Contracting over multiple parameters: Capacity allocation in semiconductor manufacturing

European Journal of Operational Research 182 (2007) 174–193

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Production, Manufacturing and Logistics

Contracting over multiple parameters: Capacity allocationin semiconductor manufacturing

Suman Mallik *

Department of Business Administration, University of Illinois, 350 Wohlers Hall, 1206 S. Sixth Street, Champaign, IL 61820, United States

Received 5 March 2006; accepted 14 July 2006Available online 27 October 2006

Abstract

This paper is a generalization of Mallik and Harker [Mallik, S., Harker, P.T., 2004. Coordinating supply chains withcompetition: Capacity allocation in semiconductor manufacturing. European Journal of Operational Research 159, 330–347] that presented an integrated model of incentive problems arising in forecasting and capacity allocation. In that model,multiple product managers and multiple manufacturing managers forecast the means of their respective demand andcapacity distributions, and a central coordinator allocates capacities based on these forecasts. A mechanism that elicitstruthful information from the managers was the main contribution of that paper. The objective of this paper is to gener-alize our previous results to multiple statistics reporting. This work assumes that the central coordinator can ask the man-agers to report multiple statistics (mean and variance, for example) about their respective distributions. We propose a gametheoretic model and design a mechanism (a bonus scheme and an allocation rule) that elicits truthful reporting of all sta-tistics by all managers. It turns out that the structure of the optimal bonus schemes are rather simple with easily calculableparameters. We also show that a large class of allocation rules are manipulable. A bonus is often required for elicitation oftruthful information. We compare our results of multiple statistics reporting with those from Mallik and Harker (2004).We also characterize under what conditions the reporting of the extra information is of limited use.� 2006 Elsevier B.V. All rights reserved.

Keywords: Supply chain management; Demand forecasting; Game theory; Capacity allocation; Incentives; Mechanism design

1. Introduction

This paper, motivated by the experiences of a major US-based semiconductor manufacturer, presents ageneralization of our previous work, Mallik and Harker (2004). The said firm is in the business of manufac-turing and marketing of telecommunications, electronic and computer equipment. The firm operates five waferfabrication facilities (fabs) and produces 36 major product lines. A large proportion of the product mix of thefirm is of relatively short life cycle (one or two years). A semiconductor chip loses 60% of its value within firstsix or seven months of its life cycle. The operating environment of the firm is characterized by volatile demand,high manufacturing leadtime (2–3 months, typically), and rapid change of technology. In order to respond

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2006.07.013

* Tel.: +1 217 265 0683; fax: +1 217 244 7969.E-mail address: [email protected]

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S. Mallik / European Journal of Operational Research 182 (2007) 174–193 175

quickly to changing operating environment, the firm has organized itself into a divisionalized structure. Aproduct manager (PM) heads each product group, while a manufacturing manager (MM) heads each waferfab. Because of the high manufacturing leadtime, production planning and capacity allocation decisions arebased on forecasts from the PMs and MMs. Mallik and Harker (2004), (henceforth abbreviated as MH) stud-ied the incentive problems arising under such circumstances at the said firm.

Since our current work is a generalization of MH, a brief description of the problem studied in the saidpaper is warranted. In MH, the demands of each product and the capacities of wafer fabs are treated as ran-dom and are private knowledge of the respective PMs and MMs. The capacity is a random variable as is theyield in semiconductor manufacturing. Each manager forecasts the mean of his respective demand or capacitydistribution. A central coordinator is responsible for allocating capacities to product groups based on theseforecasts. In the presence of a capacity shortage, which is often the case with the firm under consideration(see Table 1 in MH), the managers tend to behave strategically and misrepresent their respective forecasts.A PM tends to inflate his forecast with the hope of getting a greater allocation of capacity. On the other hand,a manufacturing manager knows that today’s forecast is tomorrow’s production quota! Therefore, he tends tobe conservative and understates his forecast. The central coordinator allocates capacities to the competingproduct lines based on these forecasts. MH studied this problem from the perspective of the central coordi-nator and designed a mechanism that induces truthful reporting by all managers. The word mechanism is usedto mean an allocation rule that the center follows to allocate capacities to different products and a bonusscheme for all managers. In particular, they address the following issues.

• What are the structures of the bonuses that elicits forecasts truthfully? Are the bonuses always required?• What kind of allocation rules are appropriate from the objective of truth telling?

To model the fast-changing environment of semiconductor manufacturing, MH considers an extension oftheir basic model (called the limited information case) where no prior information about the demand andcapacity distributions are available to the central coordinator. They show that it is possible to design a mech-anism for truthful reporting by all managers even under the assumption of limited information.

The forecasts of the PMs and the MMs in MH are point forecasts of the average demands and capacitiesrespectively. As a result, the MH study only considered the contracting problem over a single parameter,namely the mean of a distribution. This paper generalizes the results reported in MH by considering a similarcontracting problem over multiple parameters. In this paper, we keep the basic structure of the coordinationproblem identical to that of MH, but assume that the central coordinator can ask the PMs and the MMs toreport multiple statistics about their respective distributions (for example, mean and standard deviation). Theapproach is appealing from the perspective of the central coordinator as it now gets an estimate of the uncer-tainties around the forecasts and hence will do a better job in allocating capacities to the competing productlines. In fact, the said firm is currently pushing the managers to provide rough estimates of the uncertaintiesassociated with the forecasts of the demands and capacities. We propose a game theoretic model and seek todesign a mechanism that elicits truthful information about multiple statistics of the unknown distributions.Since only the first two moments (mean and variance) of a distribution are useful for any practical purpose,we restrict a bulk of our analyses of multiple statistics reporting to two statistics reporting, namely, the meanand the variance. However, as shown in Section 4, our results can be generalized to any finite number of sta-tistics reporting. To our knowledge, this paper is the first to report contracting over multiple parameters in theOperations Management literature.

Our work relates to the general stream of literature in operations management studying contracting prob-lems in supply chain management. A reader is referred to the excellent papers by Tsay and Agrawal (2004)and Cachon (2003) for a review of this research. More specifically, our work relates to two streams of literaturewithin this paradigm: the inventory competition problems and the capacity allocation problems. Inventorycompetition typically arises out of product/demand substitution. Substitution has been studied extensively innewsvendor context. The typical examples are Parlar and Goyal (1984), Parlar (1988), Lippman and McCardle(1997), Ernst and Kouvelis (1999) and Netessine and Rudi (2003). Smith and Agrawal (2000) study the problemof jointly deciding the stocking levels and assortment under probabilistic substitution. In supply chain coordi-nation literature, Van Ryzin and Mahajan (2000) consider a single retailer trading with an oligopoly of

176 S. Mallik / European Journal of Operational Research 182 (2007) 174–193

manufacturers whose products are substitutable. They show that competitive overstocking due to substitutioncounteracts the understocking induced by double marginalization. McGillivray and Silver (1978), Ernst andKouvelis (1999) and Netessine (2004) discuss various examples of inventory competition in a single-firm con-text. The focus of this stream of literature is to study how the competition for inventory affects the over-all sup-ply chain performance. As a result, available manufacturing capacity is often treated as infinite.

The capacity allocation literature, on the other hand, treats the available capacity as finite and studies thesupply chain dynamics introduced by this constraint. Apart for the MH study cited earlier, our work probablycome closest to that of Deshpande and Schwartz (2002). They consider a single supplier with limited capacityselling to several retailers who are privately informed of their optimal stocking levels and develop optimal pric-ing and allocation rule for the supplier. In order to implement the optimal allocation rule they design an auc-tion mechanism where the retailers bid for supplier capacity. Their work is a generalization of Cachon andLariviere (1999a,b) who only study the allocation rules. Our work differ from that of Deshpande and Schwartz(2002) and Mallik and Harker (2004) by studying a contracting problem over multiple parameters. We showthat it is possible to design a contract that elicits truthful information about multiple parameters of anunknown distribution. We compare and contrast our current results of multiple statistics reporting with thosefrom MH (single statistics reporting). Finally, Geng and Mallik (2006) attempt to bring the two streams ofliterature together by considering inventory competition and allocation issues simultaneously in a single-man-ufacturer–single-retailer supply chain. The focus of this study, however, is whether a manufacturer shouldundercut a retailer’s order. Thus, neither the contracting over multiple parameters nor the truth-eliciting allo-cation is the focus of this study.

It is important to emphasize that while we describe our capacity allocation problem in the context of semi-conductor manufacturing, similar problems arise in various other industries as well. In fact, similar problemsarise whenever the demand of a product exceeds its capacities. As reported in MH, going on ‘‘allocation’’ is acommon occurrence in industries in which capacity expansion is costly and time consuming (e.g. steel, paper,etc.). It also occurs with popular new products such as hot automobiles, popular toys during Christmas time,and initial public offering of stocks. In an intra-firm context, allocation issues can also arise when several pro-jects (e.g. R&D, new product development, etc.) within a firm compete for the scarce resources.

