator game gift-exchange game trust game and the empirical study ofresearchers all clearly showed that participantsrsquo fair prefer-ence was very compelling Since fair preference motivationhad not been subordinated to theoretical frame work ofmainstream economics many new theoretical models werebased on it

Fairness preference was mainly from two aspects Firstplayers concerned about whether the final outcome wasfairness or not Fehr and Schmidt [21] proposed simplelinear utility function model the characteristics of the model

Discrete Dynamics in Nature and Society 3

were that the player was faced with trade-off between hisown interests and the other playerrsquos interests that is to sayplayers would maximize its utility in the material benefitsand the distribution results of fairness preference Secondthese theoretical models were mainly Rabinrsquos [22] reciprocitymotivation theory and it was also called reciprocity intentionReciprocity motivation means the following (1) In responseto friendly actions people would like to sacrifice their ownbenefits to help them (2) Conversely in response to hostileactions people would like to sacrifice their own benefits topunish them (3) The lower the cost of sacrifice was thestronger the power of (1) and (2)was [22] Rabin [22] believedthat people often reacted to othersrsquo actions (friendly or hos-tile) If they perceived the kindness of others they themselveswould like to be friendly in return that is returning thefavor Conversely they would punish others (tit for tat) evenif they had to pay for it Rabinrsquos theory undoubtedly made aground breaking contribution Rabinrsquos theoretical model wasthe first to accurately describe the behavior process based onreciprocity intention and to discuss the significance of thisbehavior Based on the F-S [21] model there had been manyresearch results [23ndash25] in this paper we only consideredthe influence of reciprocity preference on venture investmentRelated research was as follows

Fehr and Falk [26] proved that it would lower the agentrsquoseffort level if we adopted pure material incentives to anagent with reciprocity motivation Fehr and Fischbacher [27]found that the incentive contract which was optimal for pureself-interest people had a worse effect on the agent withreciprocitymotivation instead On the contrary the incentivecontract which behaved poorly for pure self-interest peoplehad a stronger incentive for the agentwith reciprocitymotiva-tion Falk and Ichino [28] and Bartling and Von Siemens [29]found that nonpreference for pure self-interest could improvethe teamrsquos effort level and promote the level of collaborationamong team members and reduce the free-rider problemWei and Pu [30] studied the problem of choices betweenpositive incentive and negative incentive and complete con-tract incentive and incomplete contract incentive under reci-procity motivation Pu [31] successfully incorporated Rabinrsquosidea into classical principal-agent model assuming that theprincipal knew that the agent was irrational and presenteda property of reciprocity in behavior when the principalsacrificed his own interests to give more benefits to the agentthe agent would be willing to sacrifice his own interests toreturn the principalrsquos kindness Pu [32] introduced Rabinrsquosmodel [22] which considered the utility function of bothmaterial well-being and reciprocity motivation into existingprincipal-agent model and derived a new principal-agentmodel in which the agentrsquos behavior represented reciprocaland irrational These studies mainly incorporated reciprocitymotivation into the existing principal-agent model

How did reciprocity motivation affect the behavior ofthe decision-makers Based on the sequential reciprocitygame model of Dufwenberg and Kirchsteiger [33] Wan etal [34] confirmed that managerrsquos credible behavior had apotent incentive effect on employees with reciprocity Shi andPu [35] investigated the incentive effect when participantshad reciprocity preference in different information condition

according to sequential reciprocity model Results clearlyshowed that under complete information when the agenthad sufficient reciprocity preference the principal would giveup enforceable contracts and give the agent more optionsWhen the principal had a reciprocity preference the agentwould have motivation to satisfy the principal before thefinal game and this motivation would effectively inducethe agent to work hard While under incomplete informa-tion compared with the optimal case under the rationalassumption the introduction of reciprocity preference couldmake it possible for the principal to improve his materialutility without reducing the agentrsquos effort level To study thedynamic incentive effect of managersrsquo reciprocity preferenceon employees in twoperiods Pu and Shi [36] built a two-stagesequential game model based on the behaviors of reciprocalprincipals and employees with interest conflicts The resultshowed that under the circumstance of dynamic strategyreciprocity still had significant incentive effect Reciprocitypreference of managers would force employeesrsquo optimalstrategy selection to converge locally and to some extent itreplaced the contract in effect and the principalrsquos benefitunder certain conditions would be greater than that underrational conditions

Recently the amount of literatures about venture capitalunder bounded rationality was small Fairchild [37] built abehavior game model to study the choice of entrepreneursbetween angel investment and venture capital Zheng andWu [24] thought that in the venture capital market the VCsand the ENs presented bounded rationality in the contractdesign especially the ENs In addition to revenue the ENsmight also take individual leisure requirement personalfairness preference and other factors into account Whatthey were pursuing was satisfactory solution rather than theoptimal solution Aiming at venture investors and venturecapitalists with fairness preferences Zheng and Xu [25] builta new payment pattern By designing an effective incentivemechanism they provided a basis for venture capitalistsrsquoestablishing a reasonable payment contract However noneof the prior theoretical papers considered the reciprocitymotivation

This paper focuses on designing behavior capital con-tracts based on reciprocity motivation through introducingreciprocity motivation theory into the design of multistageventure capital contract Our starting point is mainly basedon the following three aspects First multistage venturecapital is pervasive in practice Instead of providing all theinvestment upfront the VC invests in stages to control risksand makes refinancing decisions according to the projectcondition There are some research results currently butfurther study is still needed Second in practice the VC andthe EN are bounded rationality and reciprocity motivationis also a common psychological preference If the EN inputsmore effort the VC may offer a higher share in returnOf course this reciprocity is bilateral Third majority ofresearch results study the design of capital structure contractsunder perfectly rationality and derive corresponding utilityor returns Thus if we can find a capital structure contractunder reciprocitymotivation inwhich the utility or returns ofboth sides are Pareto improvement of perfect rationality our

4 Discrete Dynamics in Nature and Society

research perspective will have more practical and theoreticalsignificance

Our main ideas are as follows If the VC cooperates withthe EN the ENrsquos effort will achieve the optimal We call theeffort in this case the first-best Therefore the first step is tocalculate the first-best solution of optimal effort during thecooperationThen secondly we calculate the optimal decisionof both sides under perfect rationality in the assumption thatthe two parties make their decisions independently We callthe solution here the second-best The first-best solution isthe optimal upper limit of both sides and the second-bestsolution is the lower limit of both sides The third step is toconstruct a multistage game model on venture capital basedon reciprocity motivation through introducing Rabinrsquos [22]model

We have two goals The first is to find the optimal effortlevel between the first-best solution and the second-bestsolution and to confirm that the capital structure contract isalso between the first-best and the second-best solution Thesecond goal is to confirm that under reciprocity motivationthe utility or returns of both sides are Pareto improvementcompared with that under perfect rationality

This paper proceeds as follows In the next section we putforward the assumptions notations and basic descriptionsof the model As a basis for comparison we consider themultistage decision-making model and solutions under per-fect rationality in Section 3 Sections 4 and 5 are respectivelythe theory and the construction of the reciprocity motivationpreferences utility function Section 6 is the solution of theventure capital decision-making model under reciprocitymotivation Given the complexity of the model we usesimulation to confirm the existence of the venture capitalincentive contract under reciprocity motivation in Section 7Finally Section 8 concludes the paper and directions forfuture research are proposed

3 Model Description and Assumptions

31 Model Description Consider an innovative entrepreneur(EN) who relies on a venture capitalist (VC) for investmentThe VC provides a capital contract for the EN If the ENaccepts the contract will be executed If not the ENrsquosreservation utility will be zero We assume that both the ENand the VC are risk neutral The VC stages the investmentinto 119899 periods to control risks and mitigate moral hazardAnd the VC has the option to continue or exit at each endof the period The effort of the EN is different in differentperiods And the more the effort EN inputs the higher thesuccess rate of the project isTheVCrsquos goal is to select a capitalstructure contract to maximize his utility while the ENrsquos goalis to choose an effort level to maximize his utility Time lineof game is depicted in Figure 1

32 Research Assumptions

Assumptions 1 We assume that both the EN and the VC arerisk neutralThat is they have equivalent risk tolerance facingpossible profit The ENrsquos decision goal is to maximize the

End

EN does not acceptVC provides Ik 120572 120573

EN acceptsReciprocity

EN chooses effort ek

State of nature realized

Outcome pk Success

1 minus pk Failure

EN gets 120572 and the game ends

Figure 1 The time line

utility in revenue the VCrsquos decision goal is to maximize theutility of capital gains

Assumptions 2 VC invests capital to the venture company Itis a long-term process and the venture capital is divided into119899 stages (119899 ge 2)

Assumptions 3 If the VC and the EN reach the investmentagreement the VC will provide the external capital 119868119896 (119868119896 ge0) in stage 119896 119896 = 1 2 3 119899The averagemarket return rateof external capital 119868119896 is 119903 0 lt 119903 lt 1 Thus the capital cost forthe VC is (1 + 119903)119868119896

Assumptions 4 It is a long-term process after the VC inputscapital to the venture enterprise The effort of the EN isdifferent in different periods Naturally venture investmentshould be carried out in stages So we divide the long-termpartnership into 119899 stages (119899 isin 119873) The ENrsquos effort is 119890119896119896 = 1 2 3 119899 in stage 119896 119896 = 1 2 3 119899 We can regardeffort cost as monetary cost For simplicity we assume thatthe effort cost for the EN is 119888(119890119896) = 1198871198902

1198962 119887 ge 0 One

property of this function is that the cost will increase as theeffort increases and the increasing is faster and faster that is1198881015840(119890119896) gt 0 119888

10158401015840(119890119896) gt 0

Assumptions 5 The probability of success is 119901 0 le 119901 le 1

after 119899 periods The success of the venture project is relatedto ENrsquos effort The more the effort EN inputs the greaterthe probability of success is and the success in stage 119896 isinseparable from the early effort So we assume that 119901119896 =11989011198902 sdot sdot sdot 119890119896 0 le 119901119896 = 11989011198902 sdot sdot sdot 119890119896 = prod

119896

119894=1119890119894 le 1 Let 119878 be the

projectrsquos return (a fixed cash flow) if the project is successfuluntil stage 119896 In the multiperiod model the probability ofproject success or failure can be represented by the pastefforts of EN similar studies can be referred to in Yangrsquos

Discrete Dynamics in Nature and Society 5

K

e1

1

e2

2

middot middot middot

Benefits of successful projects S

EN accepts contract

Probability of success pk =

ek

e1e2 middot middot middot ek

Figure 2 Success after 119896-stage

method [38] However the Yangrsquosmodel is a continuous-timemodel while our model is a discrete-time model Figure 2shows the success probability and return of venture projects

For simplicity we assume that the return in every periodis equal that is 119878 ge 0 The return is 0 if the project is faileduntil stage 119896 So the expected return of the project in stage 119896is

119881119896 = 119901119896119878119896 + (1 minus 119901) 0 = 119901119896119878119896 = 119878

119896

prod119894=1

119890119894 (1)

Assumptions 6 After the VC invests the capital at thebeginning of the project the VC gives fixed income 120572 andthe revenue sharing coefficient 120573 (0 le 120573 le 1) in each periodto the EN We call (120572 120573) the equity contract on investment(120572 120573) are also known as financial contracts If EN acceptscontract EN andVChave long-term game relationship if ENrefuses it the game is over

Assumptions 7 For simplicity there is no consideration forthe time value of the return We can also assume thatventure capitalistrsquos discount factor is 120575119862 and the ventureentrepreneurrsquos discount factor is 120575119864 It will increase thecomplexity of model but will not affect the conclusions

4 Decision Model and Solution ofMultistage under Perfect Rationality

Let 120587 be the profit of the EN Given the fixed income 120572 thereturn in each stage 120573119881119896 and the total effort costsum

119899

119896=1119888(119890119896) =

sum119899

119896=1(11988711989021198962) the expected profit of the EN is

120587 = 120572 +

119899

sum119896=1

120573119881119896 minus

119899

sum119896=1

1198871198902119896

2

= 120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2

(2)

The EN chooses effort level in each stage 1198901 1198902 119890119899 tomaximize the utility namely

max11989011198902119890119899

120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2 (3)

Since the VC is risk neutral VCrsquos expected utility equalsthe expected return

Π = minus120572 +

119899

sum119896=1

(1 minus 120573)119881119896 minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896

(4)

The VC chooses appropriate equity contract (120572 120573) tomaximize the expected utility The maximization problem is

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (5)

41TheOptimal Decision of Cooperation TheEN and the VCrealize cooperation through one side completely controllingproject that is the owner of the project and the provider ofthe investment are the same decision-maker Consequentlyadding (2) to (4) we derive

Π119862=

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (6)

The decision-maker chooses effort level in each stage1198901 1198902 119890119899 to maximize the overall utility namely

max11989011198902119890119899

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (7)

