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Cooperative Game Theoretic Bid Optimizer for Sponsored Search Auctions

Sriram SomanchiElectronic Commerce Laboratory

Computer Science and AutomationIndian Institute Science, BangaloreEmail: somanchi@csa.iisc.ernet.in

Chaitanya NittalaElectronic Commerce Laboratory

Computer Science and AutomationIndian Institute Science, Bangalore

Email: chytu@csa.iisc.ernet.in

Narahari YadatiElectronic Commerce Laboratory

Computer Science and AutomationIndian Institute Science, Bangalore

Email: hari@csa.iisc.ernet.in

Abstract—In this paper, we propose a bid optimizer forsponsored keyword search auctions which leads to betterretention of advertisers by yielding attractive utilities to theadvertisers without decreasing the revenue to the search engine.The bid optimizer is positioned as a key value added toolthe search engine provides to the advertisers. The proposedbid optimizer algorithm transforms the reported values of theadvertisers for a keyword into a correlated bid profile usingmany ideas from cooperative game theory. The algorithm isbased on a characteristic form game involving the search engineand the advertisers. Ideas from Nash bargaining theory areused in formulating the characteristic form game to providefor a fair share of surplus among the players involved. Thealgorithm then computes the nucleolus of the characteristicform game since we find that the nucleolus is an apt way ofallocating the gains of cooperation among the search engineandthe advertisers. The algorithm next transforms the nucleolusinto a correlated bid profile using a linear programmingformulation. This bid profile is input to a standard generalizedsecond price mechanism (GSP) for determining the allocationof sponsored slots and the prices to be be paid by the winners.The correlated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of theunderlying game. Through detailed simulation experiments, weshow that the proposed bid optimizer retains more customersthan a plain GSP mechanism and also yields better long-runutilities to the search engine and the advertisers.

Keywords-Bid Optimizer, Sponsored Search, CooperativeGame Theory, Nash bargaining, Nucleolus.

I. I NTRODUCTION

Sponsored search auctions have been studied extensivelyin the recent years due to the advent of targeted advertisingand its role in generating large revenues. With a huge com-petition in providing the sponsored search links, the searchengines face an imminent problem which can be called as theretention problem. If an advertiser (or alternatively bidder)does not get satisfied because of not getting the right numberof clicks or the anticipated payoff, he could drop out of theauction and try sponsored links at a different search engine.

A. Motivation: Retention of Advertisers in Sponsored SearchAuctions

Our motivation to study the retention problem is drivenby the compulsions faced by both the search engine and theadvertisers.

From the advertisers’ perspective, choosing theirmaximum-willingness-to-pay such that they get an attractiveslot subject to their budget constraints is a challengingproblem. The search engines can use various mechanismsfor the sponsored search auction as described in [6], [11]but the most popular mechanism is the generalized secondprice (GSP) auction since it is simple and yields betterrevenue to the search engine. In the most simple versionof GSP, where there arek slots andn advertisers (forsimplicity assumek ≤ n), the allocation and payment ruleare as follows. The allocation rule is thatn advertisers areranked in descending order based on their bids, with tiesbroken appropriately, and topk advertisers’ advertisementsare displayed. The payment rule is that every advertiserneeds to pay bid amount of the advertiser who is justbelow his slot and last advertiser is charged the highestbid that has not won any slot. If the non-truthful GSPauction [13] is used by the search engine, the bidderswill have an incentive to shade their bids. The bidderswould not want to use complicated and computationallyintensive bidding strategies as the bidding process is donemany times (typically thousands of times) in a day. Theseadvertisers generally build their own software agents oremploy third party software agents, which adjust andreadjust the bid values on behalf of these advertisers. Thebidders typically specify their maximum willingness-to-payfor their keywords for any given day. Hence each keywordhas a specific set of bidders bidding on it for the wholeday. This scenario constitutes a repeated game between allthe bidders bidding for that keyword. In this game, bidderswho cannot plan their budget effectively may experienceless utilities and thus may drop out of the system.

We now turn to the search engine’s perspective of theretention problem. When the bidders try to know eachothers’ valuations by submitting and resubmitting bids, theymay find a set of strategy profiles which may yield allof them better payoffs. This may lead to collusion amongthe bidders. Folk theorems [4] suggest that players maybe able to increase individual profits by colluding therebydecreasing the search engine’s revenue. Even though thebidders in the keyword auctions are competitors, this collu-sion against the search engine could be stable. Vorobeychik

and Reeves [10] studied this phenomenon and illustrateda particular collusive strategy which is better for all thebidders (hence worst for the search engine) and can besustainable over a range of settings. Feng and Zhang [3]showed that dynamic price competition between competingadvertisers can lead to collusion among them. However,in this dynamic scenario, when the discounted payoffs ofthe bidders under the collusive strategy are considered, thestability of collusion depends inversely on the number ofbidders [4]. That is, the lower the number of bidders inthe system, the higher is the stability of the collusion. Thismotivates us to study the bidder retention problem for thesearch engine.

Also, due to exponential growth in the space of onlineadvertising and intense competition among the search enginecompanies, the switching cost for the advertisers to changefrom one search engine to another is almost zero [6]. Hence,it is imperative for the search engine companies to retaintheir advertisers to safeguard their market share. Driven bythis, the search engine companies have introduced manyvalue added tools, such as bid optimizer, to maximize thebang-per-buck for the bidders. In what follows, we describethe bid optimizer’s role in solving the retention problem.

B. Bid Optimizers

A bid optimizeris a software agent provided by the searchengine in order to assist the advertisers. The bidders arerequired to provide to the bid optimizer a target budget forthe day and a maximum willingness-to-pay. Bid optimizers,currently provided by the search engines, promise to maxi-mize the revenue of advertisers by adjusting the bid amountin each round of the auction based on the projected keywordtraffic and remaining budget.