The remainder of the paper is organized as follows. We describe our model and analyses in Sections 2 and 3.Section 4 generalizes our results of two statistics reporting to n-statistics reporting. Section 5 summarizes andconcludes the paper.

2. The model and analyses

We discuss the model specifications in Section 2.1 while the analyses are presented in Section 2.2. To facilitatethe comparison of our work with MH, we keep the model setup in our current work identical to that of MH.

2.1. Model specification

The players in our model are the individual managers (i.e., the PMs and the MMs). The respective strategiesare the reports of the statistics about the respective distributions. The strategies belong to a strategy set definedby some transformation of the support of the respective distributions. All our equilibrium results are underdominant strategy equilibrium where a player has a dominant strategy for all possible actions by other players.The following subsections formally define the model.

2.1.1. Assumptions

The assumptions of our model are identical to those of MH. The MH study provides detailed justificationfor each assumption. The following assumptions will be used throughout the remainder of the paper:

1. Single period model with no inventory carryover. Short product life cycle justifies this assumption.2. The center as well as all managers is risk neutral. This assumption is standard in operations management

literature.


3. Demands are independent of each other. This assumption is required for analytical tractability. This is areasonable assumption given the large number of products and large number of suppliers (of the each prod-uct) in semiconductor industry.

4. Capacity of each plant is independent of the others. The capacities are independent of the demands.5. Problem parameters (excess and shortage costs, production costs, product revenues etc.) are common

knowledge. This is a valid assumption as all managers are part of the same firm. There is no uncertaintyregarding the accounting procedures and parameters.

6. All managers are treated equally in the allocation process; i.e., personal clout of the managers does notinfluence the allocation decisions.

7. A product manager does not have any preference for a specific manufacturing plants.

2.1.2. The timing

The timing of the events is as follows:

1. The managers privately learn the distribution of their respective demands or capacities.2. The center announces the allocation rule for the product managers and the bonus schemes for both the

product and manufacturing managers for all possible realizations of demands and capacities. All managersobserve the announcement.

3. The PMs and MMs give their forecasts simultaneously to the center.4. Allocations are made by the center (based on the forecasts) as per the announced rules and are observed by

all parties.5. Production takes place. Capacities are realized, and are seen by all.6. Demand is realized and is seen by all. The managers are rewarded as per announced incentive schemes.

Given the high manufacturing leadtime in semiconductor manufacturing, it is imperative for the firm toallocate capacities to the products based on forecasts. The presence of multiple products prevents the alloca-tion to be made after the capacities are realized.

2.1.3. Notation

We define the following quantities:

i = index on product, i = 1,2, . . . ,n,k = index on plant, k = 1,2, . . . ,m,hi = unit salvage (overage) cost of product i,pi = unit shortage cost for product i,ei = unit revenue for product i,ai = unit production cost for product i,Zi = random demand for product i, with cumulative distribution function Gi(Æ), density gi(Æ), and mean mi;

zi is a realization of Zi,Bk = random capacity for fab k, with cumulative distribution function Fk(Æ), density fk(Æ), and mean lk; bk is

a realization of Bk.

The center requests the expected values of two transformations, T(Æ) = (T1(Æ),T2(Æ)), of the random variables Zi,i = 1,2, . . . ,n, and Bk, k = 1,2, . . . ,m. Given that only the first two moments of any random variable are usefulfor any practical purposes, we restrict our analyses to the reporting of the mean and the variance only (or thefirst two moments of a distribution). This implies that T1(Zi) = Zi and T 2ðZiÞ ¼ Z2

i 8i. Similarly, T1(Bk) = Bk

and T 2ðBkÞ ¼ B2k 8k. Section 4 of the paper will describe how our results are generalizable to an arbitrary num-

ber of moments. Let

aij = report of jth transformation from PM i, j = 1,2;bkj = report of jth transformation from MM k, j = 1,2;


bk = (bk1,bk2) = the reported forecast of MM k;ai = (ai1,ai2) = the reported forecast of PM i;Sk(bk,bk) = bonus for MM k;ri(a,zi) = bonus for PM i.

Let b = (b1,b2, . . . ,bm) be the m-vector of reported capacities and a = (a1,a2, . . . ,an) be the n-vector ofreported demands. Let b�k denote the m � 1 vector (b1,b2, . . . ,bk�1,bk+1, . . . ,bm), and a�i denote the n � 1vector (a1,a2, . . . ,ai�1,ai+1, . . . ,an). Also, let (x,a�i) denote the vector a with its ith element replaced by x. Sim-ilar definition applies for (y,b�k). Finally, let (x)+ denote max{x, 0}.

2.1.4. Information structure

We assume each of the true demand and capacity distributions to be normal. Thus, Zi � Nðmi; r2i Þ; 8i, and

Bk � Nðlk; �r2kÞ; 8k. These distributions are private knowledge of the respective managers. All parties, how-

ever, do possess a common prior over each of the unknown distributions. The common priors are assumedto be normally distributed with unknown parameters. We will let h denote the joint distribution of the com-mon priors.

2.1.5. Utility maximization problems

We will first describe the utility maximization problem of a product manager. The payoff of PM i, Ri(a,b,zi),is the sum of two components: the bonus payment ri(Æ) received for participating in the forecasting process; anda pre-specified fraction, wi, of the profit pi(Æ) arising from selling the product to the customer. Thus,

Riða; b; ziÞ ¼ riða; ziÞ þ wipi½‘iða; bÞ; zi�; wi 2 ð0; 1Þ; ð1Þ
where ‘i(a,b) is the allocation rule and denotes the allocation received by product manager i, given the vectorof reported forecasts a and b. We consider a newsboy type profit function for each product manager i,i = 1,2, . . . ,n. Thus,
pið‘iða; bÞ; ZiÞ ¼ ei minf‘iða; bÞ; Zig þ hi½‘iða; bÞ � Zi�þ � pi½Zi � ‘iða; bÞ�þ � ai‘iða; bÞ:
Rearranging the above equation using standard newsboy calculations we get
pið‘iða; bÞ; ZiÞ ¼ ðei � aiÞZi � ðai � hiÞ½‘iða; bÞ � Zi�þ � ðpi þ ei � aiÞ½Zi � ‘iða; bÞ�þ: ð2Þ
We consider the wi’s to be given. The utility maximization problem of the PM is given by
Maxai

EGi EhRiða; b; ZiÞ: ð3Þ

We next look at the utility maximization problem for a manufacturing manager. Note that our formulation in(2) and (3) assumes that each PM is guaranteed to receive an allocation of ‘i(a,b). Thus, when the realizationsof capacities are such that each PM can not be allocated ‘i(a,b), the center needs to outsource some manufac-turing capacity (which is more expensive). As described in MH, the practice in the said firm is such that a man-ufacturing manager is penalized only when the capacity realization is substantially less than the forecast.Otherwise the center bears the extra cost (the center recognizes that the MMs are forecasting the means ofrandom variables). The payoff of MM k, thus, is given by

Skðbk; bkÞ ¼ skðbk; bkÞ � c bk1 � bk � dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q� �þ: ð4Þ

The first term in the above formulation, sk(bk,bk), is the bonus for MM k and is a function of the reportedcapacity and the realized capacity of MM k. The second term in the above expression represents the penalty

faced by an MM when the demand realization is less than the forecast by at least dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q. Recall that bk1

and bk2 is the reported expected values of Bk and B2k . Thus,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

qrepresents the reported standard devi-

ation of the capacity of MM k. The formulation in (4), thus allows the center to control the accuracy of fore-cast in units of the reported standard deviation. The center announces c and d, which are common knowledge.As discussed in MH, the firm did not charge any explicit penalty when the realized capacity was more than the


forecast. Neither did it provide an incentive for it. To cover for the uncertainties in the production process, theMMs tended to understate the capacities. The utility maximization problem of MM k is given by

Maxbk

EF k Skðbk;BkÞ: ð5Þ

The difference between the structures of the utility maximization problems of the PMs and the MMs meritsfurther discussion. In line with the practice in the firm, the utility maximization problem of a PM describedin (1) allocates a fraction of profit from selling a product to the respective PM. As described in Section 1,the firm has 36 product lines, each headed by a single PM. Thus, by allocating a fraction of profit to aPM, the center is effectively providing an incentive to a product manger to sell more. Simultaneously, thebonus function ri(Æ) provides an incentive to do accurate forecasting. However, the firm views a similararrangement of profit sharing to be impractical for a manufacturing manager because of the following rea-sons. The firm operates 4 fabs, each headed by a MM, that produce the 36 product lines. Thus, multiple prod-ucts are produced in a single plant. The yields in semiconductor manufacturing varies depending upon the typeof technology (for example, micron/sub-micron/nano) and learning effects. Clearly, learning improves theyield. The management of the firm feels that allocating a fraction of profit from producing a product to aMM will hinder the introduction of new technologies as the yields are lower with new technologies. In addi-tion, it might introduce an unhealthy competition among to MMs to get allocation of products for which theyields are higher. Consistent with this practice, the utility maximization problems of a MM in Eqs. (4) and (5)do not include a profit-sharing term.