We derive the optimal solution 119890119862lowast119896= (119878119887)1(119899minus2)

42 The Optimal Decision of Noncooperation Under nonco-operation the VC and the EN make independent decisionsto maximize their own utility or return The VCrsquos goal is themaximization of the future return Thus the VC designs acapital structure contract to maximize the total return at theend of stage 119899

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (8)

However the EN also cares about the future utilitymaximization while focusing on maximizing the currentutility [39] That means the EN will choose the maximumeffort level 1198901 tomaximize the return120572+1205731198901119878minus119887119890

2

12 in stage 1

Then in stage 2 the ENwill choose the maximum effort level1198902 to maximize the total return 120572 + 120573prod2

119896=1119890119896119878 minus sum

2

119896=1(11988711989021198962)

of the two stagesIn stage 119899 the EN will choose the maximum effort level

119890119899 to maximize the total return 120572 + 120573prod119899119896=1119890119896119878 minus sum

119899

119896=1(11988711989021198962)

of the former 119899 stages

6 Discrete Dynamics in Nature and Society

In this game the VC provides capital structure contractfirstly and then the EN chooses his own effort level There-fore this game is Stackelberg game It can be described asfollows (I)

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (9)

st max1198901

120572 + 1205731198901119878 minus11988711989021

2 (10a)

max1198902

120572 + 120573

2

prod119896=1

119890119896119878 minus

2

sum119896=1

1198871198902119896

2

(10b)

max119890119899

120572 + 120573

119899

prod119896=1

119890119896119878 minus

119899

sum119896=1

1198871198902119896

2 (10c)

In model (I) the first-order condition about 1198901 1198902 119890119899in (10a) (10b) and (10c) is

120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(11)

By introducing the first-order condition to model (I) theVCrsquos problem becomes

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (12)

st 120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(13)

Put formula (13) into objective function (12) we derivethe following

max120572120573

minus 120572 +119878(1198992+119899+2)2

119887119899(119899+1)2(1 minus 120573) 120573

119899(119899+1)2minus

119899

sum119896=1

(1 + 119903) 119868119896 (14)

From formula (14) the first-order condition about 120572 and120573 is

120572lowast= 0

120573lowast=

1198992 + 119899

1198992 + 119899 + 2

(15)

Note 1 In (15) specially when 119899 = 1 120573 = 12 this wouldtie in exactly with a one-period single-sided moral hazardmodel the VC with no value-creating ability would give halfof the equity to the EN tomotivate the EN To be sure VC hasno value-creating ability why does VC give half of the equityto the EN The reasons are as follows (1) VCrsquos investment

needs to be paid and ENrsquos projects and efforts need to berewarded because the modelrsquos parameters are set differentlythe result is precisely 120573 = 12 (2) Our results have somedifferences with Fairchild [37] which is because themodelingmechanism is differentWe are sure that Fairchild [23 37 40]provides a lot of important research results in the field ofventure capital

Therefore the ENrsquos optimal effort level in each stage is

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896 119896 = 1 2 119899 (16)

Put (15) and (16) into the function we derive

120587lowast

VC =2119878

1198992 + 119899 + 2

119899

prod119896=1

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

=2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2

119899

prod119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

minus119887

2

119899

sum119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

2

=(1198992+ 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992+ 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(17)

So there is Proposition 1

Proposition 1 If the VC cooperates with the EN the ENrsquosoptimal effort level is 119890119862lowast

119896= (119887119878)

1(119899minus2) (first-best) If the VCand the EN make decisions independently the ENrsquos optimaleffort level is 119890lowast

119896= (119878119896119887119896)((1198992 + 119899)119896(1198992 + 119899 + 2)119896) (second-

best) The optimal venture capital structure contract is 120572lowast = 0120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) The optimal benefits on both sidesrespectively are as follows

120587lowast

VC =2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(18)

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

will continue investment if the venture project is of goodquality and will withdraw from the project if the projectfails Thirdly VC and ENrsquos reciprocity need to be extendedinto the long term Therefore in this paper we will study theventure investmentrsquos decision-making from two perspectivesmultistages and the reciprocal fairness preference as well

2 Literature Review

Maintaining and increasing the value of venture capital wasan important task in the management of venture capital VCshoped to choose venture projects which had good prospectsin the future and also hoped to select the qualified enterpriseand the manager operated the venture capital to make itappreciated However at the beginning of venture capitalthe VCs usually did not clearly know the ENsrsquo managementability the effort level and the return and risk condition of theproject As a result they often made decisions according tothe principle of maximizing their own benefit such as capitalabuse and over investment which result in moral hazardproblem [1ndash8] Given this situation designing an appropriateequity contract was necessary for the VC to control moralhazard Aiming to ensure that the ENrsquos effort benefited theVC the equity contract realized benefit and risk sharingbetween both sides through coordination mechanism [1ndash3 9ndash11] Sahlman [10] believed that the VC should offerincentive restrictions to the EN based on the observedinformation so that the expected benefit of both sides couldconverge Shleifer and Vishny [11] argued that if controlrights of venture capital institutions were mainly assigned tominor shareholders theVCrsquos benefit to some extent could beprotected against potential losses but it simultaneouslymightcause the issue of insider control

According to current literatures on venture capital stagedfinancing could mitigate moral hazard [12] Dahiya and Ray[12] thought that the VC regarded staging as a mechanismStaged financing provided the VC with the option of endingprojects with low early returns This sorted ventures into twogroups stay or quit It was efficient to quit if the early returnswere weak and to stay otherwise If the ENrsquos early output waslow the VC and the EN might not work hard instead theycollected their respective outside options Venture capitalusually experienced several rounds of investment from seedcapital to final exit whichwas a long-term process Amit et al[13] and Gompers and Lerner [14] researched the multistageissue on venture capital Neher [15] believed that upfrontfinancing was feasible in early time but once the investmentwas sunk the EN was insolvent to deal with the VCrsquosclaim However staged financing could solve these problemsbecause early rounds of investment created collateral forlater financing Ramy and Arieh [9] studied the relationshipbetween the VC and the EN and offered a multistage gamemodel based onmoral hazardThemodel considered the ENrsquoseffort in different stages and designed a multistage incentivecontract to mitigate moral hazard Research showed that theVC should postpone the ENrsquos incentive and the optimalform of contract was debt financing Since Ramy and Arieh[9] only researched the single moral hazard of the EN andignored the moral hazard of the VC Zhang and Wei [16]

made an improvement considering doublemoral hazard andthe effort of both sides in different stages They deduced theoptimal incentive contract analyzed several factors affectingthe contract design andworked out the optimal exit point fortheVC Jin et al [8] supposed that the output functionwas thefunction of the effort in each stage and they studied the VCrsquosfinancing problem in two stages

Early researches only considered the efforts of EN andestablished the principal-agent model called a unilateralmoral hazard model to motivate EN to pay more effortsHowever the success of the project required both ENrsquosexpertise and the need for VCrsquos rich experience in marketingTherefore both sides needed to pay more efforts As withthe previous analysis EN and VC might hide their effortswhich might lead to the double moral hazard and somescholars had established a double moral hazard model Basedon Ramy and Ariehrsquos research Zhang and Wei [16] took thedouble moral hazard into consideration and next were ableto get the design of the contract in multistage investmentWu et al [17] thought personal bounded rationality and theuncertainty of the future benefits of the investment projectmade VCs and ENs face the risk of double moral hazardthey divided the venture capital project into early days andthe development product market stage De Bettignies andBrander [18] examined the ENrsquos choice between bank financeand venture capital With venture capital finance there wasa two-sided moral hazard problem as both EN and VCprovided unverifiable effort The EN benefited from the VCrsquosmanagerial input butmust surrender partial ownership of theventure thus diluting the ENrsquos incentive to provide effort Inincomplete contracts framework De Bettignies [19] thoughtthe EN could design contracts contingent on three possiblecontrol right allocations entrepreneur control investor con-trol and joint control with each allocation inducing differenteffort levels by both the EN and the investorThere were otherrich results see [1 20]

From above we might see that the efforts of VC and ENcould be divided into the single moral hazard and the doublemoral hazard in the management of venture capital

Then these research literatures on multistage venturecapital were all involved with a hypothesis the VC andthe EN were both perfectly rational In recent years manyresearchers had been challenging this hypothesis of tradi-tional economics They argued that not all the behaviorscould be explained by utility maximization of neoclassicaleconomics that is decision-makers were bounded rational-ityThebehavior of decision-makerswith bounded rationalitycould be presented in many aspects and one of them wasreciprocity motivation Ultimatum game dictator game gift-exchange game trust game and the empirical study ofresearchers all clearly showed that participantsrsquo fair prefer-ence was very compelling Since fair preference motivationhad not been subordinated to theoretical frame work ofmainstream economics many new theoretical models werebased on it

Fairness preference was mainly from two aspects Firstplayers concerned about whether the final outcome wasfairness or not Fehr and Schmidt [21] proposed simplelinear utility function model the characteristics of the model

Discrete Dynamics in Nature and Society 3

were that the player was faced with trade-off between hisown interests and the other playerrsquos interests that is to sayplayers would maximize its utility in the material benefitsand the distribution results of fairness preference Secondthese theoretical models were mainly Rabinrsquos [22] reciprocitymotivation theory and it was also called reciprocity intentionReciprocity motivation means the following (1) In responseto friendly actions people would like to sacrifice their ownbenefits to help them (2) Conversely in response to hostileactions people would like to sacrifice their own benefits topunish them (3) The lower the cost of sacrifice was thestronger the power of (1) and (2)was [22] Rabin [22] believedthat people often reacted to othersrsquo actions (friendly or hos-tile) If they perceived the kindness of others they themselveswould like to be friendly in return that is returning thefavor Conversely they would punish others (tit for tat) evenif they had to pay for it Rabinrsquos theory undoubtedly made aground breaking contribution Rabinrsquos theoretical model wasthe first to accurately describe the behavior process based onreciprocity intention and to discuss the significance of thisbehavior Based on the F-S [21] model there had been manyresearch results [23ndash25] in this paper we only consideredthe influence of reciprocity preference on venture investmentRelated research was as follows

Fehr and Falk [26] proved that it would lower the agentrsquoseffort level if we adopted pure material incentives to anagent with reciprocity motivation Fehr and Fischbacher [27]found that the incentive contract which was optimal for pureself-interest people had a worse effect on the agent withreciprocitymotivation instead On the contrary the incentivecontract which behaved poorly for pure self-interest peoplehad a stronger incentive for the agentwith reciprocitymotiva-tion Falk and Ichino [28] and Bartling and Von Siemens [29]found that nonpreference for pure self-interest could improvethe teamrsquos effort level and promote the level of collaborationamong team members and reduce the free-rider problemWei and Pu [30] studied the problem of choices betweenpositive incentive and negative incentive and complete con-tract incentive and incomplete contract incentive under reci-procity motivation Pu [31] successfully incorporated Rabinrsquosidea into classical principal-agent model assuming that theprincipal knew that the agent was irrational and presenteda property of reciprocity in behavior when the principalsacrificed his own interests to give more benefits to the agentthe agent would be willing to sacrifice his own interests toreturn the principalrsquos kindness Pu [32] introduced Rabinrsquosmodel [22] which considered the utility function of bothmaterial well-being and reciprocity motivation into existingprincipal-agent model and derived a new principal-agentmodel in which the agentrsquos behavior represented reciprocaland irrational These studies mainly incorporated reciprocitymotivation into the existing principal-agent model

How did reciprocity motivation affect the behavior ofthe decision-makers Based on the sequential reciprocitygame model of Dufwenberg and Kirchsteiger [33] Wan etal [34] confirmed that managerrsquos credible behavior had apotent incentive effect on employees with reciprocity Shi andPu [35] investigated the incentive effect when participantshad reciprocity preference in different information condition

according to sequential reciprocity model Results clearlyshowed that under complete information when the agenthad sufficient reciprocity preference the principal would giveup enforceable contracts and give the agent more optionsWhen the principal had a reciprocity preference the agentwould have motivation to satisfy the principal before thefinal game and this motivation would effectively inducethe agent to work hard While under incomplete informa-tion compared with the optimal case under the rationalassumption the introduction of reciprocity preference couldmake it possible for the principal to improve his materialutility without reducing the agentrsquos effort level To study thedynamic incentive effect of managersrsquo reciprocity preferenceon employees in twoperiods Pu and Shi [36] built a two-stagesequential game model based on the behaviors of reciprocalprincipals and employees with interest conflicts The resultshowed that under the circumstance of dynamic strategyreciprocity still had significant incentive effect Reciprocitypreference of managers would force employeesrsquo optimalstrategy selection to converge locally and to some extent itreplaced the contract in effect and the principalrsquos benefitunder certain conditions would be greater than that underrational conditions