It can be seen that the decisions made by the bid opti-mizer are crucial to both the search engine and the set ofadvertisers, who choose to use the bid optimizer. Hence, theobjective of a typical bid optimizer is to strike a balancebetween reduction in revenue of the search engine companyversus increase the retention of advertisers. This objective isachieved by providing enhanced utilities to the advertisers,thus ensuring retention of customers, thereby sustaining highlevels of revenue to the search engine company in the longrun. Designing such intelligent bid optimizers is the subjectof this paper.

There are some problems involved in designing bid opti-mizers.

1) For the search engine, maximizing its short-term rev-enue (that is, its payoff in a one-shot game) seemsto be a viable option. But here, the lower valuationbidders are denied slots due to allocative efficiencyconcerns. For the bidders, as shown by Caryet al [1],where all the high valuation bidders use a particulargreedy strategy, it has been proved that none of thebidders except the topk bidders get the slots after a

certain number of rounds of the auction. The abovephenomenon can permanently drive away low valua-tion bidders from the search engine.

2) Dropping out of the search engine to get better utilitiesin another search engine is a possible option for thebidders. The low valuation bidders drop out afternot getting slots for a certain period of time. Thehigher valuation bidders can observe this trend andshade their maximum-willingness-to-pay or colludeto get better utilities. This may result in the searchengine losing revenue. This is a threat to the searchengine from the bidders. However, if a large numberof bidders remain in the system, the collusion is notstable. The intuition for this is that, high valuationbidders cannot reduce their bids sharply, since theywill have the fear of undercutting the lower valuationbidders present in the system and thus losing out ontheir slots.

Hence, we propose that retaining more number of bidderssolves all the problems discussed above. The dependence ofthe search engine and the bidders on each other for mutualbenefit motivates us to use a cooperative approach in general.The above threat model naturally directs us towards usinga Nash bargaining model in particular. Our solution can beseen as associating the bid optimizer to a keyword ratherthan bidders as done by the existing bid optimizers. Theoverall model of the bid optimizer is depicted in Figure 1.

C. Contributions and Outline of the Paper

In this paper, we propose a bid optimizer that uses manyideas from cooperative game theory. The bid optimizer isshown in Figure 1.

• The inputs to the bid optimizer are the willingness-to-pay values (or valuations) of the bidders.

• The output of the bid optimizer is a correlated bidprofile, which, when input to a standard GSP auctionmechanism, yields utilities to the search engine and theadvertisers satisfying the goals set forth in the paper.

• The bid optimizer first formulates a characteristic formgame involving the search engine and the advertisers.The value for each coalition is defined based on anovel Nash Bargaining formulation with the searchengine as one player and a virtual player aggregatingall advertisers in that coalition as the other player. Theidea of using Nash Bargaining is to ensure a fair sharefor the search engine and the advertisers.

• The nucleolus of the above characteristic form game isselected as the utility profile for the search engine andthe advertisers. The choice of nucleolus is based on keyconsiderations such as, bidder retention, stability, andefficiency.

• The utility profile represented by the nucleolus ismapped to a correlated bid profile that satisfies indi-vidual rationality, retention, stability and efficiency. A

Figure 1. Proposed Bid Optimizer

linear programming based algorithm is suggested forthis purpose.

We carry out experiments to demonstrate the viabilityand efficacy of the proposed bid optimizer. We show, usinga credible bidder drop out model, that the proposed bidoptimizer has excellent bidder retention properties and alsoyields higher long-run revenues to the search engine, whencompared to the plain GSP mechanism.

The outline of the paper is as follows. Section II-A1presents the details of the bid optimizer and introducesthe model. In Section II-A3, we present a bid optimizationalgorithm which uses the Nash Bargaining approach forensuring the retention of bidders in the system. We thenmap this fair share for the aggregated bidder to a correlatedbid profile in Section II-C. We analyze the properties of ourmethod in Section II-D. We present our experimental resultsin Section III and conclude the paper in Section IV.

II. OUR APPROACH TOBID OPTIMIZATION

In this section, we present our algorithm for bid opti-mization. The algorithm can be divided into three phases asshown in Figure 1: (1) Characteristic form game definitionusing Nash bargaining, (2) Computing the utility vectors forthe players and (3) Inverse mapping of the utility vectorinto a correlated bid profile. These are discussed in thefollowing sections. The notation in the remainder of thepaper is presented in Table I.

A. Characteristic Form Game

1) The Model:The sponsored search auction scenario weconsider hasn bidders competing fork slots of a keyword.We assume that the probability that a bidderi gets clicked onthejth slot (or the click-through rateCTRij) is independentof the bidderi,that is,CTRij = βj and we also assume thatβ1 ≥ β2 ≥ . . . ≥ βk. Each bidderi specifies his maximumwillingness-to-paysi to the bid optimizer. The bid optimizertakes as input all thesi’s of the bidders and suggests them acorrelated bid profile. This bid optimization algorithm needsto be invoked only when the number of bidders in the systemor their willingness-to-pay change. We also assume that thebid of the playeri could be any real number in[0, si].

Given the above model, we define a bargaining problem1 between the search engine and the aggregated bidder andanalyze its properties which will help us in formulating acharacteristic form game.

2) Characterization of the Nash Bargaining Solution:The motivation for a cooperative approach is the dependenceof the search engine and bidders on each other for theirmutual benefit. Given this, the motivation behind choosinga bargaining approach is that the amount of short-term loss(or in other words, the investment of the search engine) forthe auctioneer should be chosen based on the bidders presentin the system. The Nash bargaining approach provides a

1Refer Appendix for the definition of Nash bargaining problem

Notation ExplanationA AuctioneerB Aggregated biddern Total number of playersk Total number of slots. We assumek < n

N Set of bidders{1, 2, . . . , n}K Set of slots{1, 2, . . . , k}si Maximum willingness-to-pay of advertiseriSi Strategy set of bidderi, [0, si]S Set of all bid profilesS1 × S2 × . . .× Sn

s Bid profile (s1, s2, . . . , sn) ∈ S

ui(s) Utility of bidder i on bid profilesUA(s) Utility of the auctioneer in the

Nash bargaining formulation for bid profilesUB(s) Utility of the aggregated bidder in the

Nash bargaining formulation for bid profilesUA maxs∈S UA(s)

UB maxs∈S UB(s)

βj Click through rate of any bidder in thejth slotTable I

NOTATION

framework for this amount to be chosen by the search engineby considering all the bidders as one aggregate agent whosebargaining power depends on all the maximum willingness-to-pay of all the bidders present in the system.