The aim of the central coordinator is to design a mechanism such that truth telling is equilibrium strategyfor all managers. By the word mechanism we mean the allocation rule ‘i(a,b) for the product managers and thebonus schemes (or the incentive schemes), sk(bk,bk) and ri(a,zi). Note that we did not specify the issue of opti-mality of allocation while specifying the objective of the central coordinator. Seeking optimal allocation is one

objective of this paper, though not the sole objective. Within the broad framework of truth-telling, we alsoseek to study the feasibility of implementing an optimal allocation. We call this the optimal allocation problem.

2.1.5.1. The optimal allocation problem. An optimal allocation maximizes the total profit of the firm less theincentive payments. The following is a formulation of the optimal allocation problem:

max‘i;ri;sk

Eh

Xn

i¼1

ð1�wiÞpið‘iða;bÞ;ZiÞ�Xn

i¼1

riða;ZiÞ�Xm

k¼1

skðb;BkÞ�Xm

k¼1

�c dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2� b2

k1

q� bk1þBk

� �þ( ): ð6Þ

Subject toXn

i¼1

‘iða;bÞ6 KðbÞ; ð7Þ

where KðbÞ ¼P

kbk1 is the total reported capacity. Under the assumption that the revelation principle holds,the central coordinator can restrict itself to searching for only those mechanisms under which each managerreveals their respective private information truthfully. Thus,

mi ¼ arg maxai1

EhEGi Riðai; a�i; b; ZiÞ; ð8aÞ

ri ¼ arg maxai2

EhEGi Riðai; a�i; b; ZiÞ ð8bÞ

lk ¼ arg maxbk1

EhEF k Skðbk; b�k; BkÞ; ð9aÞ

�rk ¼ arg maxbk2

EhEF k Skðbk; b�k; BkÞ ð9bÞ

skð�ÞP 0; rið�ÞP 0; ‘iða; bÞP 0: ð10Þ

Eqs. (6)–(10) define the optimal allocation problem. The last term in (6) denotes the cost incurred by the center

when bk falls short of bk by an amount less than dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

qor delta times the reported standard deviation.


The center incurs a cost of �c for each unit of shortage. However, when the shortfall is more than dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q,

the cost is allocated to an MM through Eq. (4). Note that MH used a similar approach while studying thecontracting problem over a single parameter (the mean). Eq. (7) is the capacity constraint. Eqs. (8) and (9)are the incentive compatibility constraints that ensure truth telling. The participation constraint, (10), ensuresthat a manager will not receive a negative bonus.

2.2. Analyses

The results and insights from of model are described in this section. Following MH, we will seek to deriveinsights on following items.

1. The structure of the truth-eliciting bonus function for manufacturing managers and product managers.2. The structure of truthful allocation rules.3. The possibility of implementing an individually responsive (IR) allocation.4. The possibility of implementing an optimal allocation.

Items 1 and 2 noted above constitute the truth telling mechanism in our model. Under an individually

responsive (IR) allocation rule, if one is receiving a positive allocation but wants more, one gets more unlessone has already claimed all of capacity. An example of IR allocation rule is the proportional allocation wherethe total capacity is allocated to competing product lines in proportion to the respective forecasts. As dis-cussed in MH, the current allocation process in the semiconductor manufacturer bears a strong resemblanceto the proportional allocation. Thus, by studying the possibility of implementing an IR allocation, we seek todevelop insight on the current allocation process. The optimal allocation is of great interest since it is a bench-mark against the centralized system. Throughout the current section we will compare and contrast our resultswith MH which deals with contracting over a single parameter (the unknown means of the demand and capac-ity distributions). The proofs of all results are included in Appendix.

Theorem 1. A bonus function sk(Æ) for manufacturing manager k will elicit truthful information about both thestatistics of the unknown capacity distribution at equilibrium if it satisfies the following relationships:

Z 1

�1

oskðbk; xÞobk1

dF kðxÞ � c 1þ dbk1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q0B@

1CAU

bk1 � dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q� lk

�rk

0@

1A ¼ 0; and

Z 1

�1

oskðbk; xÞobk2

dF kðxÞ þcd

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q Ubk1 � d


k1

q� lk

�rk

0@

1A ¼ 0;

where U(y) denote the area under the standard normal distribution curve up to point y.

Theorem 1 specifies the necessary condition for truthful reporting of both the statistics by MM k. Note thatthe exact distributions, Fk, are unknown to the center. However, depending upon the available prior informa-tion, the center can design the bonus sk(Æ) such that truthful reporting is the equilibrium strategy for the man-agers. For example, when the common prior on the unknown capacity distributions is normally distributedwith unknown parameters, it can be shown that the following bonus function will elicit truthful informationabout both the statistics at equilibrium

skðb; bkÞ ¼ c0bk1 þ c1bk1T 1ðbkÞ � c2b2k1 � c3


k1

qþ c4bk2T 2ðbkÞ � c5b

2k2; ð11Þ

where co, . . . ,c5, are constants independent of bk such that c0 = cU(�d), c1 = 2c2, c3 = dc0, c4 = 2c5. Note thatthe constants, co, . . . ,c5, are common knowledge. Interesting observations can be made from Theorem 1 andEq. (11). First, a mechanism for truthful reporting of multiple statistics can, in deed, be designed even whenvery little prior information about the unknown distribution is available. Second, the bonus function in (11) israther simple in structure with easily calculable parameters. Finally, it is also interesting to compare the bonus


functions for single statistics reporting and multiple statistics reporting cases. For the single statistics case,MH shows that the following bonus function elicits truthful information.

TableData f

PM 1PM 2PM 3MM 1MM 2

skðb; bkÞ ¼ c1bk � c2ðbk � bkÞþ � c3ðbk � bkÞþ; ð12Þ

where c1, c2 and c3 are some appropriate constants and bk is the report of the mean of the random capacity Bk.In the single statistics case, the center requested the expected value of the random variable Bk. Thus, the centercould observe the outcome and penalize the individual managers such that a risk neutral manager will reporthis respective mean truthfully; giving rise to the bonus function in (12). However, under multiple statisticsreporting, the center requests the expected values of both Bk and B2

k . Note that the random variable B2k is never

observed by the center, neither is it realized. Therefore, a bonus of the form (12) will not be appropriate formultiple statistics reporting. Eqs. (11) and (12) together compare and contrast the structure of bonus functionsfor the MMs under single statistics and multiple statistics reporting. The following theorem summarizes thestructure of the bonus and allocation rules for truthful reporting by the PMs.

Theorem 2. A bonus function ri(Æ) and an allocation rule ‘i(Æ) for product manager i will elicit truthful information

about both the statistics at equilibrium if ri(Æ) and ‘i(Æ) satisfy the following relationships:
Z 1
�1

oriða; giÞoaij

dGiðgiÞ þ wiðpi þ ei � aiÞZ 1

�1

o‘iða; bÞoaij

hðbÞdb

� wiðpi þ ei � hiÞU‘iða; bÞ � mi

ri

� � Z 1

�1

o‘iða; bÞoaij

hðbÞdb ¼ 0; j ¼ 1; 2; ð13Þ

where U is the area under the standard Normal curve.

Theorem 2 specifies the necessary condition for a bonus function and an allocation to elicit truthful infor-mation at equilibrium. In fact the equilibrium also happens to be dominant equilibrium. Note that the param-eters mi and ri are not known to the central coordinator. Thus, the central coordinator needs to select a bonusfunction and an allocation rule such that the above conditions are satisfied at ai1 ¼ mi; ai2 ¼ r2

i þ m2i . In other

words, ai1 ¼ mi; ai2 ¼ r2i þ m2

i are the solutions to the system of equations defined in (13). The quadratic bonusfunction and the modified lexicographic allocation defined below are the examples which satisfy suchcondition.

riða; ziÞ ¼ ci½2ai1T 1ðziÞ � a2i1 þ 2ai2T 2ðziÞ � a2

i2�; and ð14Þ

‘iðaÞ ¼ min KðbÞ �Xi�1

j¼1

‘jðaÞ; ai1 þ ½ai2 � a2i1�

1=2U�1 pi þ ei � ai

pi þ ei � hi

� �( ): ð15Þ

Note that under a modified lexicographic rule, the PMs are ranked in some random manner independent oftheir forecast (say alphabetically) and are allocated production in accordance with this ranking. A PM withrank i receives the minimum of yet unallocated capacity and his ‘‘modified’’ forecast, as described in (15). Notethat all parameters in (14) and (15) are common knowledge. We illustrate the above allocation rule and bonusfunction using the following numerical example.

Example 1. This example will involve three PMs and two MMs. Table 1 below shows the data used incomputations.