Recently the amount of literatures about venture capitalunder bounded rationality was small Fairchild [37] built abehavior game model to study the choice of entrepreneursbetween angel investment and venture capital Zheng andWu [24] thought that in the venture capital market the VCsand the ENs presented bounded rationality in the contractdesign especially the ENs In addition to revenue the ENsmight also take individual leisure requirement personalfairness preference and other factors into account Whatthey were pursuing was satisfactory solution rather than theoptimal solution Aiming at venture investors and venturecapitalists with fairness preferences Zheng and Xu [25] builta new payment pattern By designing an effective incentivemechanism they provided a basis for venture capitalistsrsquoestablishing a reasonable payment contract However noneof the prior theoretical papers considered the reciprocitymotivation

This paper focuses on designing behavior capital con-tracts based on reciprocity motivation through introducingreciprocity motivation theory into the design of multistageventure capital contract Our starting point is mainly basedon the following three aspects First multistage venturecapital is pervasive in practice Instead of providing all theinvestment upfront the VC invests in stages to control risksand makes refinancing decisions according to the projectcondition There are some research results currently butfurther study is still needed Second in practice the VC andthe EN are bounded rationality and reciprocity motivationis also a common psychological preference If the EN inputsmore effort the VC may offer a higher share in returnOf course this reciprocity is bilateral Third majority ofresearch results study the design of capital structure contractsunder perfectly rationality and derive corresponding utilityor returns Thus if we can find a capital structure contractunder reciprocitymotivation inwhich the utility or returns ofboth sides are Pareto improvement of perfect rationality our

4 Discrete Dynamics in Nature and Society

research perspective will have more practical and theoreticalsignificance

Our main ideas are as follows If the VC cooperates withthe EN the ENrsquos effort will achieve the optimal We call theeffort in this case the first-best Therefore the first step is tocalculate the first-best solution of optimal effort during thecooperationThen secondly we calculate the optimal decisionof both sides under perfect rationality in the assumption thatthe two parties make their decisions independently We callthe solution here the second-best The first-best solution isthe optimal upper limit of both sides and the second-bestsolution is the lower limit of both sides The third step is toconstruct a multistage game model on venture capital basedon reciprocity motivation through introducing Rabinrsquos [22]model

We have two goals The first is to find the optimal effortlevel between the first-best solution and the second-bestsolution and to confirm that the capital structure contract isalso between the first-best and the second-best solution Thesecond goal is to confirm that under reciprocity motivationthe utility or returns of both sides are Pareto improvementcompared with that under perfect rationality

This paper proceeds as follows In the next section we putforward the assumptions notations and basic descriptionsof the model As a basis for comparison we consider themultistage decision-making model and solutions under per-fect rationality in Section 3 Sections 4 and 5 are respectivelythe theory and the construction of the reciprocity motivationpreferences utility function Section 6 is the solution of theventure capital decision-making model under reciprocitymotivation Given the complexity of the model we usesimulation to confirm the existence of the venture capitalincentive contract under reciprocity motivation in Section 7Finally Section 8 concludes the paper and directions forfuture research are proposed

3 Model Description and Assumptions

31 Model Description Consider an innovative entrepreneur(EN) who relies on a venture capitalist (VC) for investmentThe VC provides a capital contract for the EN If the ENaccepts the contract will be executed If not the ENrsquosreservation utility will be zero We assume that both the ENand the VC are risk neutral The VC stages the investmentinto 119899 periods to control risks and mitigate moral hazardAnd the VC has the option to continue or exit at each endof the period The effort of the EN is different in differentperiods And the more the effort EN inputs the higher thesuccess rate of the project isTheVCrsquos goal is to select a capitalstructure contract to maximize his utility while the ENrsquos goalis to choose an effort level to maximize his utility Time lineof game is depicted in Figure 1

32 Research Assumptions

Assumptions 1 We assume that both the EN and the VC arerisk neutralThat is they have equivalent risk tolerance facingpossible profit The ENrsquos decision goal is to maximize the

End

EN does not acceptVC provides Ik 120572 120573

EN acceptsReciprocity

EN chooses effort ek

State of nature realized

Outcome pk Success

1 minus pk Failure

EN gets 120572 and the game ends

Figure 1 The time line

utility in revenue the VCrsquos decision goal is to maximize theutility of capital gains

Assumptions 2 VC invests capital to the venture company Itis a long-term process and the venture capital is divided into119899 stages (119899 ge 2)

Assumptions 3 If the VC and the EN reach the investmentagreement the VC will provide the external capital 119868119896 (119868119896 ge0) in stage 119896 119896 = 1 2 3 119899The averagemarket return rateof external capital 119868119896 is 119903 0 lt 119903 lt 1 Thus the capital cost forthe VC is (1 + 119903)119868119896

Assumptions 4 It is a long-term process after the VC inputscapital to the venture enterprise The effort of the EN isdifferent in different periods Naturally venture investmentshould be carried out in stages So we divide the long-termpartnership into 119899 stages (119899 isin 119873) The ENrsquos effort is 119890119896119896 = 1 2 3 119899 in stage 119896 119896 = 1 2 3 119899 We can regardeffort cost as monetary cost For simplicity we assume thatthe effort cost for the EN is 119888(119890119896) = 1198871198902

1198962 119887 ge 0 One

property of this function is that the cost will increase as theeffort increases and the increasing is faster and faster that is1198881015840(119890119896) gt 0 119888

10158401015840(119890119896) gt 0

Assumptions 5 The probability of success is 119901 0 le 119901 le 1

after 119899 periods The success of the venture project is relatedto ENrsquos effort The more the effort EN inputs the greaterthe probability of success is and the success in stage 119896 isinseparable from the early effort So we assume that 119901119896 =11989011198902 sdot sdot sdot 119890119896 0 le 119901119896 = 11989011198902 sdot sdot sdot 119890119896 = prod

119896

119894=1119890119894 le 1 Let 119878 be the

projectrsquos return (a fixed cash flow) if the project is successfuluntil stage 119896 In the multiperiod model the probability ofproject success or failure can be represented by the pastefforts of EN similar studies can be referred to in Yangrsquos

Discrete Dynamics in Nature and Society 5

K

e1

1

e2

2

middot middot middot

Benefits of successful projects S

EN accepts contract

Probability of success pk =

ek

e1e2 middot middot middot ek

Figure 2 Success after 119896-stage

method [38] However the Yangrsquosmodel is a continuous-timemodel while our model is a discrete-time model Figure 2shows the success probability and return of venture projects

For simplicity we assume that the return in every periodis equal that is 119878 ge 0 The return is 0 if the project is faileduntil stage 119896 So the expected return of the project in stage 119896is

119881119896 = 119901119896119878119896 + (1 minus 119901) 0 = 119901119896119878119896 = 119878

119896

prod119894=1

119890119894 (1)

Assumptions 6 After the VC invests the capital at thebeginning of the project the VC gives fixed income 120572 andthe revenue sharing coefficient 120573 (0 le 120573 le 1) in each periodto the EN We call (120572 120573) the equity contract on investment(120572 120573) are also known as financial contracts If EN acceptscontract EN andVChave long-term game relationship if ENrefuses it the game is over

Assumptions 7 For simplicity there is no consideration forthe time value of the return We can also assume thatventure capitalistrsquos discount factor is 120575119862 and the ventureentrepreneurrsquos discount factor is 120575119864 It will increase thecomplexity of model but will not affect the conclusions

4 Decision Model and Solution ofMultistage under Perfect Rationality

Let 120587 be the profit of the EN Given the fixed income 120572 thereturn in each stage 120573119881119896 and the total effort costsum

119899

119896=1119888(119890119896) =

sum119899

119896=1(11988711989021198962) the expected profit of the EN is

120587 = 120572 +

119899

sum119896=1

120573119881119896 minus

119899

sum119896=1

1198871198902119896

2

= 120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2

(2)

The EN chooses effort level in each stage 1198901 1198902 119890119899 tomaximize the utility namely

max11989011198902119890119899

120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2 (3)

Since the VC is risk neutral VCrsquos expected utility equalsthe expected return

Π = minus120572 +

119899

sum119896=1

(1 minus 120573)119881119896 minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896

(4)

The VC chooses appropriate equity contract (120572 120573) tomaximize the expected utility The maximization problem is

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (5)

41TheOptimal Decision of Cooperation TheEN and the VCrealize cooperation through one side completely controllingproject that is the owner of the project and the provider ofthe investment are the same decision-maker Consequentlyadding (2) to (4) we derive

Π119862=

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (6)

The decision-maker chooses effort level in each stage1198901 1198902 119890119899 to maximize the overall utility namely

max11989011198902119890119899

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (7)

We derive the optimal solution 119890119862lowast119896= (119878119887)1(119899minus2)

42 The Optimal Decision of Noncooperation Under nonco-operation the VC and the EN make independent decisionsto maximize their own utility or return The VCrsquos goal is themaximization of the future return Thus the VC designs acapital structure contract to maximize the total return at theend of stage 119899

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (8)

However the EN also cares about the future utilitymaximization while focusing on maximizing the currentutility [39] That means the EN will choose the maximumeffort level 1198901 tomaximize the return120572+1205731198901119878minus119887119890

2

12 in stage 1

Then in stage 2 the ENwill choose the maximum effort level1198902 to maximize the total return 120572 + 120573prod2

119896=1119890119896119878 minus sum

2

119896=1(11988711989021198962)

of the two stagesIn stage 119899 the EN will choose the maximum effort level

119890119899 to maximize the total return 120572 + 120573prod119899119896=1119890119896119878 minus sum

119899

119896=1(11988711989021198962)

of the former 119899 stages

6 Discrete Dynamics in Nature and Society

In this game the VC provides capital structure contractfirstly and then the EN chooses his own effort level There-fore this game is Stackelberg game It can be described asfollows (I)

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (9)

st max1198901

120572 + 1205731198901119878 minus11988711989021

2 (10a)

max1198902

120572 + 120573

2

prod119896=1

119890119896119878 minus

2

sum119896=1

1198871198902119896

2

(10b)

max119890119899

120572 + 120573

119899

prod119896=1

119890119896119878 minus

119899

sum119896=1

1198871198902119896

2 (10c)

In model (I) the first-order condition about 1198901 1198902 119890119899in (10a) (10b) and (10c) is

120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(11)

By introducing the first-order condition to model (I) theVCrsquos problem becomes

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (12)

st 120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(13)

Put formula (13) into objective function (12) we derivethe following

max120572120573

minus 120572 +119878(1198992+119899+2)2

119887119899(119899+1)2(1 minus 120573) 120573

119899(119899+1)2minus

119899

sum119896=1

(1 + 119903) 119868119896 (14)

From formula (14) the first-order condition about 120572 and120573 is

120572lowast= 0

120573lowast=

1198992 + 119899

1198992 + 119899 + 2

(15)

Note 1 In (15) specially when 119899 = 1 120573 = 12 this wouldtie in exactly with a one-period single-sided moral hazardmodel the VC with no value-creating ability would give halfof the equity to the EN tomotivate the EN To be sure VC hasno value-creating ability why does VC give half of the equityto the EN The reasons are as follows (1) VCrsquos investment

needs to be paid and ENrsquos projects and efforts need to berewarded because the modelrsquos parameters are set differentlythe result is precisely 120573 = 12 (2) Our results have somedifferences with Fairchild [37] which is because themodelingmechanism is differentWe are sure that Fairchild [23 37 40]provides a lot of important research results in the field ofventure capital

Therefore the ENrsquos optimal effort level in each stage is

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896 119896 = 1 2 119899 (16)

Put (15) and (16) into the function we derive

120587lowast

VC =2119878

1198992 + 119899 + 2

119899

prod119896=1

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

=2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2

119899

prod119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

minus119887

2

119899

sum119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

2

=(1198992+ 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992+ 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(17)

So there is Proposition 1

Proposition 1 If the VC cooperates with the EN the ENrsquosoptimal effort level is 119890119862lowast

119896= (119887119878)

1(119899minus2) (first-best) If the VCand the EN make decisions independently the ENrsquos optimaleffort level is 119890lowast

119896= (119878119896119887119896)((1198992 + 119899)119896(1198992 + 119899 + 2)119896) (second-

best) The optimal venture capital structure contract is 120572lowast = 0120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) The optimal benefits on both sidesrespectively are as follows

120587lowast

VC =2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(18)

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

were that the player was faced with trade-off between hisown interests and the other playerrsquos interests that is to sayplayers would maximize its utility in the material benefitsand the distribution results of fairness preference Secondthese theoretical models were mainly Rabinrsquos [22] reciprocitymotivation theory and it was also called reciprocity intentionReciprocity motivation means the following (1) In responseto friendly actions people would like to sacrifice their ownbenefits to help them (2) Conversely in response to hostileactions people would like to sacrifice their own benefits topunish them (3) The lower the cost of sacrifice was thestronger the power of (1) and (2)was [22] Rabin [22] believedthat people often reacted to othersrsquo actions (friendly or hos-tile) If they perceived the kindness of others they themselveswould like to be friendly in return that is returning thefavor Conversely they would punish others (tit for tat) evenif they had to pay for it Rabinrsquos theory undoubtedly made aground breaking contribution Rabinrsquos theoretical model wasthe first to accurately describe the behavior process based onreciprocity intention and to discuss the significance of thisbehavior Based on the F-S [21] model there had been manyresearch results [23ndash25] in this paper we only consideredthe influence of reciprocity preference on venture investmentRelated research was as follows