The utility of the aggregated bidder is the sum of theutilities of all the bidders over all possible allocations of slots(outcomes). Now, the bargaining utility space becomes thetwo dimensional Cartesian space which consists of the utilityof auctioneer on one axis and the aggregate bidder’s utilityon the other axis. Hence a bargaining solution on this spaceprovides a good compromise for the search engine fromits maximum possible revenue and thus gives the requiredinvestment of the search engine.

The bargaining space is defined in two dimensional Carte-sian space, with utility of auctioneerUA(s) along thex−axisand the utility of aggregated bidderUB(s) =

∑ni=1 ui(s)

along they−axis. LetUA andUB be the maximum possibleutilities of the auctioneer and the aggregated bidder respec-tively. It can be clearly seen that the valueUA is attainedfor the bid profiles = (s1, s2, . . . , sn) for which the cor-

respondingUB(s) =∑n

i=1

(

∑kj=1 βjyij(s)

)

(si − si+1) =

U′

B (say). Similarly, the bid profiles = (0, . . . , 0) yieldsthe utility pair (0, UB). Since it is theoretically possiblethat all the bidders can collude and bid(0, 0, . . . , 0), wechoose the point(0, 0) in this Nash bargaining space asthe disagreement point. Ramakrishnanet. al studied thisproblem in [8] and characterized the solution(U∗

A, U∗B) to

this Nash bargaining(NBS) as

(U∗A, U

∗B) = (UA, U

′

B) if UA ≤UB

2

=

(

UB

2,UB

2

)

otherwise

3) Definition of the Characteristic Form Game:Weuse the above model to define Nash bargaining solutionNBS(N) = U∗

A + U∗B where N is the set of bidders

participating in the auction.Let N = {1, 2, . . . , n} be the set of all bidders and let

0 represent the search engine. The characteristic form gameν : 2N∪{0} → ℜ for each coalitionC ⊆ N ∪ {0} is nowdefined as

ν(C) = NBS(C) if 0 ∈ C

= 0 otherwise

whereNBS(C) is defined as above. If the search engine isnot a part of the coalition, its worth is zero since the playerscannot gain anything without the search engine displayingtheir ads. Otherwise, we associate the sum of utilities in thecorresponding Nash bargaining bid profile for that coalitionwith the search engine as the worth of each coalition. Thischaracteristic functionν defines the bargaining power ofeach coalition with the search engine.

B. Computing a Utility Vector for the Players: Use ofNucleolus

Since there is an aggregation of the bidders’ revenuetaking place in the NBS, we map the utility of the aggregatedbidder in the Nash bargaining solution to a correlated bidprofile. The NBS gives an aggregate amount of investmentthe search engine has to make on all the bidders. Thisinvestment increases the utility of the aggregated bidder.This utility has to be distributed to the bidders in a waythat our goal of retention is reached. Ideally, we would likethe allocation to have the following properties.

• The bidders must not have incentive for not participat-ing in the bid optimizer (individual rationality-IR).

• It must retain as many bidders as possible(retention).• The bidders must not have the incentive to shade their

maximum willingness to pay (incentive compatibility).• It should be stable both in the one-shot game of GSP

and in the cooperative analysis (stability).• It should divide the entire worth of the grand coalition

among all the bidders (efficiency).

There are several solution concepts in cooperative gametheory that one could employ here,for example, the core, theShapley value, the nucleolus, etc. We believe the nucleolusis clearly the best choice that satisfies a majority of theabove properties. Since nucleolus is defined as the uniqueutility vector which makes the unhappiest coalition as lessunhappy as possible [9], and given that the nucleolus isalways in a non-empty core, it is the utility vector thatretains the most number of bidders if the core is emptyand is the most stable one retaining all the bidders if thecore is non-empty. We compute the nucleolus by solving aseries of linear programs [4], [5] and obtain the utility vector(x1, x2, . . . , xn) for then players and the search engine (x0).

C. Mapping the Utility Vector to a Correlated Bid Profile

1) Obtaining a locally envy-free bid profile for eachvalid coalition: To satisfy the stability criterion in the non-cooperative sense, and ensure truthful participation of all thebidders in the proposed bid optimizer, we aim to find outlocally envy-free bids for each of the

(

nk

)

possible sets ofwinning bidders. For finding these bids, consider a subsetof (k + 1) bidders and allocate slots to the bidders inthis subset in the sorted order of their willingness-to-payvalues to satisfy the requirement for the locally envy-freeequilibrium. Now, the bids can be calculated as follows. The(k + 1)th bidder bids the reserve price (assumed to be0here without loss of generality). The bid of thekth bidder(who payssk+1) is now calculated by solving forbk inβk(sk−bk+1) = βk−1(sk−bk) to satisfy the envy-freeness.Once we obtainbk, we proceed recursively by replacing thebk+1 by bk andk by (k − 1) in the above equation to getbk−1 and so on till we get the bids of all thek players. Notethat the bid of the first player does not have a role here aslong as it is greater than the next highest bid. Thus we obtaina set of bids which are in locally envy-free equilibrium.

2) Obtaining a correlated bid profile:The solution givenby the nucleolus provides a utility for each bidder. This can-not be used directly in the GSP auction of the search engine.Towards this end, we map the nucleolus to a correlated bidprofile which defines the required rotation among the biddersfor occupying the slots. This correlated bid profile is whatis finally suggested by the bid optimizer, which retains themaximum number of advertisers without hurting the searchengine.