1or Example 1

Actualmean

ActualSD

Actual EðZ2i Þ

or EðB2kÞ

pi + ei � ai

(Underage)ai � hi

(Overage)ci (Eq. (4)) wi Optimal

order

10 3 109 35 4 0.005 10% 1422 8 548 20 6 0.0005 10% 2817 6 325 22 4 0.0005 10% 2320 6 436 N/A N/A N/A N/A N/A15 8 289 N/A N/A N/A N/A N/A


The first three data columns of Table 1 represent the actual values of the parameters of the demand andcapacity distributions which are the private knowledge of respective managers. The next four data columnsrepresent the problem parameters which are common knowledge. The last column of Table 1 representsthe newsvendor optimal order quantity for each PM, which is again the private knowledge of a PM. Obvi-ously, a PM will earn the highest expected profit when the capacity allocated to him matches this quantity.

Note that the quantity U�1 piþei�ai

piþei�hi

� �in (15) is common knowledge. Thus, the central coordinator can announce

an allocation rule based on this quantity before receiving the forecasts. We consider three different scenarios ofreporting of private information by the managers as described below.

Scenario 1: a11 = 10, a12 = 109; a21 = 22, a22 = 548; a31 = 17, a32 = 325,Scenario 2: a11 = 10, a12 = 109; a21 = 26, a22 = 712; a31 = 20, a32 = 423, andScenario 3: a11 = 14, a12 = 218; a21 = 31, a22 = 1096; a31 = 24, a32 = 650,Lexicographic order: PM 2, PM 3, PM 1.

Recall that in our model setup, a PM reports E(Zi) and EðZ2i Þ. Comparing the three scenarios above with

the true values from Table 1, it is easy to see that all PMs report private information truthfully under Scenario1. Under Scenario 2, PM 1 still reports information truthfully, while both PM2 and PM3 inflate their respec-tive forecasts (roughly by 20% for the first statistic and by 30% for the second statistic), while in Scenario 3 allmanagers inflate their forecasts (roughly by 40% for the first statistic and by 100% for the second statistic).Table 2 below provides the computations of expected bonus and the expected revenue share for three differentvalues of total reported Capacity K(b).

It is worth emphasizing that under (14) and (15) truth telling (i.e. Scenario 1) is the unique equilibrium. Thecomputations for Scenarios 2 and 3 simply gives a reader an idea about how much of expected reward a man-ager might lose by not acting rationally. As a result, bulk of our discussion will focus on Scenario 1. Severalinteresting observations can be made from Table 2. First, note from Table 1 that the sum of the newsvendoroptimal order quantities for the three PMs is 65. Thus, when K(b) P 65, the capacity constraint is no longerbinding. Therefore, when the PMs report their information truthfully, the quantity allocated to each managerthrough (15) equals the newsvendor optimal order quantity (Scenario 1). Second, as described in Theorem 2,truth telling is the equilibrium strategy under (14) and (15). It is easy to see from Table 2 that the expectedvalues the bonus as well as the profit share (i.e., Ewipi[‘i(a,b), zi]) goes down as the reports deviate from thetruthful information (Scenarios 2 and 3, for all values of the reported capacity). In any of the three scenarios,the bonus of a PM does not change when the reported capacity K(b) changes. Per (14), the bonus of a PMdepends only on the reports and outcomes involving the demand distributions. The capacity allocation of aPM, however, changes as the reported capacity K(b) changes. Third, the capacity constraint becomes bindingfor K(b) = 45 or for K(b) = 35. Hence the managers need to go on allocation. Consider the case K(b) = 45 andScenario 1. Under the lexicographic allocation, PM 2 is ranked highest and is followed by PM 3. Thus, per(15), PM 2 receives an allocation of

Table 2Illustration bonus and allocation schemes in Eqs. (14) and (15)

K(b) Scenario 1 Scenario 2 Scenario 3

PM 1 PM 2 PM 3 PM 1 PM 2 PM 3 PM 1 PM 2 PM 3

65 Qty allocated 14 28 23 9.69 30 25 0 40 25Exp. bonus 60 150 53 60 137 48 0.42 0.20 0.12Exp. profit sh. 33 38 34 30 37 34 0 33 33

45 Qty allocated 0 28 17 0 30 15 0 40 5Exp. bonus 60 150 53 60 137 48 0.42 0.20 0.12Exp. profit sh. 0 38 31 0 37 29 0 33 12

35 Qty allocated 0 28 7 0 30 5 0 35 0Exp. bonus 60 150 53 60 137 48 0.42 0.20 0.12Exp. profit sh. 0 38 15 0 37 10 0 36 0


min KðbÞ ¼ 45; ai1 þ ½ai2 � a2i1�


pi þ ei � hi

� �¼ 28

¼ 28;

which is also his newsvendor optimal order quantity. After allocating 28 units to PM 2, the center is left with17 units. The allocation of PM 3 (2nd in rank) thus is given by

min KðbÞ �X1

j¼1

‘jðaÞ ¼ 17; ai1 þ ½ai2 � a2i1�


pi þ ei � hi

� �¼ 23

( )¼ 17:

PMs 2 and 3, thus, have consumed all available capacity resulting in zero allocation for PM 1. The other sce-narios have similar interpretations. Table 2 also quantifies the expected profit share for each manager. Givenhis rank, PM2 clearly gains at the expense of PM3 and PM1. The computations of expected bonuses for theMMs are straightforward using (11). It is easy to verify that for c2 = 0.005, c5 = 0.0005, d = 2 and c = 2, theexpected bonus of MM 1 and MM 2 at equilibrium (i.e. under truthful reporting, implying K(b) = 35) are97.41 and 42.84 respectively. Note that the choice of the parameter d = 2 implies that the center imposes extrapenalty on an MM when the realization of a capacity falls short of forecast by more than two times the re-ported standard deviation. Otherwise, as explained earlier, the center bears the cost of shortage, the expectedvalue of which is 54.37 for �c ¼ 2. Finally, at equilibrium (i.e., under Scenario 1 with K(b) = 35) expected resid-ual profit for the central coordinator after making all contractual payments to all managers is 159.43.

Theorem 2 tells us that by designing an appropriate bonus and an allocation rule, the center can extractinformation about the entire distribution when the form of the distribution is known. Theorems 1 and 2 alsoyield several important insights, which are stated in the following corollaries.

Corollary 1. A bonus payment is essential to elicit the information truthfully from the manufacturing managers atequilibrium; while it is not essential to elicit information truthfully from the product managers at equilibrium.

The proof is straightforward, and hence, is omitted. This result is similar to the corresponding result ofthe single statistics case (Corollary 1 of MH). We showed in MH that it is possible for the central coordi-nator to design an allocation rule such that a PM will report information truthfully in absence of a bonus.Our current result establishes the same result to be true even when the variance of the Normal distribution isunknown. An example of such allocation rule is the modified lexicographic allocation rule described byEq. (15). Under such an allocation rule the dominant strategy of each of the product managers is to reportthe expected values of requested statistics truthfully. Thus, in this situation, profit sharing alone is enoughfor truthful reporting.

It is also instructive to highlight the difference between the modified lexicographic rules described in (15)and that in MH (single statistics reporting). The allocation rule described in MH for single statistics reportingrequires the knowledge of the standard deviation of the normal distributions, while that described in (15)does not require the knowledge of the standard deviation. Thus, our current corollary generalizes the findingsin MH. The intuitive explanation of why modified lexicographic allocation is truthful is the following. WhenPM i is ranked sufficiently high such that he will get the allocation of capacity, reporting ai1 ¼ mi; ai2 ¼ r2

i þ m2i

gives him the newsvendor optimal quantity. Thus, he has no incentive to over-report. On the other hand, whenthe rank of PM i is such that he will not be allocated any capacity, a PM is indifferent between reporting truthor the otherwise. Corollary 1 also implies that the modified lexicographic allocation rule is quite robust in thesense that it retains its truth-revealing property even when the parameters of the Normal distribution areunknown. We next look at the possibility of implementing an individually responsive (IR) allocation ruleby revelation.

Corollary 2. An individually responsive (IR) allocation rule can elicit truthful information at equilibrium.

However, a bonus is required for truthful reporting.

The proof is by construction and is included in Appendix. We have demonstrated that a linear allocationrule (which is IR) can elicit information about both the statistics truthfully in conjunction with an appropriatebonus function. The presence of a bonus function, in addition to the profit function, in our model allows theimplementation of an IR mechanism by revelation. The intuition behind this result is that the center can set


the bonus for truthful forecasting to be sufficiently high so that it compensates a PM for the forgone profit interms of product allocation.

We again compare our current result about IR allocation with the corresponding result of MH. MH showsa similar result for the IR allocation rule. However, there is an important distinction between our current workand MH: a larger class of allocation rules (not just the IR allocation) will require a bonus for implementationby revelation for the multiple statistics reporting. We state this result formally in the following Corollary.