Fehr and Falk [26] proved that it would lower the agentrsquoseffort level if we adopted pure material incentives to anagent with reciprocity motivation Fehr and Fischbacher [27]found that the incentive contract which was optimal for pureself-interest people had a worse effect on the agent withreciprocitymotivation instead On the contrary the incentivecontract which behaved poorly for pure self-interest peoplehad a stronger incentive for the agentwith reciprocitymotiva-tion Falk and Ichino [28] and Bartling and Von Siemens [29]found that nonpreference for pure self-interest could improvethe teamrsquos effort level and promote the level of collaborationamong team members and reduce the free-rider problemWei and Pu [30] studied the problem of choices betweenpositive incentive and negative incentive and complete con-tract incentive and incomplete contract incentive under reci-procity motivation Pu [31] successfully incorporated Rabinrsquosidea into classical principal-agent model assuming that theprincipal knew that the agent was irrational and presenteda property of reciprocity in behavior when the principalsacrificed his own interests to give more benefits to the agentthe agent would be willing to sacrifice his own interests toreturn the principalrsquos kindness Pu [32] introduced Rabinrsquosmodel [22] which considered the utility function of bothmaterial well-being and reciprocity motivation into existingprincipal-agent model and derived a new principal-agentmodel in which the agentrsquos behavior represented reciprocaland irrational These studies mainly incorporated reciprocitymotivation into the existing principal-agent model

How did reciprocity motivation affect the behavior ofthe decision-makers Based on the sequential reciprocitygame model of Dufwenberg and Kirchsteiger [33] Wan etal [34] confirmed that managerrsquos credible behavior had apotent incentive effect on employees with reciprocity Shi andPu [35] investigated the incentive effect when participantshad reciprocity preference in different information condition

according to sequential reciprocity model Results clearlyshowed that under complete information when the agenthad sufficient reciprocity preference the principal would giveup enforceable contracts and give the agent more optionsWhen the principal had a reciprocity preference the agentwould have motivation to satisfy the principal before thefinal game and this motivation would effectively inducethe agent to work hard While under incomplete informa-tion compared with the optimal case under the rationalassumption the introduction of reciprocity preference couldmake it possible for the principal to improve his materialutility without reducing the agentrsquos effort level To study thedynamic incentive effect of managersrsquo reciprocity preferenceon employees in twoperiods Pu and Shi [36] built a two-stagesequential game model based on the behaviors of reciprocalprincipals and employees with interest conflicts The resultshowed that under the circumstance of dynamic strategyreciprocity still had significant incentive effect Reciprocitypreference of managers would force employeesrsquo optimalstrategy selection to converge locally and to some extent itreplaced the contract in effect and the principalrsquos benefitunder certain conditions would be greater than that underrational conditions

Recently the amount of literatures about venture capitalunder bounded rationality was small Fairchild [37] built abehavior game model to study the choice of entrepreneursbetween angel investment and venture capital Zheng andWu [24] thought that in the venture capital market the VCsand the ENs presented bounded rationality in the contractdesign especially the ENs In addition to revenue the ENsmight also take individual leisure requirement personalfairness preference and other factors into account Whatthey were pursuing was satisfactory solution rather than theoptimal solution Aiming at venture investors and venturecapitalists with fairness preferences Zheng and Xu [25] builta new payment pattern By designing an effective incentivemechanism they provided a basis for venture capitalistsrsquoestablishing a reasonable payment contract However noneof the prior theoretical papers considered the reciprocitymotivation

This paper focuses on designing behavior capital con-tracts based on reciprocity motivation through introducingreciprocity motivation theory into the design of multistageventure capital contract Our starting point is mainly basedon the following three aspects First multistage venturecapital is pervasive in practice Instead of providing all theinvestment upfront the VC invests in stages to control risksand makes refinancing decisions according to the projectcondition There are some research results currently butfurther study is still needed Second in practice the VC andthe EN are bounded rationality and reciprocity motivationis also a common psychological preference If the EN inputsmore effort the VC may offer a higher share in returnOf course this reciprocity is bilateral Third majority ofresearch results study the design of capital structure contractsunder perfectly rationality and derive corresponding utilityor returns Thus if we can find a capital structure contractunder reciprocitymotivation inwhich the utility or returns ofboth sides are Pareto improvement of perfect rationality our

4 Discrete Dynamics in Nature and Society

research perspective will have more practical and theoreticalsignificance

Our main ideas are as follows If the VC cooperates withthe EN the ENrsquos effort will achieve the optimal We call theeffort in this case the first-best Therefore the first step is tocalculate the first-best solution of optimal effort during thecooperationThen secondly we calculate the optimal decisionof both sides under perfect rationality in the assumption thatthe two parties make their decisions independently We callthe solution here the second-best The first-best solution isthe optimal upper limit of both sides and the second-bestsolution is the lower limit of both sides The third step is toconstruct a multistage game model on venture capital basedon reciprocity motivation through introducing Rabinrsquos [22]model

We have two goals The first is to find the optimal effortlevel between the first-best solution and the second-bestsolution and to confirm that the capital structure contract isalso between the first-best and the second-best solution Thesecond goal is to confirm that under reciprocity motivationthe utility or returns of both sides are Pareto improvementcompared with that under perfect rationality

This paper proceeds as follows In the next section we putforward the assumptions notations and basic descriptionsof the model As a basis for comparison we consider themultistage decision-making model and solutions under per-fect rationality in Section 3 Sections 4 and 5 are respectivelythe theory and the construction of the reciprocity motivationpreferences utility function Section 6 is the solution of theventure capital decision-making model under reciprocitymotivation Given the complexity of the model we usesimulation to confirm the existence of the venture capitalincentive contract under reciprocity motivation in Section 7Finally Section 8 concludes the paper and directions forfuture research are proposed

3 Model Description and Assumptions

31 Model Description Consider an innovative entrepreneur(EN) who relies on a venture capitalist (VC) for investmentThe VC provides a capital contract for the EN If the ENaccepts the contract will be executed If not the ENrsquosreservation utility will be zero We assume that both the ENand the VC are risk neutral The VC stages the investmentinto 119899 periods to control risks and mitigate moral hazardAnd the VC has the option to continue or exit at each endof the period The effort of the EN is different in differentperiods And the more the effort EN inputs the higher thesuccess rate of the project isTheVCrsquos goal is to select a capitalstructure contract to maximize his utility while the ENrsquos goalis to choose an effort level to maximize his utility Time lineof game is depicted in Figure 1

32 Research Assumptions

Assumptions 1 We assume that both the EN and the VC arerisk neutralThat is they have equivalent risk tolerance facingpossible profit The ENrsquos decision goal is to maximize the

End

EN does not acceptVC provides Ik 120572 120573

EN acceptsReciprocity

EN chooses effort ek

State of nature realized

Outcome pk Success

1 minus pk Failure

EN gets 120572 and the game ends

Figure 1 The time line

utility in revenue the VCrsquos decision goal is to maximize theutility of capital gains

Assumptions 2 VC invests capital to the venture company Itis a long-term process and the venture capital is divided into119899 stages (119899 ge 2)

Assumptions 3 If the VC and the EN reach the investmentagreement the VC will provide the external capital 119868119896 (119868119896 ge0) in stage 119896 119896 = 1 2 3 119899The averagemarket return rateof external capital 119868119896 is 119903 0 lt 119903 lt 1 Thus the capital cost forthe VC is (1 + 119903)119868119896

Assumptions 4 It is a long-term process after the VC inputscapital to the venture enterprise The effort of the EN isdifferent in different periods Naturally venture investmentshould be carried out in stages So we divide the long-termpartnership into 119899 stages (119899 isin 119873) The ENrsquos effort is 119890119896119896 = 1 2 3 119899 in stage 119896 119896 = 1 2 3 119899 We can regardeffort cost as monetary cost For simplicity we assume thatthe effort cost for the EN is 119888(119890119896) = 1198871198902

1198962 119887 ge 0 One

property of this function is that the cost will increase as theeffort increases and the increasing is faster and faster that is1198881015840(119890119896) gt 0 119888

10158401015840(119890119896) gt 0

Assumptions 5 The probability of success is 119901 0 le 119901 le 1

after 119899 periods The success of the venture project is relatedto ENrsquos effort The more the effort EN inputs the greaterthe probability of success is and the success in stage 119896 isinseparable from the early effort So we assume that 119901119896 =11989011198902 sdot sdot sdot 119890119896 0 le 119901119896 = 11989011198902 sdot sdot sdot 119890119896 = prod

119896

119894=1119890119894 le 1 Let 119878 be the

projectrsquos return (a fixed cash flow) if the project is successfuluntil stage 119896 In the multiperiod model the probability ofproject success or failure can be represented by the pastefforts of EN similar studies can be referred to in Yangrsquos

Discrete Dynamics in Nature and Society 5

K

e1

1

e2

2

middot middot middot

Benefits of successful projects S

EN accepts contract

Probability of success pk =

ek

e1e2 middot middot middot ek

Figure 2 Success after 119896-stage

method [38] However the Yangrsquosmodel is a continuous-timemodel while our model is a discrete-time model Figure 2shows the success probability and return of venture projects

For simplicity we assume that the return in every periodis equal that is 119878 ge 0 The return is 0 if the project is faileduntil stage 119896 So the expected return of the project in stage 119896is

119881119896 = 119901119896119878119896 + (1 minus 119901) 0 = 119901119896119878119896 = 119878

119896

prod119894=1

119890119894 (1)

Assumptions 6 After the VC invests the capital at thebeginning of the project the VC gives fixed income 120572 andthe revenue sharing coefficient 120573 (0 le 120573 le 1) in each periodto the EN We call (120572 120573) the equity contract on investment(120572 120573) are also known as financial contracts If EN acceptscontract EN andVChave long-term game relationship if ENrefuses it the game is over

Assumptions 7 For simplicity there is no consideration forthe time value of the return We can also assume thatventure capitalistrsquos discount factor is 120575119862 and the ventureentrepreneurrsquos discount factor is 120575119864 It will increase thecomplexity of model but will not affect the conclusions

4 Decision Model and Solution ofMultistage under Perfect Rationality

Let 120587 be the profit of the EN Given the fixed income 120572 thereturn in each stage 120573119881119896 and the total effort costsum

119899

119896=1119888(119890119896) =

sum119899

119896=1(11988711989021198962) the expected profit of the EN is

120587 = 120572 +

119899

sum119896=1

120573119881119896 minus

119899

sum119896=1

1198871198902119896

2

= 120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2

(2)

The EN chooses effort level in each stage 1198901 1198902 119890119899 tomaximize the utility namely

max11989011198902119890119899

120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2 (3)

Since the VC is risk neutral VCrsquos expected utility equalsthe expected return

Π = minus120572 +

119899

sum119896=1

(1 minus 120573)119881119896 minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896

(4)

The VC chooses appropriate equity contract (120572 120573) tomaximize the expected utility The maximization problem is

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (5)

41TheOptimal Decision of Cooperation TheEN and the VCrealize cooperation through one side completely controllingproject that is the owner of the project and the provider ofthe investment are the same decision-maker Consequentlyadding (2) to (4) we derive

Π119862=

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (6)

The decision-maker chooses effort level in each stage1198901 1198902 119890119899 to maximize the overall utility namely

max11989011198902119890119899

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (7)

We derive the optimal solution 119890119862lowast119896= (119878119887)1(119899minus2)

42 The Optimal Decision of Noncooperation Under nonco-operation the VC and the EN make independent decisionsto maximize their own utility or return The VCrsquos goal is themaximization of the future return Thus the VC designs acapital structure contract to maximize the total return at theend of stage 119899

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (8)

However the EN also cares about the future utilitymaximization while focusing on maximizing the currentutility [39] That means the EN will choose the maximumeffort level 1198901 tomaximize the return120572+1205731198901119878minus119887119890

2

12 in stage 1

Then in stage 2 the ENwill choose the maximum effort level1198902 to maximize the total return 120572 + 120573prod2

119896=1119890119896119878 minus sum

2

119896=1(11988711989021198962)

of the two stagesIn stage 119899 the EN will choose the maximum effort level

119890119899 to maximize the total return 120572 + 120573prod119899119896=1119890119896119878 minus sum

119899

119896=1(11988711989021198962)

of the former 119899 stages

6 Discrete Dynamics in Nature and Society

In this game the VC provides capital structure contractfirstly and then the EN chooses his own effort level There-fore this game is Stackelberg game It can be described asfollows (I)