The characterization of a correlated bid profile corre-sponds to assigning the probabilities associated with eachof the bid profiles associated with the bidders. There existseveral algorithms in general, for finding the correlated bidprofile. But we would like to exploit the structure of theproblem and obtain a simpler solution without going intothe complex details about modifying the ellipsoid algorithmas done in most of the work in this area. See [7] for example.Any correlated strategy we consider here has a subset of sizek bidders bidding their corresponding LEF (locally envy-free) bids (obtained in the previous section) and all otherbidders bidding the reserve price. Considering only these(

nk

)

strategy profiles corresponding to each subset of sizek

bidders winning the slots would suffice since they exhaustall the possible outcomes of the underlying GSP auction.

The probability distribution which yields the utilitiessuggested by the nucleolus to the players is any distributionwhich satisfies the constraints that it is a probability distri-bution, it is individually rational for each player and it mustyield the payoffs suggested by the nucleolus to the bidderssubject to their budget constraints. This can be obtained bysolving a linear program as follows.

min∑

i∈N∪{0}

zi+∑

i∈N

si

xi −

∑

C⊆N, |C|=ki∈C, j=yiC

pCβj(si − bj)

subject to

∀i ∈ N zi ≥

∑

C⊆N, |C|=ki∈C, j=yiC

pCβj(si − bj)

− xi

∀i ∈ N z0 ≥

∑

C⊆N, |C|=k

pC∑

i∈C, j=yiC

βjbj+1

− x0

∀i ∈ N z0 ≥ x0 −

∑

C⊆N, |C|=k

pC∑

i∈C, j=yiC

βjbj+1

∀i ∈ N zi ≥ xi −

∑

C⊆N, |C|=ki∈C, j=yiC

pCβj(si − bj)

pCβj(si − bj) ≥ 0 ∀C ⊆ N ∀i ∈ C ∀j ∈ K∑

C⊂N

pC = 1

∀C ⊂ N pC ≥ 0

whereyiC denotes the slot that playeri wins in a locallyenvy-free allocation if only the setC of players were to winall the slots.

The linear program maps the utility vector suggested bythe nucleolus into a correlated bid profile. The objectivefunction minimizes the difference between the utility sug-gested by nucleolus and the expected utility in the correlatedbid profile for each player. The minimization of differenceleads to two constraints for each player. This is because forany two variablesx andy,

min | x− y |

is the same asmin z

subject to

z ≥ x− y

z ≥ y − x

In the minimization, the higher valuation bidders are givena preference over the lower valuation bidders. This is doneby weighting each player’s difference from the nucleolus inthe objective function by their valuation. This is a heuristicto ensure that the error in the inverse mapping of the utilityvector to a correlated bid profile is biased towards the

higher valuation bidders so that they voluntarily participatein the bid optimizer. Since the only problem to Individualrationality is when the higher valuation bidders shade theirwillingness-to-pay, this weighing gives the incentive forthem to reveal their true valuations.

In the objective function, we minimize thedifference(thisis done by the first4 constraints of the linear program)between the utility vector and the obtained expected utilityin the above linear program since the restriction of the bidprofiles to the set of locally envy-free equilibria may nothave a feasible correlated bid profile. The minimization isdone in such a way that the higher valuation bidders obtainrelatively higher utility (due to the weights given to thedifference in the objective function) than the lower valuationbidders in case the optimal value of the objective functionis non-zero. This is a heuristic to ensure that the error inthe inverse mapping of the utility vector to a correlated bidprofile is biased towards the higher valuation bidders so theyvoluntarily participate in the bid optimizer.

D. Properties of the Proposed Solution

The properties of the proposed solution are as follows:

• The proposed solution has the bidders participating vol-untarily in the bid optimizer for the following reasons.(i) The auctioneer is benefited since he has a guaranteedrevenue of at least what is suggested by the nucleolus.(ii) The high valuation bidders are benefited sincethey are offered the same slots at a relatively lowerprice. Also, since the nucleolus tries to retain the grandcoalition intact, it will be individually rational for thehigh valuation bidders to participate in the bid optimizerrather than to deviate and bid higher. (iii) The lowervaluation bidders are benefited because they get moreslots and hence more clicks and their campaign is moreeffective. Thus, the utility of every player increases andthe individual rationality (IR) condition is satisfied.

• The bids suggested are in a locally envy-free equilib-rium of the game and also are in the core since thenucleolus is always in core if the core is non-empty.This indicates that the proposed solution is strategicallystable. In other words, no one can profitably deviateunilaterally from the solution proposed by the bidoptimizer.

• Retention and efficient division of the worth of thegrand coalition are guaranteed by the nucleolus since itis the allocation which tries to retain the grand coalition.

• Truthfulness is difficult to satisfy, given that the GSPmechanism is non-truthful. But note that the lowervaluation bidders have no incentive to shade their bids.If they do so, they may lose their slots or run intonegative utilities. Hence there is a problem only whenthe higher valuation bidders do not participate in the bidoptimizer or they understate their willingness-to-pay.The higher valuation bidders cannot understate their

valuations by a large amount since they have a threatof losing their slots to lower valuation bidders whoare retained in the system. Also, the higher valuationbidders are given more benefits to participate in thebid optimizer and it is individually rational for them toparticipate in the bid optimizer.

Hence this solution satisfies all the properties which werementioned in Section II-B.

III. E XPERIMENTAL RESULTS

This section presents simulation based experimental re-sults to explore the effectiveness of the approach presentedin the paper. First we start with a model for the drop outsof bidders.

A. Bidder drop out model

The bidders drop out if they do not get enough slots (oralternatively clicks) consistently over a period of time. Theconditional probability that a bidder drops out given that hedid not get a slot in a round may vary from bidder to bidder.Also, the positions of slots occupied by the bidders in theprevious few rounds of auction could play an important rolein the dropping out of a bidder.