Corollary 2a. Under multiple statistics reporting, a bonus payment is essential to elicit truthful information from

a PM for any allocation rule that is solely a function of the outcome; i.e., the first moment of the unknowndistribution.

Note that IR allocation rules (for example, the linear and the proportional allocation rules) are special casesof the allocation rules defined by Corollary 2a. This corollary tells us that a larger class of allocation rules willrequire a bonus for truthful reporting under the multiple statistics case. The findings of Corollary 1 are alsosupported by this corollary. In Corollary 1, we stated that the modified lexicographic allocation defined by(15) does not require a bonus for truth elicitation. This allocation rule is a function of the first two moments.We have not addressed the issues of optimal allocation thus far. We do so now in Corollary 3 below.

Corollary 3. Given appropriate bonus functions for the PMs and MMs, an optimal allocation can be implemented

by revelation at equilibrium.

We have defined the optimal allocation in Eqs. (6)–(10). Given the appropriate bonus functions (as sug-gested by Corollary 3), ri(Æ) and Sk(Æ) are no longer the decision variables in the optimal allocation problemand the problem of the center reduces to finding an allocation rule ‘i(a) that maximizes (6). Note that witha given ri(a,Zi) and Sk(b,Bk), the solution, ‘�i , of the optimal allocation problem is given by

U‘�i ða; bÞ � mi

ri

� �¼ ð1� wiÞðpi þ ei � aiÞ � kðbÞ

ð1� wiÞðpi þ ei � hiÞ; ð16Þ

where k(b) is the shadow price of the capacity constraint (7) and U is the area under the standard normaldistribution curve. Note that Eq. (16) is also the optimal newsvendor solution under full information. Next,consider the following allocation rule:

‘iða; bÞ ¼ max 0; ai1 þ ½ai2 � a2i1�

1=2U�1 ð1� wiÞðpi þ ei � aiÞ � kðbÞð1� wiÞðpi þ ei � hiÞ

� � : ð17Þ

By comparing this allocation with the capacitated newsboy solution of (16), we can say that allocation rule(17) will implement an optimal allocation by revelation if ai1 = mi and. ai2 ¼ r2

i þ m2i ; 8i. Thus, to achieve

an optimal solution, the center needs to design a bonus function ri(Æ) such that it, along with the allocationrule in (17), maximizes the expected payoff of PM i at ai1 ¼ mi; ai2 ¼ r2

i þ m2i ; 8i. As shown in the proof of

Corollary 3, the following reward function achieves an optimal allocation:

riða; ziÞ ¼ kðbÞ wi

1� wi

ðzi � ai1Þ � fai2 � a2i1g

1=2U�1 ð1� wiÞðpi þ ei � aiÞ � kðbÞð1� wiÞðpi þ ei � hiÞ

� �� : ð18Þ

Cachon and Lariviere (1999b) show that there does not exist a truth inducing mechanism that maximizes thetotal profit. What is the difference between their work and our work? The allocation decisions obtained with(17) and (18) are newsvendor optimal. However, the resulting sum of profits from our model is not. In fact, noclaims can be made about the resulting profit. We have demonstrated the implementation of an optimal allo-cation for a given bonus function, as suggested by Corollary 3. To claim that the resulting profit is optimal forthe firm, we have to prove that the chosen reward functions result in minimum coordination loss among allpossible reward functions. However, solving the optimization problem for arbitrary classes of functions ri(Æ),Sk(Æ) and ‘(Æ) is extremely difficult. Therefore, we had to restrict Corollary 3 for given reward functions. TheMH study also made a similar assumption. Cachon and Lariviere (1999b) consider a problem involving a sin-gle manufacturer and multiple retailers. Their truth-telling mechanism involves only an allocation rule. On thecontrary, our truth-telling mechanism involves an allocation rule as well as a bonus. This lets us achieve an


optimal allocation of capacity for a given bonus function, as described in Corollary 3. We illustrate the opti-mal allocation in the following example.

Example 2. We use the data provided in Table 1 to illustrate our result. When truth telling is the equilibriumstrategy for all managers, the total reported capacity K(b) = 35, which is no longer sufficient to satisfy all PMswith their newsvendor optimal quantities. Under such a situation we solve the optimal allocation problem fora given bonus function (Eq. (18)) for the allocation rule described in (17) using standard optimizationprocedures. This gives us k(b) = k(35) = 15 as the shadow price of the capacity constraint (7). Table 4 belowsummarizes other quantities of interest.

It is easy to verify that for k(b) = k(35) = 15 and under the bonus and allocation rules described by (17) and(18), the equilibrium strategy for all managers is to report all information truthfully. The resulting allocationfor the PMs is first best. To compare the optimal allocation with the lexicographic allocation, we next comparethe allocation decisions and expected bonus and profits of Table 3 with those of Table 1 (under Scenario 1 andK(b) = 35). Under lexicographic allocation PM2 is arbitrarily given priority over the other PMs. As a result,PM2 is allocated 28 units of capacity (which is also his newsvendor optimal), PM3 is allocated 17 units whilePM 1 is denied any capacity. The Scheme in Table 3 maximizes the total profit of the firm for a given bonusfunction. As a result, the capacity allocation decisions are different under optimal allocation and that all PMsare allocated capacity. PM1 and PM3 under optimal allocation make more expected profit than under lexico-graphic allocation at the expense of PM 2. The expected bonus for all three managers is identical under thetwo allocation rules. The bonus of a PM depends only on the forecasts and outcomes of the demand distri-bution statistics. As a result it does not change with a change in allocation rule. How does the expected resid-ual expected profit for the central coordinator after making all contractual payments to all managers compareunder the two allocation rules? We showed in Example 1 that this profit is 159.43 under lexicographic alloca-tion. A similar calculation using identical parameters (note that the rewards of the MMs do not change as anallocation rule affects the PMs only) show that the same is 252.56 under the optimal allocation.

Given our current work is an extension to and generalization of the MH study, a comparison between ourwork and MH is warranted. Table 4 below accomplishes this. The most significant observation from Table 4 isthat the piece-wise linear structure of the truth-eliciting bonus is lost under multiple statistics reporting. Theintuitive reasoning behind this is as follows. Under multiple statistics reporting, the center requests the fist tomoments of the unknown distributions. However, a random variable such as z2

i or B2k is never realized in

Table 3Illustration of optimal allocation for K(b) = 35

PM 1 PM 2 PM 3

Quantity allocated 10 13 12Expected bonus 60 150 53Expected profit share 30 25 24

Table 4Comparison of our equilibrium results with those of MH study

Item of comparison Single statistics reporting(Mallik and Harker, 2004)

Multiple statistics reporting(this paper)

Structure of truth-eliciting bonus • Piece-wise linear in report for bothPM and MM

• Piece-wise linearity does not hold• Quadratic in reports

Bonus for MM Required for truthful reporting Required for truthful reportingBonus for PM Not required for truthful reporting Not required for truthful reportingIR allocation Truthful with bonus • Truthful with bonus

• A larger class requires bonusOptimal allocation • Truthful with bonus

• Bonus is piece-wise linear in report• Truthful with bonus• Bonus is not linear in reports

Truthful allocation (modified

lexicographic)• Knowledge of ri required • Knowledge of ri not required


practice. On the other hand, under single statistics reporting, the center requested only the first moment of theunknown distributions and that the outcome of the random variables zi and Bk were actually realized andobserved. This resulted in a piece-wise linear structure of the bonus function which penalized a managerfor over- or under-reporting.

Another interesting observation from Table 4 is that the modified lexicographic allocation is quite robust inthe sense that it is truth-eliciting even when the parameters of the normal distributions are unknown. Thus, asshown in Table 4, it is the truth-eliciting allocation under both single and multiple statistics reporting. We havealso shown that, unlike the single statistics reporting, any allocation rule that is solely a function of the out-come is manipulable under multiple statistics reporting. The importance of multiple statistics reporting is thatit allows the central coordinator to extract all information about an unknown distribution with minimal priorinformation.

3. Limited information: The case of no priors

Citing examples from the semiconductor manufacturer under consider as well as from a satellite telephonemanufacturer, the MH study argued that modeling of highly uncertain environment in the volatile high techindustry often requires the assumption that no prior information exists about the uncertain environment. Theycall this scenario as the scenario with limited information. In the context of our capacity allocation problemthe limited information case referred to the scenario where the central coordinator does not possess any priorinformation about the demands and capacities. The MH study showed that it is possible, even under limitedinformation, to design bonuses and allocation rules such that all managers report their private informationtruthfully. These results are robust and distribution-independent. However, their work was again limited tothe reporting of a single statistics (the mean) of the unknown distribution. In the current section we extendand generalize the findings of MH to multiple statistics reporting, namely reporting both mean and the stan-dard deviation. A reader is referred to the MH study for the motivating examples.