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (9)

st max1198901

120572 + 1205731198901119878 minus11988711989021

2 (10a)

max1198902

120572 + 120573

2

prod119896=1

119890119896119878 minus

2

sum119896=1

1198871198902119896

2

(10b)

max119890119899

120572 + 120573

119899

prod119896=1

119890119896119878 minus

119899

sum119896=1

1198871198902119896

2 (10c)

In model (I) the first-order condition about 1198901 1198902 119890119899in (10a) (10b) and (10c) is

120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(11)

By introducing the first-order condition to model (I) theVCrsquos problem becomes

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (12)

st 120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(13)

Put formula (13) into objective function (12) we derivethe following

max120572120573

minus 120572 +119878(1198992+119899+2)2

119887119899(119899+1)2(1 minus 120573) 120573

119899(119899+1)2minus

119899

sum119896=1

(1 + 119903) 119868119896 (14)

From formula (14) the first-order condition about 120572 and120573 is

120572lowast= 0

120573lowast=

1198992 + 119899

1198992 + 119899 + 2

(15)

Note 1 In (15) specially when 119899 = 1 120573 = 12 this wouldtie in exactly with a one-period single-sided moral hazardmodel the VC with no value-creating ability would give halfof the equity to the EN tomotivate the EN To be sure VC hasno value-creating ability why does VC give half of the equityto the EN The reasons are as follows (1) VCrsquos investment

needs to be paid and ENrsquos projects and efforts need to berewarded because the modelrsquos parameters are set differentlythe result is precisely 120573 = 12 (2) Our results have somedifferences with Fairchild [37] which is because themodelingmechanism is differentWe are sure that Fairchild [23 37 40]provides a lot of important research results in the field ofventure capital

Therefore the ENrsquos optimal effort level in each stage is

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896 119896 = 1 2 119899 (16)

Put (15) and (16) into the function we derive

120587lowast

VC =2119878

1198992 + 119899 + 2

119899

prod119896=1

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

=2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2

119899

prod119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

minus119887

2

119899

sum119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

2

=(1198992+ 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992+ 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(17)

So there is Proposition 1

Proposition 1 If the VC cooperates with the EN the ENrsquosoptimal effort level is 119890119862lowast

119896= (119887119878)

1(119899minus2) (first-best) If the VCand the EN make decisions independently the ENrsquos optimaleffort level is 119890lowast

119896= (119878119896119887119896)((1198992 + 119899)119896(1198992 + 119899 + 2)119896) (second-

best) The optimal venture capital structure contract is 120572lowast = 0120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) The optimal benefits on both sidesrespectively are as follows

120587lowast

VC =2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(18)

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

research perspective will have more practical and theoreticalsignificance

Our main ideas are as follows If the VC cooperates withthe EN the ENrsquos effort will achieve the optimal We call theeffort in this case the first-best Therefore the first step is tocalculate the first-best solution of optimal effort during thecooperationThen secondly we calculate the optimal decisionof both sides under perfect rationality in the assumption thatthe two parties make their decisions independently We callthe solution here the second-best The first-best solution isthe optimal upper limit of both sides and the second-bestsolution is the lower limit of both sides The third step is toconstruct a multistage game model on venture capital basedon reciprocity motivation through introducing Rabinrsquos [22]model

We have two goals The first is to find the optimal effortlevel between the first-best solution and the second-bestsolution and to confirm that the capital structure contract isalso between the first-best and the second-best solution Thesecond goal is to confirm that under reciprocity motivationthe utility or returns of both sides are Pareto improvementcompared with that under perfect rationality

This paper proceeds as follows In the next section we putforward the assumptions notations and basic descriptionsof the model As a basis for comparison we consider themultistage decision-making model and solutions under per-fect rationality in Section 3 Sections 4 and 5 are respectivelythe theory and the construction of the reciprocity motivationpreferences utility function Section 6 is the solution of theventure capital decision-making model under reciprocitymotivation Given the complexity of the model we usesimulation to confirm the existence of the venture capitalincentive contract under reciprocity motivation in Section 7Finally Section 8 concludes the paper and directions forfuture research are proposed

3 Model Description and Assumptions

31 Model Description Consider an innovative entrepreneur(EN) who relies on a venture capitalist (VC) for investmentThe VC provides a capital contract for the EN If the ENaccepts the contract will be executed If not the ENrsquosreservation utility will be zero We assume that both the ENand the VC are risk neutral The VC stages the investmentinto 119899 periods to control risks and mitigate moral hazardAnd the VC has the option to continue or exit at each endof the period The effort of the EN is different in differentperiods And the more the effort EN inputs the higher thesuccess rate of the project isTheVCrsquos goal is to select a capitalstructure contract to maximize his utility while the ENrsquos goalis to choose an effort level to maximize his utility Time lineof game is depicted in Figure 1

32 Research Assumptions

Assumptions 1 We assume that both the EN and the VC arerisk neutralThat is they have equivalent risk tolerance facingpossible profit The ENrsquos decision goal is to maximize the

End

EN does not acceptVC provides Ik 120572 120573

EN acceptsReciprocity

EN chooses effort ek

State of nature realized

Outcome pk Success

1 minus pk Failure

EN gets 120572 and the game ends

Figure 1 The time line

utility in revenue the VCrsquos decision goal is to maximize theutility of capital gains

Assumptions 2 VC invests capital to the venture company Itis a long-term process and the venture capital is divided into119899 stages (119899 ge 2)

Assumptions 3 If the VC and the EN reach the investmentagreement the VC will provide the external capital 119868119896 (119868119896 ge0) in stage 119896 119896 = 1 2 3 119899The averagemarket return rateof external capital 119868119896 is 119903 0 lt 119903 lt 1 Thus the capital cost forthe VC is (1 + 119903)119868119896

Assumptions 4 It is a long-term process after the VC inputscapital to the venture enterprise The effort of the EN isdifferent in different periods Naturally venture investmentshould be carried out in stages So we divide the long-termpartnership into 119899 stages (119899 isin 119873) The ENrsquos effort is 119890119896119896 = 1 2 3 119899 in stage 119896 119896 = 1 2 3 119899 We can regardeffort cost as monetary cost For simplicity we assume thatthe effort cost for the EN is 119888(119890119896) = 1198871198902

1198962 119887 ge 0 One

property of this function is that the cost will increase as theeffort increases and the increasing is faster and faster that is1198881015840(119890119896) gt 0 119888

10158401015840(119890119896) gt 0

Assumptions 5 The probability of success is 119901 0 le 119901 le 1

after 119899 periods The success of the venture project is relatedto ENrsquos effort The more the effort EN inputs the greaterthe probability of success is and the success in stage 119896 isinseparable from the early effort So we assume that 119901119896 =11989011198902 sdot sdot sdot 119890119896 0 le 119901119896 = 11989011198902 sdot sdot sdot 119890119896 = prod

119896

119894=1119890119894 le 1 Let 119878 be the

projectrsquos return (a fixed cash flow) if the project is successfuluntil stage 119896 In the multiperiod model the probability ofproject success or failure can be represented by the pastefforts of EN similar studies can be referred to in Yangrsquos

Discrete Dynamics in Nature and Society 5

K

e1

1

e2

2

middot middot middot

Benefits of successful projects S

EN accepts contract

Probability of success pk =

ek

e1e2 middot middot middot ek

Figure 2 Success after 119896-stage

method [38] However the Yangrsquosmodel is a continuous-timemodel while our model is a discrete-time model Figure 2shows the success probability and return of venture projects

For simplicity we assume that the return in every periodis equal that is 119878 ge 0 The return is 0 if the project is faileduntil stage 119896 So the expected return of the project in stage 119896is

119881119896 = 119901119896119878119896 + (1 minus 119901) 0 = 119901119896119878119896 = 119878

119896

prod119894=1

119890119894 (1)

Assumptions 6 After the VC invests the capital at thebeginning of the project the VC gives fixed income 120572 andthe revenue sharing coefficient 120573 (0 le 120573 le 1) in each periodto the EN We call (120572 120573) the equity contract on investment(120572 120573) are also known as financial contracts If EN acceptscontract EN andVChave long-term game relationship if ENrefuses it the game is over

Assumptions 7 For simplicity there is no consideration forthe time value of the return We can also assume thatventure capitalistrsquos discount factor is 120575119862 and the ventureentrepreneurrsquos discount factor is 120575119864 It will increase thecomplexity of model but will not affect the conclusions

4 Decision Model and Solution ofMultistage under Perfect Rationality

Let 120587 be the profit of the EN Given the fixed income 120572 thereturn in each stage 120573119881119896 and the total effort costsum

119899

119896=1119888(119890119896) =

sum119899

119896=1(11988711989021198962) the expected profit of the EN is

120587 = 120572 +

119899

sum119896=1

120573119881119896 minus

119899

sum119896=1

1198871198902119896

2

= 120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2

(2)

The EN chooses effort level in each stage 1198901 1198902 119890119899 tomaximize the utility namely

max11989011198902119890119899

120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2 (3)

Since the VC is risk neutral VCrsquos expected utility equalsthe expected return

Π = minus120572 +

119899

sum119896=1

(1 minus 120573)119881119896 minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896

(4)

The VC chooses appropriate equity contract (120572 120573) tomaximize the expected utility The maximization problem is

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (5)

41TheOptimal Decision of Cooperation TheEN and the VCrealize cooperation through one side completely controllingproject that is the owner of the project and the provider ofthe investment are the same decision-maker Consequentlyadding (2) to (4) we derive

Π119862=

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (6)

The decision-maker chooses effort level in each stage1198901 1198902 119890119899 to maximize the overall utility namely

max11989011198902119890119899

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (7)

We derive the optimal solution 119890119862lowast119896= (119878119887)1(119899minus2)

42 The Optimal Decision of Noncooperation Under nonco-operation the VC and the EN make independent decisionsto maximize their own utility or return The VCrsquos goal is themaximization of the future return Thus the VC designs acapital structure contract to maximize the total return at theend of stage 119899

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (8)

However the EN also cares about the future utilitymaximization while focusing on maximizing the currentutility [39] That means the EN will choose the maximumeffort level 1198901 tomaximize the return120572+1205731198901119878minus119887119890

2

12 in stage 1

Then in stage 2 the ENwill choose the maximum effort level1198902 to maximize the total return 120572 + 120573prod2

119896=1119890119896119878 minus sum

2

119896=1(11988711989021198962)

of the two stagesIn stage 119899 the EN will choose the maximum effort level

119890119899 to maximize the total return 120572 + 120573prod119899119896=1119890119896119878 minus sum

119899

119896=1(11988711989021198962)

of the former 119899 stages

6 Discrete Dynamics in Nature and Society

In this game the VC provides capital structure contractfirstly and then the EN chooses his own effort level There-fore this game is Stackelberg game It can be described asfollows (I)

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (9)

st max1198901

120572 + 1205731198901119878 minus11988711989021

2 (10a)

max1198902

120572 + 120573

2

prod119896=1

119890119896119878 minus

2

sum119896=1

1198871198902119896

2

(10b)

max119890119899

120572 + 120573

119899

prod119896=1

119890119896119878 minus

119899

sum119896=1

1198871198902119896

2 (10c)

In model (I) the first-order condition about 1198901 1198902 119890119899in (10a) (10b) and (10c) is

120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(11)

By introducing the first-order condition to model (I) theVCrsquos problem becomes

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (12)

st 120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(13)

Put formula (13) into objective function (12) we derivethe following

max120572120573

minus 120572 +119878(1198992+119899+2)2

119887119899(119899+1)2(1 minus 120573) 120573

119899(119899+1)2minus

119899

sum119896=1

(1 + 119903) 119868119896 (14)

From formula (14) the first-order condition about 120572 and120573 is

120572lowast= 0

120573lowast=

1198992 + 119899

1198992 + 119899 + 2

(15)

Note 1 In (15) specially when 119899 = 1 120573 = 12 this wouldtie in exactly with a one-period single-sided moral hazardmodel the VC with no value-creating ability would give halfof the equity to the EN tomotivate the EN To be sure VC hasno value-creating ability why does VC give half of the equityto the EN The reasons are as follows (1) VCrsquos investment

needs to be paid and ENrsquos projects and efforts need to berewarded because the modelrsquos parameters are set differentlythe result is precisely 120573 = 12 (2) Our results have somedifferences with Fairchild [37] which is because themodelingmechanism is differentWe are sure that Fairchild [23 37 40]provides a lot of important research results in the field ofventure capital

Therefore the ENrsquos optimal effort level in each stage is

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896 119896 = 1 2 119899 (16)

Put (15) and (16) into the function we derive

120587lowast

VC =2119878

1198992 + 119899 + 2

119899

prod119896=1

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

=2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2

119899

prod119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

minus119887

2

119899

sum119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

2

=(1198992+ 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992+ 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(17)

So there is Proposition 1

Proposition 1 If the VC cooperates with the EN the ENrsquosoptimal effort level is 119890119862lowast

119896= (119887119878)