To model this behavior of the bidder dropping out basedon the outcomes of the previous auctions and giving moreimportance to recent outcomes, we propose a discountedweighting of the outcomes of the previous auctions tocompute the probability that a bidder will continue in thenext round of the auction. This model also captures themyopic human behavior that the bidder’s choice is dependenton only the recent outcomes. That is, the history the bidderlooks into, before taking a decision to continue or not forthe next round is limited. The amount of history howeverdepends on the bidder in the form of his discounting factor.Let x−i ∈ {0, 1} denote the outcome of theith previousround. A “1“ denotes that the bidder received a click in theith previous auction with a zero indicating otherwise. Wepropose that the probability that the bidder will participate inthe next round is given by

P

∞

i=1γix−i

P

∞

i=1γi = (1−γ)

∑∞i=1 γ

ix−i

whereγ is the discount factor of the bidder. To see how themyopic nature of the bidders is captured, suppose that thebidder’s discount factorγ = 0.95. The discount factor forthe 101st round will be 0.006 which is negligible. Hencethe bidder’s decision is dependent on at most100 previousauctions. Thus, the discount factor decides the nature of thebidder.

B. Experimental Setup

Given a fixed set of CTRs, the valuations of the biddersare chosen close enough to each other, to analyse ourmodel in competitive environment. The retention problem,is fundamental in competitive environment, as search engineneeds to retain the bidders and allow them compete in furtherauctions. The results indicate that the proposed bid optimizer

not only retains a higher number of bidders than the normalGSP but also yields better cumulative revenue to the searchengine in the long run.

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700

800

900

1000

Rounds of Auction

Searc

h E

ngin

e’s

Reve

nue

GSPOur Approach

Figure 2. Cumulative revenue of the search engine

C. Cumulative Revenue of the Search Engine

First we consider the cumulative revenue of the searchengine for comparing the non-cooperative bidding and usingcooperative bid optimizer. We consider10 advertisers with5slots to be allocated.We run successive auctions and find thecumulative revenue of the search engine after each auctionusing the two approaches. We run this experiment until thechange in the average cumulative revenue after each auctionbecomes acceptably small. Figure 2 shows a comparisonof the cumulative revenue of the search engine under theproposed approach with that of a standard GSP auction.

It can be seen in Figure 2 that though initially the non-cooperative approach (GSP) yields more revenue, after a fewruns, the cooperative bid optimizer, with all the solutionvectors, starts outperforming. Initially the GSP outcomeis better, as the advertisers are bidding their maximumwillingness to pay, and hence the search engine gets highlevels of revenue. However, as the utilities of the advertisersare less in the case of the non-cooperative approach, theystart dropping out of the auction and hence in a long run,the cumulative revenue starts declining compared to thecooperative bid optimizer. This is the adverse effect ofthe dropping out of the advertisers leading to the retentionproblem.

D. Number of Bidders Retained in the System

After each auction, we compute the average number ofadvertisers retained, to analyze the retention dynamics inthe system. Figure 3 gives a comparison between using thecooperative bid optimizer and the GSP approach.

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

Rounds of Auction

Nu

mb

er

of

Bid

de

rs R

eta

ine

d

Our ApproachGSP

Figure 3. Number of bidders retained in the system

In Figure 3, it can be observed that there are considerablymore number of advertisers retained in the system whenusing the solution vector suggested by the cooperative bidoptimizer in comparison to the GSP approach. This explainsthe reason for the dip in the non-cooperative cumulativerevenue of the search engine.

Through experimentation, we are able to demonstraterevenue increase for the search engine in the long runand also reduction of the retention problem considerably. Itshould be noted that even though the raise is not substantial,its impact on retaining and attracting the advertisers andthereby other indirect advantages are immense.

IV. SUMMARY AND FUTURE WORK

We have proposed a bid optimizer for sponsored keywordsearch auctions which leads to better retention of advertisersby yielding attractive utilities to the advertisers without de-creasing the long-run revenue to the search engine. The bidoptimizer is a value added tool the search engine provides tothe advertisers which transforms the reported values of theadvertisers for a keyword into a correlated bid profile. Thecorrelated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of theunderlying game. Through detailed simulation experiments,we have shown that the proposed bid optimizer retainsmore customers than a plain GSP mechanism and alsoyields better long-run utilities to the search engine and theadvertisers.

The experiments were carried out with a model thatcaptures the phenomenon of customer drop outs and showedthat our approach produces a better long run utility to thesearch engine and all the advertisers. The proposed bid

optimizer is beneficial for both the bidders and the searchengine in the long run.

We considered GSP auction, which is popularly run inmost of the search engines, in our analysis. However, itwould be interesting to look at the effects of other auctionmechanisms like VCG auctions on the overall process.

The other important components like budget optimizationand ad scheduling are involved in sponsored search auctions.We would further like to combine these components with ourcooperative bid optimizer.

REFERENCES

[1] M Cary, A Das, B Edelman and I Giotis, K Heimerl, A RKarlin, C Mathieu, and M Schwarz. Greedy bidding strategiesfor keyword auctions. InProceedings of the Eighth ACMConference on Electronic Commerce, pages 57–58, 2007.

[2] B Edelman and M Ostrovsky. Strategic bidder behaviorin sponsored search auctions.Decision Support Systems,43(1):192–198, 2007.

[3] J Feng and X Zhang. Dynamic price competition on theInternet: advertising auctions. InProceedings of the EighthACM Conference on Electronic Commerce, pages 57–58,2007.

[4] A Mas-Colell, M D Whinston, and J R Green.MicroeconomicTheory. Oxford University Press, Oxford, 1995.

[5] R B Myerson. Game Theory: Analysis of Conflict. HarvardUniversity Press, Cambridge, Massachusetts, 1997.

[6] N Nisan, T Roughgarden, E Tardos and V V Vazirani.Algorithmic Game Theory. Cambridge University Press,2007.