In accordance with our discussion, we, in this section, assume that the center does not know the distribu-tions of demands or capacities and no prior information is available. Our objective is to design a skim fortruthful reporting of both the statistics about the unknown distribution. The results presented in this section,thus, are valid for any distribution of demands and capacities. Note that a great majority of the contractingpapers in the operations management literature assumes the existence of a prior on the part of the contractdesigner. In contrast, our current analyses will not require this assumption. The following lemma forms thebasis of our analysis.

Lemma 1. Let X 2 [a,b] be a random variable with distribution function F(Æ). Let T1(x) and T2(x) be two

continuous and strictly monotonic functions of x and T(x) = (T1(x),T2(x)). Also let t = (t1, t2) denote another

vector of finite dimension. A continuously differentiable function gðxÞ : ½a; b� ! R, satisfies the conditionR ba gðxÞdF ðxÞ ¼ 0; 8F such that

R ba T ðxÞdF ðxÞ ¼ t, if and only if there exists a vector k = (k1,k2), independent of

x, such that g(x) = k1[T1(x) � t1] + k2[T2(x) � t2].

Lemma 1 is a generalization of a similar result of the MH study for single statistics reporting. This lemma isquite powerful in the sense that it is valid for all distributions. This lemma allows us to prove the followingresult for the limited information case.

Theorem 3. Truth telling is the equilibrium strategy under limited information if and only if the bonus function for

manufacturing manager k is of the following form:

skðbk; bkÞ ¼ ckðbkÞ þockðbkÞobk1

½T 1ðbkÞ � bk1� þockðbkÞobk2

½T 2ðbkÞ � bk2� ð19Þ

for some appropriate function ck(Æ) and for c = 0.

Theorem 1 gives the necessary and sufficient condition for truth telling by a manufacturing manager. Thefirst term in the above equations represents the surplus of a manager and solely depends on the reported infor-mation. Hence it can be considered as an ex-ante payment. The manager is penalized (or rewarded) ex-post inproportion to the difference between actual and projected values of the requested statistics. The proportion-


ality factors are given by the gradient of the surplus function. Theorem 1 in Section 3 specified the necessarycondition for truth telling by a manufacturing manager in presence of prior information. Under limited infor-mation no prior on Fk(Æ) is available. Therefore, the only choice for the center to satisfy the first order neces-sary condition of the utility maximization problem of a MM is to set c = 0. This means that the center is not ina position to charge the MMs an extra penalty (and it bears the cost of shortfall), but encourages truth tellingthrough the bonus function. Theorem 3 also illustrates two important features that any truth-eliciting bonusfunction must satisfy. First, the bonus is separable and additive in the two requested statistics. Second, thebonus is linear in both the requested statistics. An appropriate choice of the function ck(Æ) in (19) isck ¼ b2

k1 þ b2k2. Substituting this into (19) yields the following truth-eliciting bonus function for MM k.

skðbk; bkÞ ¼ ½2bk1T 1ðbkÞ � b2k1� þ ½2bk2T 2ðbkÞ � b2

k2�: ð20Þ

The elegance of this bonus function lies in its simplicity. In addition, (20) has the nice property of beingstrongly quasiconcave in bkj, j = 1,2. This means that the bonus payment is highest when the forecast matchesexactly with the outcome, i.e. bkj = Tj(bk), j = 1,2 and decreases monotonically as the deviation between thetwo increases. The MH study reported the following bonus function to be truth-eliciting for single statisticsreporting under limited information

skðbk; bkÞ ¼ c½2bkbk � ðbkÞ2�; ð21Þ

where c is any constant independent of bk. Interesting insight can be obtained by comparing (20) and (21).First, the bonus function for the multiple statistics case reduces to that of single statistics case if only the firstmoment of the unknown distribution is requested from the individual managers. Second, the class of bonusfunctions does not seem to ‘‘shrink’’ as more information is elicited. The reason for this is that as the setof reported parameters of a distribution increases, the central coordinator has a wider choice of bonus func-tions. Third, the bonus function for the multiple statistics case is additive in the requested moments, with asimilar functional form in each of the requested moments. The MH study shows that additional classes oftruth-eliciting bonus functions can be obtained if additional information about the distributions is available.For example, when the supports of the distributions are finite and known, they show the following bonus to betruth-eliciting.

skðb; bkÞ ¼ c ðbk � BkÞ þ ðBk � bkÞ lnBk � bk

Bk � Bk

" #; ð22Þ

where ½Bk; �Bk�, with 0 6 Bk < �Bk <1, is the support of Bk, for all k. A simple calculation shows that adding asecond term to the RHS of (22) involving the second moment of the unknown distribution that is similar infunctional form of the first moment will be truth eliciting for the multiple statistics case. However, as (20) and(21) demonstrate, the assumption of finite support is required.

We next look at the product managers. The following theorem describes the structure of proper bonus func-tions and allocation rules for the product managers.

Theorem 4. Under limited information, a bonus function ri(Æ) and an allocation rule ‘i(Æ) for product manager i will

elicit truthful information about both the statistics at equilibrium if ri(Æ) and ‘i(Æ) satisfy the following relationships

for any distribution function Gi(Æ):

Z 1 Z 1

�1

oriða; giÞoaij

dGiðgiÞ þ wiðpi þ ei � aiÞ�1

o‘iða; bÞoaij

hðbÞdb

� wiðpi þ ei � hiÞGi½‘iða; bÞ�Z 1

�1

o‘iða; bÞoaij

hðbÞdb ¼ 0; j ¼ 1; 2: ð23Þ

The proof of Theorem 4 is straightforward and is omitted. This theorem gives the necessary condition fortruthful reporting by a PM. Note that under limited information, the distributions Gi(Æ) and h(Æ) are unknown


to the central coordinator. Under such a situation, the combination of classes of functions ri(Æ) and ‘i(Æ) thatsatisfy the conditions of Eq. (23) are quite narrow. Examples of such functions are

riða; ziÞ ¼ ½2ai1T 1ðziÞ � a2i1� þ ½2ai2T 2ðziÞ � a2

i2�; ð24Þ

and an allocation rule ‘i(a,b) independent of aij. We call this allocation rule the constant allocation. One exam-ple of such allocation is ‘i(a,b) = K(b)/n, where K(b) is the total reported capacity and n is the number of prod-uct managers. We were unable to find any other combination of reward and allocation rule that satisfiescondition (23). In fact, under limited information, the only way the center can satisfy (23) is to seto‘iða;bÞ

oaij¼ 0; j ¼ 1; 2 ¼ 0. Note that when the bonus is given by (24),

R1�1

oriða;giÞoaij

dGiðgiÞ ¼ 0; j ¼ 1; 2, by Lemma

1, for all distributions Gi(Æ).Theorem 4 is quite similar to that of the corresponding result of MH. In addition, the following results are

also similar to those of the corresponding results of MH. We, therefore, state them without providing any clar-ifying discussions.

Corollary 4. For multiple statistics reporting under limited information, a bonus payment is required to elicit theinformation truthfully from the manufacturing managers at equilibrium. However, a bonus is not essential for the

product managers. Under such a situation, the center does not take any action based on the reports of the product

managers.

Corollary 5. An individually responsive (IR) allocation rule does not elicit truthful reporting at equilibrium under

limited information.

Corollary 6. An optimal allocation cannot be achieved at equilibrium under limited information.

The proofs of Corollaries 4–6 are straightforward given that no prior information is available under limitedinformation. It is clear from our discussion so far that the presence of prior information allows the centralcoordinator significant flexibility over the limited information regarding the implementation of different allo-cation rules. However, it is still interesting to note that it is possible to get truthful reporting even under limitedinformation.

4. Discussion

Several interesting insights can be obtained by comparing our results with prior information and thoseunder limited information and by comparing our results with those from the MH study. By comparing ourresults for multiple statistics reporting under prior information with those for single statistics reporting (fromthe MH study), we find the following differences. The truth eliciting bonus functions in multiple statisticsreporting have different functional form than those from the single statistics reporting. In addition, and as dis-cussed earlier, there are other differences between single and multiple statistics reporting results in terms of thestructure of the modified lexicographic allocation rule and the possibility of an IR allocation being truth-elic-iting under prior information. On the other hand, by comparing our current work with the MH study wehardly see any difference in the results between the single and multiple statistics reporting under limited infor-

mation. What contributes to this anomaly? Under limited information, the central coordinator has no infor-mation about the unknown distributions. Under such scenario, in order to estimate the distribution of arandom variable from its moments, one needs to know infinite number of moments. Thus, requesting an extramoment of the unknown distributions does not make any significant difference when the distribution is com-pletely unknown. As a result, we do not see any significant difference in the limited information resultsbetween our study and the MH study.