1(119899minus2) (first-best) If the VCand the EN make decisions independently the ENrsquos optimaleffort level is 119890lowast

119896= (119878119896119887119896)((1198992 + 119899)119896(1198992 + 119899 + 2)119896) (second-

best) The optimal venture capital structure contract is 120572lowast = 0120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) The optimal benefits on both sidesrespectively are as follows

120587lowast

VC =2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(18)

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

K

e1

1

e2

2

middot middot middot

Benefits of successful projects S

EN accepts contract

Probability of success pk =

ek

e1e2 middot middot middot ek

Figure 2 Success after 119896-stage

method [38] However the Yangrsquosmodel is a continuous-timemodel while our model is a discrete-time model Figure 2shows the success probability and return of venture projects

For simplicity we assume that the return in every periodis equal that is 119878 ge 0 The return is 0 if the project is faileduntil stage 119896 So the expected return of the project in stage 119896is

119881119896 = 119901119896119878119896 + (1 minus 119901) 0 = 119901119896119878119896 = 119878

119896

prod119894=1

119890119894 (1)

Assumptions 6 After the VC invests the capital at thebeginning of the project the VC gives fixed income 120572 andthe revenue sharing coefficient 120573 (0 le 120573 le 1) in each periodto the EN We call (120572 120573) the equity contract on investment(120572 120573) are also known as financial contracts If EN acceptscontract EN andVChave long-term game relationship if ENrefuses it the game is over

Assumptions 7 For simplicity there is no consideration forthe time value of the return We can also assume thatventure capitalistrsquos discount factor is 120575119862 and the ventureentrepreneurrsquos discount factor is 120575119864 It will increase thecomplexity of model but will not affect the conclusions

4 Decision Model and Solution ofMultistage under Perfect Rationality

Let 120587 be the profit of the EN Given the fixed income 120572 thereturn in each stage 120573119881119896 and the total effort costsum

119899

119896=1119888(119890119896) =

sum119899

119896=1(11988711989021198962) the expected profit of the EN is

120587 = 120572 +

119899

sum119896=1

120573119881119896 minus

119899

sum119896=1

1198871198902119896

2

= 120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2

(2)

The EN chooses effort level in each stage 1198901 1198902 119890119899 tomaximize the utility namely

max11989011198902119890119899

120572 +

119899

sum119896=1

[120573119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2 (3)

Since the VC is risk neutral VCrsquos expected utility equalsthe expected return

Π = minus120572 +

119899

sum119896=1

(1 minus 120573)119881119896 minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896

(4)

The VC chooses appropriate equity contract (120572 120573) tomaximize the expected utility The maximization problem is

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (5)

41TheOptimal Decision of Cooperation TheEN and the VCrealize cooperation through one side completely controllingproject that is the owner of the project and the provider ofthe investment are the same decision-maker Consequentlyadding (2) to (4) we derive

Π119862=

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (6)

The decision-maker chooses effort level in each stage1198901 1198902 119890119899 to maximize the overall utility namely

max11989011198902119890119899

119899

sum119896=1

[119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

1198871198902119896

2minus

119899

sum119896=1

(1 + 119903) 119868119896 (7)

We derive the optimal solution 119890119862lowast119896= (119878119887)1(119899minus2)

42 The Optimal Decision of Noncooperation Under nonco-operation the VC and the EN make independent decisionsto maximize their own utility or return The VCrsquos goal is themaximization of the future return Thus the VC designs acapital structure contract to maximize the total return at theend of stage 119899

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (8)

However the EN also cares about the future utilitymaximization while focusing on maximizing the currentutility [39] That means the EN will choose the maximumeffort level 1198901 tomaximize the return120572+1205731198901119878minus119887119890

2

12 in stage 1

Then in stage 2 the ENwill choose the maximum effort level1198902 to maximize the total return 120572 + 120573prod2

119896=1119890119896119878 minus sum

2

119896=1(11988711989021198962)

of the two stagesIn stage 119899 the EN will choose the maximum effort level

119890119899 to maximize the total return 120572 + 120573prod119899119896=1119890119896119878 minus sum

119899

119896=1(11988711989021198962)

of the former 119899 stages

6 Discrete Dynamics in Nature and Society

In this game the VC provides capital structure contractfirstly and then the EN chooses his own effort level There-fore this game is Stackelberg game It can be described asfollows (I)

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (9)

st max1198901

120572 + 1205731198901119878 minus11988711989021

2 (10a)

max1198902

120572 + 120573

2

prod119896=1

119890119896119878 minus

2

sum119896=1

1198871198902119896

2

(10b)

max119890119899

120572 + 120573

119899

prod119896=1

119890119896119878 minus

119899

sum119896=1

1198871198902119896

2 (10c)

In model (I) the first-order condition about 1198901 1198902 119890119899in (10a) (10b) and (10c) is

120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(11)

By introducing the first-order condition to model (I) theVCrsquos problem becomes

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (12)

st 120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(13)

Put formula (13) into objective function (12) we derivethe following

max120572120573

minus 120572 +119878(1198992+119899+2)2

119887119899(119899+1)2(1 minus 120573) 120573

119899(119899+1)2minus

119899

sum119896=1

(1 + 119903) 119868119896 (14)

From formula (14) the first-order condition about 120572 and120573 is

120572lowast= 0

120573lowast=

1198992 + 119899

1198992 + 119899 + 2

(15)

Note 1 In (15) specially when 119899 = 1 120573 = 12 this wouldtie in exactly with a one-period single-sided moral hazardmodel the VC with no value-creating ability would give halfof the equity to the EN tomotivate the EN To be sure VC hasno value-creating ability why does VC give half of the equityto the EN The reasons are as follows (1) VCrsquos investment

needs to be paid and ENrsquos projects and efforts need to berewarded because the modelrsquos parameters are set differentlythe result is precisely 120573 = 12 (2) Our results have somedifferences with Fairchild [37] which is because themodelingmechanism is differentWe are sure that Fairchild [23 37 40]provides a lot of important research results in the field ofventure capital

Therefore the ENrsquos optimal effort level in each stage is

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896 119896 = 1 2 119899 (16)

Put (15) and (16) into the function we derive

120587lowast

VC =2119878

1198992 + 119899 + 2

119899

prod119896=1

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

=2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2

119899

prod119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

minus119887

2

119899

sum119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

2

=(1198992+ 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992+ 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(17)

So there is Proposition 1

Proposition 1 If the VC cooperates with the EN the ENrsquosoptimal effort level is 119890119862lowast

119896= (119887119878)

1(119899minus2) (first-best) If the VCand the EN make decisions independently the ENrsquos optimaleffort level is 119890lowast

119896= (119878119896119887119896)((1198992 + 119899)119896(1198992 + 119899 + 2)119896) (second-

best) The optimal venture capital structure contract is 120572lowast = 0120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) The optimal benefits on both sidesrespectively are as follows

120587lowast

VC =2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(18)

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

In this game the VC provides capital structure contractfirstly and then the EN chooses his own effort level There-fore this game is Stackelberg game It can be described asfollows (I)

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (9)

st max1198901

120572 + 1205731198901119878 minus11988711989021

2 (10a)

max1198902

120572 + 120573

2

prod119896=1

119890119896119878 minus

2

sum119896=1

1198871198902119896

2

(10b)

max119890119899

120572 + 120573

119899

prod119896=1

119890119896119878 minus

119899

sum119896=1

1198871198902119896

2 (10c)

In model (I) the first-order condition about 1198901 1198902 119890119899in (10a) (10b) and (10c) is

120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(11)

By introducing the first-order condition to model (I) theVCrsquos problem becomes

max120572120573

minus 120572 +

119899

sum119896=1

[(1 minus 120573) 119878

119896

prod119894=1

119890119894] minus

119899

sum119896=1

(1 + 119903) 119868119896 (12)

st 120573119878 = 1198871198901

1205731198781198901 = 1198871198902 120573119878

119899minus1

prod119896=1

= 119887119890119899(13)

Put formula (13) into objective function (12) we derivethe following

max120572120573

minus 120572 +119878(1198992+119899+2)2

119887119899(119899+1)2(1 minus 120573) 120573

119899(119899+1)2minus

119899

sum119896=1

(1 + 119903) 119868119896 (14)

From formula (14) the first-order condition about 120572 and120573 is

120572lowast= 0

120573lowast=

1198992 + 119899

1198992 + 119899 + 2

(15)

Note 1 In (15) specially when 119899 = 1 120573 = 12 this wouldtie in exactly with a one-period single-sided moral hazardmodel the VC with no value-creating ability would give halfof the equity to the EN tomotivate the EN To be sure VC hasno value-creating ability why does VC give half of the equityto the EN The reasons are as follows (1) VCrsquos investment

needs to be paid and ENrsquos projects and efforts need to berewarded because the modelrsquos parameters are set differentlythe result is precisely 120573 = 12 (2) Our results have somedifferences with Fairchild [37] which is because themodelingmechanism is differentWe are sure that Fairchild [23 37 40]provides a lot of important research results in the field ofventure capital

Therefore the ENrsquos optimal effort level in each stage is

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896 119896 = 1 2 119899 (16)

Put (15) and (16) into the function we derive

120587lowast

VC =2119878

1198992 + 119899 + 2

119899

prod119896=1

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

=2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2

119899

prod119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

minus119887

2

119899

sum119896=1

[

[

119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

]

]

2

=(1198992+ 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992+ 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(17)

So there is Proposition 1

Proposition 1 If the VC cooperates with the EN the ENrsquosoptimal effort level is 119890119862lowast

119896= (119887119878)

1(119899minus2) (first-best) If the VCand the EN make decisions independently the ENrsquos optimaleffort level is 119890lowast

119896= (119878119896119887119896)((1198992 + 119899)119896(1198992 + 119899 + 2)119896) (second-

best) The optimal venture capital structure contract is 120572lowast = 0120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) The optimal benefits on both sidesrespectively are as follows

120587lowast

VC =2119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

120587lowast

119864=(1198992 + 119899) 119878

1198992 + 119899 + 2[119878

119887

(1198992 + 119899)

(1198992 + 119899 + 2)]

119899(119899+1)2

minus1198871198782 (1198992 + 119899)

2

1 minus [119878 (1198992 + 119899) (1198992 + 119899 + 2)]2119899

2 [1198872 (1198992 + 119899 + 2)2minus 1198782 (1198992 + 119899)

2]

(18)

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

5 The Principle of Reciprocity MotivationFairness Utility Function

When studying the decision-making in venture capital thetraditional view assumes that theVC and the ENare pure self-interest That is they only pursue their individual maximumbenefits rather than care about the fairness of the welfareallocation or behavioral motivation Yet a series of gameexperiments (like ultimatum game trust game and gift-exchange game) in recent years indicate that fairness prefer-ences also exist in addition to self-interest preferences Thetheory believes that they will also focus on the fairness of thewelfare allocation or behavioralmotivationwhen they pursuepersonal interests As well as the self-interest preferencesfairness preferences can influence the decision-making ofthe participants in venture capital For instance people maysacrifice part of their own interests to preserve the fairnessin the revenue allocating to revenge hostile behaviors or toreward kindness

The fairness function in venture capital is based on thepsychological game framework of Geanakoplos et al Rabin[22] created a game payoff function through incorporatingfairness preferences which was published in the Americaneconomic review Rabin [22] defined the fairness preferencesas a behavior that one rewards othersrsquo kindness or punishesothersrsquo unkindness However how to define the kindness orunkindness specifically Rabin [22] believed that the behaviorof sacrificing your own utility (incomes interests and so on)to improve othersrsquo utility (incomes interests and so on) couldbe defined as kindness Similarly sacrificing your own utilityto reduce othersrsquo utility could be defined as unkindnessRabinrsquos framework about fairness preferences incorporatedthe following three stylized facts (A) People were willingto sacrifice their own material well-being to help those whowere being kind (B) People were willing to sacrifice theirown material well-being to punish those who are beingunkind (C) Both motivations (A) and (B) had a greatereffect on behavior as the material cost of sacrificing becamesmaller Rabin [22] regarded that these facts could explain notonly the behavioral motivation of fairness preference in theultimatum game but also the one of reciprocity preference inthe cooperative game

Hence Rabin [22] presented the thoughts in the utilityfunction throughout mathematical models The first stepto incorporate fairness into the analysis was to define aldquokindness functionrdquo 119891119894(119886119894 119887119895) which measured how kindplayer 119894 was being to player 119895 or the ldquodistancerdquo between twoparties If player 119894 believed that player 119895was choosing strategy119887119895 how kind was player 119894 being by choosing 119886119894 Player 119894 waschoosing the payoff pair 120587119894(119886119894 119887119895) from among the set of allpayoffs feasible 120587119894(119887119895) = 120587119894(119886119894 119887119895) | 119886119894 isin 119878119894 (119878119894 was thestrategy space of player 119894) to show the kindness level of player119894 In Rabinrsquos model let 120587max