[7] C H Papadimitriou. Computing correlated equilibria in multi-player games. InProceedings of the thirty-seventh annualACM symposium on Theory of computing, Baltimore, MD,USA, May 22-24 2005.

[8] K Ramakrishnan, D Garg, K Subbian, and Y Narahari. ANash bargaining approach to retention enhancing bid opti-mization in sponsored search auctions with discrete bids. InFourth Annual IEEE Conference on Automation Science andEngineering (IEEE CASE), Arizona, USA, 2008.

[9] P D Straffin. Game Theory and Strategy. MathematicalAssociation of America, New York, 1993.

[10] Y Vorobeychik and D M Reeves. Equilibrium analysis of dy-namic bidding in sponsored search auctions. InProceedingsof Workshop on Internet and Network Economics (WINE),2007.

[11] Y Narahari, D Garg, R Narayanam, H Prakash.GameTheoretic Problems in Network Economics and MechanismDesign Solutions. Springer Series in Advanced Informationand Knowledge Processing, 2009.

[12] J F Nash Jr.The bargaining problem. Econometrica, 18:155–162, 1950.

[13] B Edelman, M Ostrovsky, and M Schwarz.Internet advertis-ing and the generalized second price auction: Selling billonsof dollars worth of keywords. American Economic Review,97(1):242–259, 2007.

APPENDIX

Nash [12] proposed that there exists a unique solutionfunction f(F, v) for every two person bargaining problem,that satisfies the following 5 axioms -Pareto strong efficient,Individual Rationality, Symmetry, Scale Covariance, and In-dependence of Irrelevant Alternatives. The solution functionis

f(F, v) ∈ argmax(x1,x2)∈F ((x1 − v1)(x2 − v2))

where,x1 ≥ v1 and x2 ≥ v2 and the pointv = (v1, v2)is known as thepoint of disagreement. There are severalpossibilities for choosing the disagreement pointv. The threepopular choices are those based on (1) a minimax criterion,(2) focal equilibrium, and (3) rational threats. As part of thispaper, we use the rational threats to identify disagreementpoint v [5]. For more details please refer to the books [5][9].

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A Novel Bid Optimizer for Sponsored Search Auctions based onCooperative Game Theory

Sriram SomanchiElectronic Commerce Laboratory

Computer Science and AutomationIndian Institute Science, BangaloreEmail: somanchi@csa.iisc.ernet.in

Chaitanya NittalaElectronic Commerce Laboratory

Computer Science and AutomationIndian Institute Science, Bangalore

Email: chytu@csa.iisc.ernet.in

Narahari YadatiElectronic Commerce Laboratory

Computer Science and AutomationIndian Institute Science, Bangalore

Email: hari@csa.iisc.ernet.in

Abstract—In this paper, we propose a bid optimizer forsponsored keyword search auctions which leads to betterretention of advertisers by yielding attractive utilities to theadvertisers without decreasing the revenue to the search engine.The bid optimizer is positioned as a key value added toolthe search engine provides to the advertisers. The proposedbid optimizer algorithm transforms the reported values of theadvertisers for a keyword into a correlated bid profile usingmany ideas from cooperative game theory. The algorithm isbased on a characteristic form game involving the search engineand the advertisers. Ideas from Nash bargaining theory areused in formulating the characteristic form game to providefor a fair share of surplus among the players involved. Thealgorithm then computes the nucleolus of the characteristicform game since we find that the nucleolus is an apt way ofallocating the gains of cooperation among the search engineandthe advertisers. The algorithm next transforms the nucleolusinto a correlated bid profile using a linear programmingformulation. This bid profile is input to a standard generalizedsecond price mechanism (GSP) for determining the allocationof sponsored slots and the prices to be be paid by the winners.The correlated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of theunderlying game. Through detailed simulation experiments, weshow that the proposed bid optimizer retains more customersthan a plain GSP mechanism and also yields better long-runutilities to the search engine and the advertisers.

Keywords-Bid Optimizer, Sponsored Search, CooperativeGame Theory, Nash bargaining, Nucleolus.

I. I NTRODUCTION

Sponsored search auctions have been studied extensivelyin the recent years due to the advent of targeted advertisingand its role in generating large revenues. With a huge com-petition in providing the sponsored search links, the searchengines face an imminent problem which can be called as theretention problem. If an advertiser (or alternatively bidder)does not get satisfied because of not getting the right numberof clicks or the anticipated payoff, he could drop out of theauction and try sponsored links at a different search engine.

A. Motivation: Retention of Advertisers in Sponsored SearchAuctions

Our motivation to study the retention problem is drivenby the compulsions faced by both the search engine and theadvertisers.

From the advertisers’ perspective, choosing theirmaximum-willingness-to-pay such that they get an attractiveslot subject to their budget constraints is a challengingproblem. The search engines can use various mechanismsfor the sponsored search auction as described in [6], [11]but the most popular mechanism is the generalized secondprice (GSP) auction since it is simple and yields betterrevenue to the search engine. In the most simple versionof GSP, where there arem slots andn advertisers (forsimplicity assumem ≤ n), the allocation and payment ruleare as follows. The allocation rule is thatn advertisers areranked in descending order based on their bids, with tiesbroken appropriately, and topm advertisers’ advertisementsare displayed. The payment rule is that every advertiserneeds to pay bid amount of the advertiser who is justbelow his slot and last advertiser is charged the highestbid that has not won any slot. If the non-truthful GSPauction is used by the search engine, the bidders will havean incentive to shade their bids. The bidders will not havetime to adjust the bid value after the search query hasbeen initiated. These advertisers generally build their ownsoftware agents or employ third party software agents,which adjust and readjust the bid values on behalf of theseadvertisers. The bidders typically specify their maximumwillingness-to-pay for their keywords for any given day.Hence each keyword has a specific set of bidders biddingon it for the whole day. This scenario constitutes a repeatedgame between all the bidders bidding for that keyword. Inthis game, bidders who cannot plan their budget effectivelymay experience less utilities and thus may drop out of thesystem.