An interesting feature of the multiple statistics reporting is that the bonus functions are additive and sep-arable in the two requested statistics. In addition, under limited information, the single statistics results can beobtained from the multiple statistics results by simply ignoring the terms involving the second moment. How-ever, this is not the case in presence of prior information. The origin of this difference can be traced to theutility maximization problem of the MMs. The structures of the utility functions of the MMs differ slightly


between our current study and the MH study under prior information. However the requirement c = 0 makesthem identical under limited information.

5. Generalization to N-statistics reporting

In this section, we show that our results for two statistics reporting are generalizable to any arbitrary butfinite number of statistics. This is a significant result as this allows the central coordinator to get arbitrarilyclose to the unknown distributions by requesting more and more information. Our analyses for the case ofprior information assume the unknown distributions to be Normal. Thus, the question of generalization ofthese results beyond two statistics does not arise. We therefore restrict our analyses in this section to the lim-ited information case only.

We will modify our notation slightly in this Section to accommodate the case of N-statistics reporting. Wewill assume that the center now requests the expected values of N transformations, T(Æ) = (T1(Æ),T2(Æ), . . . ,TN(Æ)), of the random variables Zi, i = 1,2, . . . ,n, and Bk, k = 1,2, . . . ,m. We will let aij and bkj denotethe report of jth transformation from PM i, and MM k respectively for j = 1, . . . ,N. The definitions of thevectors of b and a will be as follows: bk = (bk1, . . . ,bkN), b = (b1, . . . ,bm), ai = (ai1, . . . ,aiN), a = (a1,a2, . . . ,an).

Following our analyses in Section 3, we will set c = 0 in the utility maximization problem of MM k, asdefined in (4). All other utility maximization problems will remain identical.

Lemma 2 below generalizes Lemma 1 and provides the foundation for the analyses n-statistics reporting.

Lemma 2. Let X 2 [a,b] be a random variable with distribution function F(Æ). Let T1(x), T2(x), . . . ,TN(x), for

finite N, be continuous and strictly monotonic functions of x and T(x) = (T1(x),T2(x), . . . ,TN(x)). Also let

t = (t1, t2, . . . , tN) denote another vector of finite dimension. A continuously differentiable function

gðxÞ : ½a; b� !R, satisfies the conditionR b

a gðxÞdF ðxÞ ¼ 0; 8F such thatR b

a T ðxÞdF ðxÞ ¼ t, if and only if there

exists a vector k = (k1,k2, . . . , kN), independent of x, such that gðxÞ ¼PN

i¼1ki½T iðxÞ � ti�.

The proof is included in Appendix. It is clear from this lemma that the additive form of bonus functions inrequested statistics will be valid for any finite generalization of the two statistics case. This result leads to thefollowing structure of the truth-eliciting bonus function for MM k.

skðbk; bkÞ ¼ ckðbkÞ � rckðbkÞ � ½T ðbkÞ � bk� ð25Þ
for some appropriate function ck(Æ), where $ck(Æ) represent the gradient of ck(Æ) and the term$ck(bk) Æ [T(bk) � bk] represents the scalar product of two vectors. Clearly, Eq. (25) is the generalization ofTheorem 3 in Section 3. An appropriate choice of the function ck(Æ) in (25) is ck ¼
PNj¼1b

2kj. Substituting this

into (25) yields the following truth-eliciting bonus function for MM k.

skðbk; bkÞ ¼XN

j¼1

½2bkjT jðbkÞ � b2kj�: ð26Þ

It is clear from Eqs. (25) and (26) that our results of Section 3 are generalizable to any finite number of sta-tistics. The analyses for the products managers are similar and hence are omitted.

6. Summary and conclusion

In this paper, we studied contracting over multiple parameters in context of forecasting and capacity allo-cation. This is a generalization of our previous work, MH, on single statistics reporting. The majority of theoperations management literature on supply chain contracting assumes only one unknown parameter overwhich a contract is made (Deshpande and Schwartz, 2002, for example). We have shown that it is possibleto design a mechanism that elicits information truthfully about multiple parameters of a distribution simulta-neously. As shown in Section 4, our results for two statistics reporting are generalizable to any arbitrary finitenumber of statistics. The most interesting feature of the truth-eliciting bonus function in multiple statisticsreporting is its additive form in the requested statistics, with a similar functional form in each of the requestedstatistics. It is shown that under the condition of limited information (when nothing is known about the


distributions and no prior is available), requesting an extra moment is of limited use. With prior information,we did not obtain a piece-wise linear bonus function in multiple statistics reporting. We have also highlightedthe major differences in results between our study and those of MH. An interesting conclusion our analyses isthat the modified lexicographic allocation is quite robust in the sense that it can be implemented by revelationeven when the parameters of the Normal distribution are unknown.

Our work was motivated by the experiences of a US-based semiconductor manufacturer. However, theframework we have proposed is quite general and can be applied to any allocation problem. We assumed riskneutrality to kep the results analytically tractable. This assumption is standard in operations management lit-erature. The main contribution of our model is in considering a contracting problem over multiple parameters.Despite our specific motivation from semiconductor manufacturing, the analytical results presented in thispaper are general enough to be used in other coordination problems.

Acknowledgement

The author graciously acknowledges the helpful comments of two anonymous reviewers.

Appendix. Proofs of results

Proof of Theorem 1. Using Eqs. (4) and (5) it is straightforward to see that the utility maximization problem of

MM k is given by Maxbk1bk2

R1�1 skðbk;bkÞdF kðbkÞ � c

R bk1�dffiffiffiffiffiffiffiffiffiffiffiffibk2�b2

k1

p0 ðbk1 � d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibk2 � b2

k1

q� xÞfkðxÞdx. The

conditions of Theorem 1 follows directly from taking the first order necessary condition of the optimizationproblem described above and noting that o2

b2kjfEF k Skðb;BkÞg < 0; j ¼ 1; 2. It is straightforward to verify that

Eq. (11) satisfies the conditions stated in Theorem 1. h

Proof of Theorem 2. The proof of Theorem 2 is straightforward and involves taking the first order necessaryconditions and noting that o2

oa2ijfERiða; ZiÞg < 0; j ¼ 1; 2. To see that (14) and (15) together are truth eliciting,

plug (14) and (15) into (13). Note that when PM i’s allocation is independent of his order (this happens whenhe is either denied capacity or is given the leftover capacity after allocating capacity up to the rank i-1),o‘iða;bÞ

oaij¼ 0; j ¼ 1; 2 and that Eq. (13) is trivially satisfied. Thus, we only need to examine the case o‘iða;bÞ

oaij6¼ 0.

Let X ¼ piþei�ai

piþei�hi, for notational simplicity. Therefore,

o‘iða; bÞoai1

¼ 1� ai1

½ai2 � a2i1�

1=2U�1ðX Þ and

o‘iða; bÞoai2

¼ 1

2½ai2 � a2i1�

1=2U�1ðX Þ:

Substituting these expressions in (13) we get two identical equations given by

Uai1 þ ½ai2 � a2

i1�1=2U�1ðX Þ � mi

ri

" #¼ X :

To see that ai1 ¼ mi; ai2 ¼ r2i þ m2

i is the unique solution to the above equation, note that ai1 ¼ mi; ai2 ¼ r2i þ m2

i

satisfies the above equation and that the LHS of the above equation is monotone increasing in ai1,ai2, while theRHS of the above equation is independent of both ai1, ai2. h

Proof of Corollary 2. The proof is by construction. We provide an example of the implementation of a linearallocation rule (which is IR) by an appropriate choice of bonus function. In a linear allocation rule the man-agers are ranked according to a decreasing order of forecasts. The allocation of a manager with rank i is givenby

‘iða; bÞ ¼ai1 �

1

~n

X~n

j¼1

aj1 � KðbÞ !

; i 6 ~n;

0 i > ~n;

8><>: ðA:1Þ


where ~n is the largest integer less than or equal to n such that ‘~nða; bÞ > 0 and ‘~n þ 1ða; bÞ 6 0. Along with theabove allocation rule, consider the following bonus function:

riða; ziÞ ¼ c1iai1 � c2iðai1 � ziÞ2 þ c3if2ai2z2i � a2

i2g; 8i; ðA:2Þ

where

c1i ¼~n� 1

~nwi ðpi þ ei � hiÞU

�P~n

j¼1aj1 � KðbÞ� �

~nri

0@

1A� ðpi þ ei � aiÞ

24

35; c2i � c1i; 8i: ðA:3Þ

From (A.1) and (A.2), the solution to the first-order conditions with respect to ai2 for the utility maximizationproblem of PM i is given by

ai2 ¼ EðZ2i Þ ¼ r2

i þ m2i :

With (A.1), (A.2) and (A.3), it can be shown that the first-order conditions with respect to ai1 for the utilitymaximization problem of PM i receiving a positive allocation are given by

Uai � 1

~n

P~nj¼1aj � KðbÞ

� �� mi

ri

0@

1A ¼ 1

wiðpi þ ei � hiÞc1i � 2c2iðai � miÞf g ~n

~n� 1þ wiðpi þ ei � aiÞ

� �:

ðA:4Þ

To see that ai1 = mi is the unique solution to (A.4), note that the LHS of (A.4) is increasing in ai1 and RHS of(A.4) is decreasing in ai1 and ai1 = mi satisfies (A.4). Thus, it is the unique solution. When the PM i is notreceiving any positive allocation of product, the condition c2i� c1i/2 ensures that the first order conditionof the PM is satisfied at ai1 = mi. h

Proof of Corollary 2a. When an allocation rule is a function of the first moment only, we must have o‘iða;bÞoai2¼ 0.