119894(119887119895) be player 119895rsquos highest payoff in

120587(119887119895) and let 120587min119894(119887119895) be player 119895rsquos lowest payoff among points

that were Pareto-efficient in 120587(119887119895) Then the equitable payoffor fairness payoff could be expressed as

120587fair119895(119887119895) =

120587max119894

(119887119895) + 120587min119894

(119887119895)

2 (19)

More generally it provided a crude reference pointagainst which to measure how generous player 119894 was beingto player 119895 From these payoffs the kindness function was

119891119894 (119886119894 119887119895) =120587119895 (119886119894 119887119895) minus 120587

fair119895(119887119895)

120587max119894

(119887119895) minus 120587min119894

(119887119895) (20)

This function captures how much more than or less thanplayer 119895rsquos equitable payoff player 119894 believes he is giving toplayer 119895 If119891119894(119886119894 119887119895) is positive it indicates that player 119894 is kindto player 119895 since player 119895rsquos actual payoff is higher than thefairness payoffOtherwise it indicates that player 119894 is not kind

Furthermore Rabin [22] defined

119891119895 (119887119895 119888119894) =120587119894 (119888119894 119887119895) minus 120587

fair119894(119888119894)

120587max119894

(119888119894) minus 120587min119894

(119888119894) (21)

This function was to represent player 119894rsquos beliefs about howkindly player 119895was treating him where 119888119894 represents player 119894rsquosbeliefs about what player 119895 believed what player 119894rsquos strategywas Because the kindness functions were normalized thevalues of 119891119894(119886119894 119887119895) and 119891119895(119887119895 119888119894) must lie in the interval[minus1 12] These kindness functions could now be used tospecify fully the playersrsquo preferences

Thus the reciprocal fairness model [22] was

119880119894 (119886119894 119887119895 119888119894) = 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894)

+ 119891119895 (119887119895 119888119894) 119891119894 (119886119894 119887119895)

= 120587119894 (119886119894 119887119895) + 119891119895 (119887119895 119888119894) [1 + 119891119894 (119886119894 119887119895)]

(22)

As was shown above 119880119894(119886119894 119887119895 119888119894) represents player 119894rsquosexpected utility player 119894 not only cared about his materialpayoff (the first item of the right side of the formula) but alsocared treated kindly (the second item of the right side of theformula)

Because these preferences form a psychological gamewe could use the concept of psychological Nash equilibriumdefined by GPS This was simply the analog of Nash equi-librium for psychological games imposing the additionalcondition that all higher-order beliefs match actual behaviorRabin [22] called that the solutionwas ldquofairness equilibriumrdquo

Definition 2 If the pair of strategies (1198861 1198862) isin 1198781 times 1198782 was afairness equilibrium for 119894 = 119895 119894 = 1 2

(1) 119886119894 isin argmax119886119894isin119878119894

119880119894(119886119894 119887119895 119888119894)

(2) 119886119894 = 119887119894 = 119888119894

6 The Construction of Reciprocity MotivationFairness Utility Function

According to Rabinrsquos reciprocity fairness preferences theorywe can construct the utility functions of the participants inventure capital based on the fairness preferences

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Based on Section 2 and practical properties of venturecapital the fairness function is

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

120587VC (120573) = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896

120587maxVC (120573) = minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896 = minus120572

+ (1 minus 120573) 119878

119899

prod119896=1

(119887

119904)

1(119899minus2)

= minus120572

+ (1 minus 120573) 119878(1198992+3119899minus4)2(119899minus2)

119887minus((1198992+119899)2(119899minus2))

120587minVC (120573) = minus120572

120587fair(120573) =

120587maxVC (120573) + 120587min

VC (120573)

2

=(1 minus 120573) 119878(119899

2+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

2minus 120572

(23)

For simplicity remark 119861 = 119878(1198992+3119899minus4)2(119899minus2)119887minus((119899

2+119899)2(119899minus2))

where 119890119896 and 120573 are controlled variables 0 le 119890119896 le 119890119862lowast

119896 0 le

120573 le 1 and 119890119862lowast119896

(119890119862lowast119896= (119878119887)

1(119899minus2)) denotes the optimal levelof cooperation

Substitute the formulas above into (20) there is anotherform

119891119864 (120573) =120587VC (120573) minus 120587

fair (120573)

120587maxVC (120573) minus 120587min

VC (120573)

=2 (1 minus 120573) 119878prod

119899

119896=1119890119896 minus (1 minus 120573) 119861

2 (1 minus 120573) 119861

(24)

And then

120587119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587max119864

(119890119896) = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

120587min119864

(119890119896) = 120572 minus

119899

sum119896=1

1198871198902k2

120587fair119864(119890119896) =

120587max119864

(119890119896) + 120587min119864

(119890119896)

2

=1

2119878

119899

prod119896=1

119890119896 + 120572 minus

119899

sum119896=1

1198871198902119896

2

(25)

Continue to substitute the expression into (21) then wehave

119891VC (119890119896) =120587119864 (119890119896) minus 120587

fair119864(119890119896)

120587max119864

(119890119896) minus 120587min119864

(119890119896)=2120573 minus 1

2 (26)

Finally we put formulas (24) and (26) into Rabinrsquos fairnessmodel (22) then we can derive the utility function of the VCand the EN respectively

119880VC = minus120572 + (1 minus 120573) 119878119899

prod119896=1

119890119896 + 119891119864 (120573) [1 minus 119891VC (119890119896)]

minus

119899

sum119896=1

(1 + 119903) 119868119896

= minus120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(27)

119880119864 = 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2+ 119891VC (119890119896) [1 minus 119891119864 (120573)]

minus

119899

sum119896=1

1198871198902119896

2

= 120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(28)

7 The Decision-Making Model in VentureInvestments and the Solutions

In Section 4 we construct the utility functions of the VC andthe EN based on reciprocity motivation Similarly accordingto the principles of model constructing in Section 4 we canderive another model (III)

max120572120573

minus 120572 + (1 minus 120573) 119878

119899

prod119896=1

119890119896

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896] minus 119861

4119861

(29)

st max1198901

120572 + 1205731198781198901 minus11988711989021

2+(2120573 minus 1) (21198781198901 + 119861)

4119861 (30a)

max1198902

120572 + 120573119878

2

prod119896=1

119890119896 minus

2

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

2

119896=1119890119896 + 119861]

4119861

(30b)

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

max119890119899

120572 + 120573119878

119899

prod119896=1

119890119896 minus

119899

sum119896=1

1198871198902119896

2

+(2120573 minus 1) [2119878prod

119899

119896=1119890119896 + 119861]

4119861

(30c)

In model (III) the first-order condition about 1198901 1198902 119890119899 of(30a) (30b) and (30c) is

120573119878 minus 1198871198901 +119878 (2120573 minus 1)

2119861= 0

1205731198781198901 minus 1198871198902 +119878 (2120573 minus 1) 1198901

2119861= 0

120573119878 = 1198871198901 1205731198781198901 = 11988711989011198902

120573119878

119899minus1

prod119896=1

119890119896 minus 119887119890119899 +119878 (2120573 minus 1)prod

119899minus1

119896=1(119890119896)

2119861= 0

120573119878

119899minus1

prod119896=1

119890119896 = 119887119890119899

(31)

There is

119890119865lowast

1=1

119887[119878 (2120573 minus 1)

2119861+ 120573119878]

119890119865lowast

2=1

1198872[119878 (2120573 minus 1)

2119861+ 120573119878]

2

119890119865lowast

119899=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

(32)

Put formulas (32) into the objective function (29) wederive the following

max120572120573

minus 120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573 minus 1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

(33)

It is complicated to work out the first-order condition of(120572 120573) in the optimal problems above Based on reciprocity

motivation preferences we remark both partiesrsquo returnrespectively as

119880119865

VC = minus120572 + (1 minus 120573) 119878119887minus119899(119899+1)2

+ [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

minus

119899

sum119896=1

(1 + 119903) 119868119896

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2minus 119861

4119861

119880119865

119864= 120572 + 120573119878119887

minus119899(119899+1)2

+ [1 minus119887

2119887minus119899(119899+1)

] [120573119878 +119878 (2120573 minus 1)

2119861]

119899(119899+1)2

+(2120573minus1) 2119878119887minus119899(119899+1)2 minus [120573119878 + 119878 (2120573 minus 1) 2119861]

119899(119899+1)2+ 119861

4119861

(34)

Does we focus onwhether there exists an appropriate pair(120572 120573) which canmake the profits of both sides higher than theoptimal profits under perfect rationality or not That is wehope to find out a pair (120572 120573) to satisfy the inequalities119880119865VC ge120587lowastVC 119880

119865

119864ge 120587lowast119864

8 The Simulation Algorithm ofExistence with Venture InvestmentReciprocity Incentive Contract (120572 120573)

Given too many parameters in formula (33) and the com-plexity of the model it is very complicated to work outthe solutions directly We program with Matlab to derivean appropriate (120572 120573) to satisfy the condition 119880119865VC ge 120587lowastVC119880119865119864ge 120587lowast119864 For simplicity we just study 3 stages that is we

take 119899 = 3 The cost coefficient of the EN is 119887 = 1 Since theprobability 119901 satisfies the condition 0 le 119901119896 = 119890

lowast

1119890lowast

2sdot sdot sdot 119890lowast

119896=

[119878(1198992 + 119899)119887(1198992 + 119899 + 2)]119899(119899+1)2 When we take 119899 = 3 119887 = 1there is 0 le 119878 le 76 so the return of the successful project119878 = 08 If the projectrsquos value is 119878 = 76 it means the projectrsquossuccess in order to generality we take 119878 = 08 it means thatthe project may succeed or also may fail Consider 119868119896 = 05119903 = 02 Substituting them into the utility function we cansimplify the utility function

119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

lowast

VC = 0143 (08)7(0857)

6

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(35)

Now the two inequalities 119880119865VC ge 120587lowastVC 119880119865

119864ge 120587lowast119864can be

simplified to

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 2160

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7

minus 0444 [1 minus (0686)6]

(36)

For convenience we remark

1198921 = 119865

VC = minus120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1)

sdot 2 (08)7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

1198922 = lowast

VC = 0143 (08)7(0857)

6

ℎ1 = 119865

119864= 120572 + 120573 (08)

7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

sdot [120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1)

sdot 2 (08)6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

ℎ2 = lowast

119864= (0686)

7minus 0444 [1 minus (0686)

6]

(37)

The objective of the research is to find a contract structure(120572 120573) in which the utility of both sides under the reciprocitymotivation is greater than that under perfect rationality thatis to say 1198921 ge 1198922 ℎ1 ge ℎ2 Intuitively there may be severalstructures which satisfy 1198921 ge 1198922 ℎ1 ge ℎ2 How to find theoptimal one The EN is the manager of the enterprise andhence has more information So firstly he will maximize hisown utility (max119880) and then maximize the utility of the VC(max119880) Thus the problem is

max120572120573

119865

VC

st max 119865119864

119865

VC ge lowast

VC

119865

119864ge lowast

119864

(38)

That is

max120572120573

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

st max120572 + 120573 (08)7 [120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7

minus 120572 + (1 minus 120573) (08)7[120573 minus 05

(08)7+ 120573]

6

+025

(08)7(2120573 + 1) 2 (08)

7[120573 minus 05

(08)7+ 120573]

6

minus (08)7 minus 18

ge 0143 (08)7(0857)

6

120572 + 120573 (08)7[120573 minus 05

(08)7+ 120573]

6

+ 05 (08)(12)

[120573 minus 05

(08)7+ 120573]

(12)

+025

(08)7(2120573 minus 1) 2 (08)

6[120573 minus 05

(08)7+ 120573]

6

+ (08)7 ge (0686)

7minus 0444 [1 minus (0686)

6]

(39)

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

We use simulation algorithm to solve the optimizationproblem The algorithm is as follows

Step 1 Let 120572 and 120573 lie in the interval [0 1] The initial valueof 120572 and 120573 is 120572 = 0 and 120573 = 0 and the step length is001

Step 2 Loop over the value of 120572 and 120573 to calculate 1198921 and ℎ1If 1198921 gt 1198922 and ℎ1 gt ℎ2 go to Step 3 If the loop is end go toStep 4

Step 3 Store the maximum value of 1198921 in the variable ldquomaxrdquoand let the initial of Max be negative infinity Store the valueof 1198921 120572 and 120573 in the 119899 lowast 3 matrix ldquo119898rdquo If 1198921 lt max returnto Step 2 If 1198921 = max add a new row to matrix119898 to store thecurrent value of [max 120572 120573] and return to Step 2 if 1198921 gt maxreplace the value of max with 1198921 and the matrix119898 and returnto Step 2

Step 4 Search in matrix 119898 to find the maximum value of 1198921and ℎ1 then input (1198921 ℎ1)

We use Matlab editing program we can work outthat the optimal venture capital structure is (lowast 120573lowast) =