We now turn to the search engine’s perspective of theretention problem. When the bidders try to know eachothers’ valuations by submitting and resubmitting bids, they

may find a set of strategy profiles which may yield allof them better payoffs. This may lead to collusion amongthe bidders. Folk theorems [4] suggest that players maybe able to increase individual profits by colluding therebydecreasing the search engine’s revenue. Even though thebidders in the keyword auctions are competitors, this collu-sion against the search engine could be stable. Vorobeychikand Reeves [10] studied this phenomenon and illustrateda particular collusive strategy which is better for all thebidders (hence worst for the search engine) and can besustainable over a range of settings. Feng and Zhang [3]showed that dynamic price competition between competingadvertisers can lead to collusion among them. However,in this dynamic scenario, when the discounted payoffs ofthe bidders under the collusive strategy are considered, thestability of collusion depends inversely on the number ofbidders [4]. That is, the lower the number of bidders inthe system, the higher is the stability of the collusion. Thismotivates us to study the bidder retention problem for thesearch engine.

Also, due to exponential growth in the space of onlineadvertising and intense competition among the search enginecompanies, the switching cost for the advertisers to changefrom one search engine to another is almost zero [6]. Hence,it is imperative for the search engine companies to retaintheir advertisers to safeguard their market share. Driven bythis, the search engine companies have introduced manyvalue added tools, such as bid optimizer, to maximize thebang-per-buck for the bidders. In what follows, we describethe bid optimizer’s role in solving the retention problem.

B. Bid Optimizers

A bid optimizeris a software agent provided by the searchengine in order to assist the advertisers. The bidders arerequired to provide to the bid optimizer a target budget forthe day and a maximum willingness-to-pay. Bid optimizers,currently provided by the search engines, promise to maxi-mize the revenue of advertisers by adjusting the bid amountin each round of the auction based on the projected keywordtraffic and remaining budget.

It can be seen that the decisions made by the bidoptimizer are crucial to both the search engine and theset of advertisers, who choose to use the bid optimizer.Hence, the objective of a typical bid optimizer is to strike abalance between reduction in revenue of the search enginecompany versus increase the retention of advertisers. Suchan objective is achieved by providing enhanced utilitiesto the advertisers, thus ensuring retention of customers,thereby sustaining high levels of revenue to the search enginecompany in the long run. Designing such intelligent bidoptimizers is the subject of this paper.

There are some problems involved in designing bid opti-mizers.

1) For the search engine, maximizing its short-term rev-enue (that is, its payoff in a one-shot game) seemsto be a viable option. But here, the lower valuationbidders are denied slots due to allocative efficiencyconcerns. For the bidders, as shown by Caryet al [1],where all the high valuation bidders use a particulargreedy strategy, it has been proved that none of thebidders except the topk bidders get the slots after acertain number of rounds of the auction. The abovephenomenon can permanently drive away low valua-tion bidders from the search engine.

2) Dropping out of the search engine to get better utilitiesin another search engine is a possible option for thebidders. The low valuation bidders drop out afternot getting slots for a certain period of time. Thehigher valuation bidders can observe this trend andshade their maximum-willingness-to-pay or colludeto get better utilities. This may result in the searchengine losing revenue. This is a threat to the searchengine from the bidders. However, if a large numberof bidders remain in the system, the collusion is notstable. The intuition for this is that, high valuationbidders cannot reduce their bids sharply, since theywill have the fear of undercutting the lower valuationbidders present in the system and thus losing out ontheir slots.

Hence, we propose that retaining more number of bidderssolves all the problems discussed above. The dependence ofthe search engine and the bidders on each other for mutualbenefit motivates us to use a cooperative approach in general.The above threat model naturally directs us towards usinga Nash bargaining model in particular. Our solution can beseen as associating the bid optimizer to a keyword ratherthan bidders as done by the existing bid optimizers. Theoverall model of the bid optimizer is depicted in Figure 1.

C. Contributions and Outline of the Paper

In this paper, we propose a bid optimizer that uses manyideas from cooperative game theory. The bid optimizer isshown in Figure 1.

• The inputs to the bid optimizer are the willingness-to-pay values (or valuations) of the bidders.

• The output of the bid optimizer is a correlated bidprofile, which, when input to a standard GSP auctionmechanism, yields utilities to the search engine and theadvertisers satisfying the goals set forth in the paper.

• The bid optimizer first formulates a characteristic formgame involving the search engine and the advertisers.The value for each coalition is defined based on anovel Nash Bargaining formulation with the searchengine as one player and a virtual player aggregatingall advertisers in that coalition as the other player. Theidea of using Nash Bargaining is to ensure a fair sharefor the search engine and the advertisers.

Figure 1. Proposed Bid Optimizer

• The nucleolus of the above characteristic form game isselected as the utility profile for the search engine andthe advertisers. The choice of nucleolus is based on keyconsiderations such as, bidder retention, stability, andefficiency.

• The utility profile represented by the nucleolus ismapped to a correlated bid profile that satisfies indi-vidual rationality, retention, stability and efficiency. Alinear programming based algorithm is suggested forthis purpose.

We carry out experiments to demonstrate the viabilityand efficacy of the proposed bid optimizer. We show, usinga credible bidder drop out model, that the proposed bidoptimizer has excellent bidder retention properties and alsoyields higher long-run revenues to the search engine, whencompared to the plain GSP mechanism.

The outline of the paper is as follows. Section II-A1presents the details of the bid optimizer and introducesthe model. In Section II-A3, we present a bid optimizationalgorithm which uses the Nash Bargaining approach forensuring the retention of bidders in the system. We thenmap this fair share for the aggregated bidder to a correlatedbid profile in Section II-C. We analyze the properties of ourmethod in Section II-D. We present our experimental resultsin Section III and conclude the paper in Section IV.