When o‘iða;bÞoai2¼ 0, Eq. (13) becomes

R1�1

oriða;ziÞoai2

dGiðziÞ ¼ 0. When an allocation rule is independent of ai2, theonly reason a PM will report the second moment truthfully is to satisfy the above equation. However, if abonus is not provided the PM does not have any incentive to report the second moment truthfully. h

Proof of Corollary 3. Consider the allocation rule and the bonus given by (17) and (18). We show that theallocation rule and the reward function together will give rise to ai1 ¼ mi; ai2 ¼ r2

i þ m2i ; 8i. For notational sim-

plicity, let Y i ¼ U�1 ð1�wiÞðpiþei�aiÞ�kðbÞð1�wiÞðpiþei�hiÞ

� �. With this notation we have

o‘iða; bÞoai1

¼ 1� ai1

½ai2 � a2i1�

1=2Y i; and

o‘iða; bÞoai2

¼ 1

2½ai2 � a2i1�

1=2Y i: ðA:5Þ

Substituting (17), (18), and (A.5) into (13) we obtain the following first-order condition for the utility maxi-mization problem of PM i:

Uai1 � mi þ ½ai2 � a2

i1�1=2U�1 ð1�wiÞðpiþei�aiÞ�kðbÞ

ð1�wiÞðpiþei�hiÞ

� �ri

24

35 ¼ ð1� wiÞðpi þ ei � aiÞ � kðbÞ

ð1� wiÞðpi þ ei � hiÞ: ðA:6Þ

To see that ai1 ¼ mi; ai2 ¼ r2i þ m2

i is the unique solution to the above equation, note that ai1 ¼ mi; ai2 ¼ r2i þ m2

i

satisfies the above equation and that the LHS of the above equation is monotone increasing in ai1,ai2, while theRHS of the above equation is independent of both ai1,ai2. h

Proof of Lemma 1. Lemma 1 is a special case of Lemma 2. The detailed proof of Lemma 2 is presentedbelow. h

Proof of Theorem 3. The proof of the ‘‘if’’ part of the Theorem follows trivially. Consider the ‘‘only if’’ part.With c = 0, the second term in (4) is zero and we re left with the first term only. Taking the first order


necessary condition of MM k’s problem, we get, oobkj

RX skðbk; nkÞdF kðnkÞ ¼ 0; or equivalently,R

Xoskðbk ;nkÞ

obkjdF kðnkÞ ¼ 0, for j = 1,2, "k. However, neither Fk nor X is unknown to the center. Therefore, the

center must setR

Xoskðbk ;nkÞ

obkjdF kðnkÞ ¼ 0 for all possible distributions, i.e., for all possible Fk(Æ). By Lemma 1,

this is possible if and only if

oskðbk; nkÞobkj

¼ q1jðbkÞ½T 1ðbkÞ � bk1� þ q2jðbkÞ½T 2ðbkÞ � bk2�; j ¼ 1; 2: ðA:7Þ

Integrating by parts the two equations in (A.7) we get,

skðbk; bkÞ ¼ ½T 1ðbkÞ � bk1�Z

q1jðbkÞdbk1 þ ½T 2ðbkÞ � bk2�Z

q2jðbkÞdbk1 þZ Z

½q1jðbkÞdbk1�dbk1;

skðbk; bkÞ ¼ ½T 1ðbkÞ � bk1�Z

q1jðbkÞdbk2 þ ½T 2ðbkÞ � bk2�Z

q2jðbkÞdbk2 þZ Z

½q2jðbkÞdbk2�dbk2:

Combining the arbitrary constants in the two equations above, we can write,

skðbk; bkÞ ¼ ckðbkÞ þ c1kðbkÞ½T 1ðbkÞ � bk1� þ c2kðbkÞ½T 2ðbkÞ � bk2�: ðA:8Þ
Since (A.8) must satisfy the two equations described in (A.7), we must have
skðbk; bkÞ ¼ ckðbkÞ þockðbkÞobk1

½T 1ðbkÞ � bk1� þockðbkÞobk2

½T 2ðbkÞ � bk2�: ðA:9Þ

Next consider ck ¼ b2k1 þ b2

k2. Substituting this into (19) and noting that c = 0 per Theorem 3, we get

Skðbk; bkÞ ¼ ½2bk1T 1ðbkÞ � b2k1� þ ½2bk2T 2ðbkÞ � b2

k2�. Therefore,

oSkðbk;BkÞobkj

¼ 2½T jðbkÞ � bkj�; j ¼ 1; 2; ando2Skðbk;BkÞ

ob2kj

¼ �2; j ¼ 1; 2:

Thus,

oSkðbk;BkÞobk1

¼ 0 ) bk1 ¼ E½T 1ðbkÞ� ¼ lk; andoSkðbk;BkÞ

obk2

¼ 0 ) bk2 ¼ E½T 2ðbkÞ� ¼ �r2k þ ðlkÞ

2: �

Proof of Lemma 2. Lemma 2 is a generalization of Lemma 1. We prove Lemma 2 below for an arbitrary N.Substituting N = 2 in the proof yields the proof for Lemma 1. The ‘‘if’’ part in both the Lemma is triviallytrue. Consider the ‘‘only if’’ part. Consider the ith component of the condition

R ba T ðxÞdF ðxÞ ¼ t,

for i = 1,2, . . . ,N. Here F(Æ) satisfiesR b

a T iðxÞdF ðxÞ ¼ ti. Choose any two points (x0,x1) 2 [a,b] such thatTi(x0) < ti < Ti(x1). Note that under the assumption that Ti(x) is continuous and strictly monotonic in x

and that ti is the expected value of Ti(x), it is clearly possible to choose (x0,x1) 2 [a,b] such that Ti(x0) < ti <Ti(x1). If possible, let, gðx1Þ

gðx0Þ6¼ T iðx1Þ�ti

T iðx0Þ�ti. Since X is any random variable, we construct the following r.v.

X ¼x1 w:p: ðti � T iðx0ÞÞ=ðT iðx1Þ � T iðx0ÞÞ;x0 w:p: ðT iðx1Þ � tiÞ=ðT iðx1Þ � T iðx0ÞÞ:

ðA:10Þ

Note that with this construction, the condition EF[Ti(x)] = ti is satisfied. With this X,EF gðX Þ ¼ gðx1Þ ti�T iðx0Þ

T iðx1Þ�T iðx0Þ þ gðx0Þ T iðx1Þ�ti

T iðx1Þ�T iðx0Þ 6¼ 0; as gðx1Þgðx0Þ 6¼

T iðx1Þ�ti

T iðx0Þ�ti. Hence we have a contradiction. Therefore

we must have gðx1Þgðx0Þ¼ T iðx1Þ�ti

T iðx0Þ�ti; implying g(x) = ki[Ti(x) � ti], i = 1,2, . . . ,N, where ki is a constant independent

of x. Suppose that it is possible to find �F ð�Þ and �gð�Þ such that E�F ½�gðxÞ� ¼ 0, but �gðxÞ 6¼ ki½T iðxÞ � ti�. For this�gðxÞ, choose F(x) to be the cdf defined by (A.10). Clearly, EF ½�gðxÞ� 6¼ 0. Note that our lemma requiresEF[g(x)] = 0, "F. Therefore the construction of �gðxÞ is incorrect. We have shown thatR b

a gðxÞdF ðxÞ ¼ 0; 8F such thatR b

a T iðxÞdF ðxÞ ¼ ti, if and only if g(x) = ki[Ti(x) � ti], "i. Now consider thatF(Æ) satisfies all N conditions simultaneously, i.e.,

R ba T ðxÞdF ðxÞ ¼ t. This means that F(Æ) satisfies the condition


Z b

a

XN

i¼1

qiT iðxÞ" #

dF ðX Þ ¼XN

i¼1

qiti ðA:11Þ

for constants q1, . . . ,qN. Repeating the argument presented above we can say thatR b

a gðxÞdF ðxÞ ¼ 0; 8F suchthat (A.11) holds if and only if: gðxÞ ¼ Q

PNi¼1qiT iðxÞ �

PNi¼1qiti

� �, where Q is a constant independent of x.

Rearranging and writing ki = qiQ, we get, gðxÞ ¼PN

i¼1ki½T iðxÞ � ti�. h

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Contracting over multiple parameters: Capacity allocation in semiconductor manufacturing

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