(10000 06400) optimal utility 119865VC = 22515 119865

119864= 17572

Thus we can conclude Proposition 3

Proposition 3 Inmultistage venture capital if the VC and theENboth have reciprocitymotivation there necessarily exists theoptimal capital structure contract (lowast 120573lowast) = (10000 06400)The utility of both sides under reciprocity motivation is Paretoimprovement compared with that under perfect rationality

Proposition 3 shows the crucial significance of this paperThe existing literatures in multistage venture capital mainlyfocus on the case of perfect rationality while we also considerthe reciprocity motivation under which the optimal utilitiesof both the VC and the EN are Pareto improvement

Proposition 4 In multistage venture capital if the VC andthe EN both have reciprocity motivation compared with thestandard game VC changes the payment structure that is120572 increases from 0 to 1 and 120573 decreases from 085 to 064At the same time ENrsquos efforts increases from 119890

lowast

119896to 119890119865lowast119896 the

utility of both sides under reciprocity motivation is Paretoimprovement

Proposition 4 is established as follows Obviously accord-ing to Proposition 1 120572 = 0 120573lowast = (1198992 + 119899)(1198992 + 119899 + 2) when119899 = 3 there is 120573lowast = 085 and according to Proposition 3(lowast 120573lowast) = (10000 06400)

Owing to

119890lowast

119896=119878119896

119887119896

(1198992 + 119899)119896

(1198992 + 119899 + 2)119896

119890119865lowast

119896=1

119887119899[119878 (2120573 minus 1)

2119861+ 120573119878]

119899

119896 = 1 2 119899

(40)

The parameter values and the optimal numerical solu-tions are brought into 119890lowast

119896 119890119865lowast119899

119890lowast

119896= 0686

119896= (0174 + 0512)

119896

= (0112

0644+ 0512)

119896

119890119865lowast

119896= (

0112

087+ 0512)

119896

119896 = 1 2 119899

(41)

Obviously

119890119865lowast

119896= (

0112

087+ 0512)

119896

ge (0112

0644+ 0512)

119896

= 119890lowast

119896

119896 = 1 2 119899

(42)

Note 2 First Propositions 3 and 4 are two numerical resultsthat show the existence of the optimal solution of the modelTherefore it cannot guarantee the modelrsquos uniqueness andthus our model does not pursue uniqueness which is adefect of this paper Second Propositions 3 and 4 reflectthe significance of this paper but in practice there is nosuch a financial contract that can motivate EN to pay thebest first-order effort but however there are also manyexamples to illustrate the positive role of the reciprocalfairness preference For example CEO quietly prepared foran employee a birthday party (the cost of which not includedin the wage contract) thus leading to the staff rsquos willing topay more efforts to the CEO Employees then pay moreeffort naturally the output is improved and both CEO andemployees benefit from this party In venture capital thereis also such a situation VC pays the fixed income to EN(120572 increases from 0 to 1) and EN will increase efforts torepay VC (ENrsquos efforts increases 119890lowast

119896to 119890119865lowast119896) Although we see

that 120573 decreases from 085 to 064 we can also witness 120572increases significantly from 0 to 1 the final revenue of ENwas improved and the mutual reciprocal of both VC and ENimproves the utility of the two sides

9 Conclusive Remarks

The moral hazard in the venture capital has been a widelyconcerned problem all over the world The multistage modelin venture capital that scholars considered could mitigatethe moral hazard problem Speaking of the models somescholars researched how to design an incentive contract usingthe principal-agent model to ensure the venture capitalrsquosappreciation and safety While some scholars considered theexit of the VC in the multistage problem based on completeinformation and explained the condition inwhich theVCwillcontinue or exit their underlying hypotheses are all perfectrationality

However it is impractical for some decision-makersto adopt this perfect rationality hypothesis In contrastdecision-makers commonly adopt bounded rationalityTherefore we can say that to study the multistage venturecapital model under bounded rationality is of crucialimportance Reciprocity motivation is one of many forms of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

bounded rationality for certain and it is also a common onein practice

In this paper we have incorporated the reciprocitymotivation into the multistage model to study the optimalcapital structure contract As a basis for comparison at firstwe offered two game models a cooperative model and anoncooperative one both under perfect rationality Then weintroduced the reciprocity motivation into the multistagemodel and built the multistage behavioral game model basedon reciprocity motivation Our study clearly showed that theutility of the VC and the EN under reciprocity motivationis Pareto improvement compared with that under perfectrationality This is where we exhibited the innovation pointof this paper

However our study has limitations in the four follow-ing aspects Firstly our model has been founded underinformation symmetry then how to combine our study ofthe reciprocity motivation with the principal-agent modelSecondly we have mainly focused on the designing of theoptimal capital structure contract but have not given the exitconditions of the VC Thirdly the model is based on therisk neutral assumption so how will the model change ifthe VC and EN have fairness preferences Fourthly we onlyconsidered the ENrsquos effort so what are the results for the ENand VCrsquos efforts These are valuable challenges in front of us

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are particularly grateful to the associate edi-tor and reviewers for thoughtful valuable discussions andsuggestions The authors acknowledge the financial supportby Humanities and Social Science Project of Ministry ofEducation of China (14XJCZH001) by Soft Science ResearchProject of Sichuan Province (2014ZR0027) and by theFundamental Research Funds for the Central Universities(JBK130401)They also thank their students Litian Xu YiqingZhao QinyuChenMengxingDu Xiao Xu Zexi Zhang FengYu and Huan Zhou Jiaci Wang contributed to this paper

References

[1] C Casamatta ldquoFinancing and advising optimal financial con-tracts with venture capitalistsrdquo Journal of Finance vol 58 no 5pp 2059ndash2086 2003

[2] K M Schmidt ldquoConvertible securities and venture capitalfinancerdquo Journal of Finance vol 58 no 3 pp 1139ndash1166 2003

[3] Y Chan D Siegel and A V Thakor ldquoLearning corporatecontrol and performance requirements in venture capital con-tractsrdquo International Economic Review vol 31 no 2 pp 365ndash381 1990

[4] J-J Zheng P Zhang X-S Hu and W-L Jiang ldquoResearchon bidding strategy and modeling of equity auction based onstochastic differential equationsrdquo System Engineering Theoryand Practice vol 33 no 4 pp 893ndash900 2013

[5] J J Zheng X Tan and W T Fan ldquoAn incentive model withstocks based on principal-agent theoryrdquo Journal of ManagementSciences in China vol 8 no 1 pp 24ndash29 2005

[6] W X Guo and Y Zeng ldquoDouble moral hazard and the theoryof capital structure of venture capital financingrdquo Journal ofManagement Sciences in China vol 12 no 3 pp 119ndash131 2009

[7] J-P Xu and S-J Chen ldquoThe study of venture capitalrsquos principal-agent model based on asymmetric informationrdquo System Engi-neering Theory and Practice vol 24 no 1 pp 19ndash24 2004

[8] Y H Jin Y Q Xi and Z X Ye ldquoStudy of multi-period dynamicfinancial model of venture capital considering reputationrdquoSystem EngineeringTheory and Practice vol 5 no 8 pp 76ndash802003

[9] E Ramy and G Arieh ldquoA multi-period game theoretic modelof venture capitalists and entrepreneursrdquo European Journal ofOperational Research vol 144 no 2 pp 440ndash453 2003

[10] W A Sahlman ldquoThe structure and governance of venture-capital organizationsrdquo Journal of Financial Economics vol 27no 2 pp 473ndash521 1990

[11] A Shleifer and R W Vishny ldquoLiquidation values and debtcapacity a market equilibrium approachrdquo The Journal ofFinance vol 47 no 4 pp 1343ndash1366 1997

[12] S Dahiya and K Ray ldquoStaged investments in entrepreneurialfinancingrdquo Journal of Corporate Finance vol 18 no 5 pp 1193ndash1216 2012

[13] R Amit J Brander and C Zott ldquoVenture capital financingof entrepreneurship in Canadardquo in Capital Markets Issues inCanada P Halpern Ed pp 237ndash277 Industry Canada OttawaCanada 1997

[14] P Gompers and J LernerTheVenture Capital Cycle MIT PressCambridge Mass USA 1999

[15] D V Neher ldquoStaged financing an agency perspectiverdquo Reviewof Economic Studies vol 66 no 2 pp 255ndash274 1999

[16] S D Zhang and D X Wei ldquoAnalysis on multi-stage investmentgame of double moral hazard in venture capitalrdquo NankaiEconomics Studies vol 54 no 6 pp 142ndash150 2008

[17] B Wu X X Xu and J M He ldquoDouble-side moral hazard andconvertible bond design in venture capital firmsrdquo Journal ofManagement Sciences in China vol 15 no 1 pp 11ndash20 2012

[18] J-E De Bettignies and J A Brander ldquoFinancing entrepreneur-ship bank finance versus venture capitalrdquo Journal of BusinessVenturing vol 22 no 6 pp 808ndash832 2007

[19] J-E De Bettignies ldquoFinancing the entrepreneurial venturerdquoManagement Science vol 54 no 1 pp 151ndash166 2008

[20] R Repullo and J Suarez ldquoVenture capital finance a securitydesign approachrdquo Review of Finance vol 8 no 1 pp 75ndash1082004

[21] E Fehr and K M Schmidt ldquoA theory of fairness competitionand cooperationrdquo The Quarterly Journal of Economics vol 114no 3 pp 817ndash868 1999

[22] M Rabin ldquoIncorporating fairness into game theory and eco-nomicsrdquo American Economic Review vol 83 no 5 pp 1281ndash1302 1993

[23] R Fairchild ldquoFairness norms and self-interest in venture cap-italentrepreneur contracting and performancerdquo ManagementOnline Review 2010

[24] J J Zheng and J F Wu ldquoResearch on moral hazard of ventureentrepreneur base on the fairness preferencerdquo Technology Eco-nomics vol 29 no 8 pp 88ndash92 2010

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 13

[25] J J Zheng and M Y Xu ldquoA research into venture financingcontract on the basis of fairness preferencerdquo Economic Reviewno 1 pp 14ndash18 2010

[26] E Fehr and A Falk ldquoPsychological foundations of incentivesrdquoEuropean Economic Review vol 46 no 4-5 pp 687ndash724 2002

[27] E Fehr and U Fischbacher ldquoWhy social preferences mattermdashthe impact of non-selfish motives on competition cooperationand incentivesrdquo Economic Journal vol 112 no 478 pp C1ndashC332002

[28] A Falk and A Ichino ldquoClean evidence on peer effectsrdquo Journalof Labor Economics vol 24 no 1 pp 39ndash57 2006

[29] B Bartling and F A Von Siemens ldquoInequity aversion andmoral hazard with multiple agentsrdquo Working Paper MimeoUniversity of Munich Munich Germany 2004

[30] G X Wei and Y J Pu ldquoFairness preference and tournamentincentivesrdquoManagement Science vol 19 no 2 pp 47ndash54 2006

[31] Y J Pu ldquoA model incorporating fairness into principal-agent acontribution from behavior economicsrdquo Contemporary Financeand Economics vol 268 no 3 pp 5ndash11 2007

[32] Y J Pu ldquoA principal-agent model with fairnessrdquo China Eco-nomic Quarterly vol 7 no 1 pp 298ndash318 2007

[33] M Dufwenberg and G Kirchsteiger ldquoA theory of sequentialreciprocityrdquo Games and Economic Behavior vol 47 no 2 pp268ndash298 2004

[34] D FWan J H Luo and J F Zhao ldquoThemanagerial trustworthybehavior and employeersquos effort levelmdashbased on a two stagessequential reciprocity game modelrdquo Systems Engineering vol 7no 1 pp 101ndash106 2009

[35] W Shi and Y J Pu ldquoDynamic reciprocity effect based on differ-ent information conditionsrdquo Journal of System Engineering vol28 no 1 pp 167ndash179 2013

[36] Y J Pu and W Shi ldquoReciprocity incentive effects based on theperspective of conict of interestrdquo Journal of Systems Engineeringvol 28 no 1 pp 28ndash37 2013

[37] R Fairchild ldquoAn entrepreneurrsquos choice of venture capitalist orangel-financing a behavioral game-theoretic approachrdquo Journalof Business Venturing vol 26 no 3 pp 359ndash374 2011

[38] J Yang ldquoAn entrepreneurrsquos choice of venture capitalist or angel-financing a behavioral game-theoretic approachrdquoManagementSciences vol 56 no 9 pp 1568ndash1583 2010

[39] M Wu and S Y Lai ldquoThe study of effect on venture entre-preneurrsquos persistent efforts to venture investment decision-makingrdquo Journal of Industrial Engineering and EngineeringManagement vol 27 no 4 pp 22ndash32 2013

[40] R Fairchild ldquoFinancial contracting between managers andventure capitalists the role of value-added services reputationseeking and bargaining powerrdquo Journal of Financial Researchvol 27 no 4 pp 481ndash495 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of