II. OUR APPROACH TOBID OPTIMIZATION

In this section, we present our algorithm for bid opti-mization. The algorithm can be divided into three phases asshown in Figure 1: (1) Characteristic form game definitionusing Nash bargaining, (2) Computing the utility vectors forthe players and (3) Inverse mapping of the utility vectorinto a correlated bid profile. These are discussed in thefollowing sections. The notation in the remainder of thepaper is presented in Table I.

A. Characteristic Form Game

1) The Model:The sponsored search auction scenario weconsider hasn bidders competing fork slots of a keyword.We assume that the probability that a bidderi gets clicked onthejth slot (or the click-through rateCTRij) is independentof the bidderi,that is,CTRij = βj and we also assume thatβ1 ≥ β2 ≥ . . . ≥ βk. Each bidderi specifies his maximumwillingness-to-paysi to the bid optimizer. The bid optimizertakes as input all thesi’s of the bidders and suggests them acorrelated bid profile. This bid optimization algorithm needsto be invoked only when the number of bidders in the systemor their willingness-to-pay change. We also assume that thebid of the playeri could be any real number in[0, si].

Given the above model, we define a bargaining problembetween the search engine and the aggregated bidder andanalyze its properties which will help us in formulating acharacteristic form game.

Notation ExplanationA AuctioneerB Aggregated biddern Total number of playersm Total number of slots. We assumem < n

N Set of bidders{1, 2, . . . , n}K Set of slots{1, 2, . . . , k}si Maximum willingness-to-pay of advertiseriSi Strategy set of bidderi, [0, si]S Set of all bid profilesS1 × S2 × . . .× Sn

s Bid profile (s1, s2, . . . , sn) ∈ S

ui(s) Utility of bidder i on bid profilesUA(s) Utility of the auctioneer in the

Nash bargaining formulation for bid profilesUB(s) Utility of the aggregated bidder in the

Nash bargaining formulation for bid profilesUA maxs∈S UA(s)

UB maxs∈S UB(s)

βj Click through rate of any bidder in thejth slotTable I

NOTATION

2) Characterization of the Nash Bargaining Solution:The motivation for a cooperative approach is the dependenceof the search engine and bidders on each other for theirmutual benefit. Given this, the motivation behind choosinga bargaining approach is that the amount of short-term loss(or in other words, the investment of the search engine) forthe auctioneer should be chosen based on the bidders presentin the system. The Nash bargaining approach provides aframework for this amount to be chosen by the search engineby considering all the bidders as one aggregate agent whosebargaining power depends on all the maximum willingness-to-pay of all the bidders present in the system.

The utility of the aggregated bidder is the sum of theutilities of all the bidders over all possible allocations of slots(outcomes). Now, the bargaining utility space becomes thetwo dimensional Cartesian space which consists of the utilityof auctioneer on one axis and the aggregate bidder’s utilityon the other axis. Hence a bargaining solution on this spaceprovides a good compromise for the search engine fromits maximum possible revenue and thus gives the requiredinvestment of the search engine.

The bargaining space is defined in two dimensional Carte-sian space, with utility of auctioneerUA(s) along thex−axisand the utility of aggregated bidderUB(s) =

∑ni=1 ui(s)

along they−axis. LetUA andUB be the maximum pos-sible utilities of the auctioneer and the aggregated bidderrespectively. It can be clearly seen that the valueUA isattained for the bid profiles = (s1, s2, . . . , sn) for which the

correspondingUB =∑n

i=1

(

∑mj=1 βjyij(s)

)

(si − si+1) =

U′

B (say). Similarly, the bid profiles = (0, . . . , 0) yieldsthe utility pair (0, UB). Since it is theoretically possiblethat all the bidders can collude and bid(0, 0, . . . , 0), wechoose the point(0, 0) in this Nash bargaining space asthe disagreement point. Ramakrishnanet. al studied this

problem in [8] and characterized the solution(U∗A, U

∗B) to

this Nash bargaining(NBS) as

(U∗A, U

∗B) = (UA, U

′

B) if UA ≤UB

2

=

(

UB

2,UB

2

)

otherwise

3) Definition of the Characteristic Form Game:Weuse the above model to define Nash bargaining solutionNBS(N) = U∗

A + U∗B where N is the set of bidders

participating in the auction.Let N = {1, 2, . . . , n} be the set of all bidders and let

0 represent the search engine. The characteristic form gameν : 2N∪{0} → ℜ for each coalitionC ⊆ N ∪ {0} is nowdefined as

ν(C) = NBS(C) if 0 ∈ C

= 0 otherwise

whereNBS(C) is defined as above. If the search engine isnot a part of the coalition, its worth is zero since the playerscannot gain anything without the search engine displayingtheir ads. Otherwise, we associate the sum of utilities in thecorresponding Nash bargaining bid profile for that coalitionwith the search engine as the worth of each coalition. Thischaracteristic functionν defines the bargaining power ofeach coalition with the search engine.

B. Computing a Utility Vector for the Players: Use ofNucleolus

Since there is an aggregation of the bidders’ revenuetaking place in the NBS, we map the utility of the aggregatedbidder in the Nash bargaining solution to a correlated bidprofile. The NBS gives an aggregate amount of investmentthe search engine has to make on all the bidders. Thisinvestment increases the utility of the aggregated bidder.This utility has to be distributed to the bidders in a waythat our goal of retention is reached. Ideally, we would likethe allocation to have the following properties.

• The bidders must not have incentive for not participat-ing in the bid optimizer (individual rationality-IR).

• It must retain as many bidders as possible(retention).• The bidders must not have the incentive to shade their

maximum willingness to pay (incentive compatibility).• It should be stable both in the one-shot game of GSP

and in the cooperative analysis (stability).• It should divide the entire worth of the grand coalition

among all the bidders (efficiency).

There are several solution concepts in cooperative gametheory that one could employ here, for example, the core, theShapley value, the nucleolus, etc. We believe the nucleolusis clearly the best choice that satisfies a majority of theabove properties. Since nucleolus is defined as the uniqueutility vector which makes the unhappiest coalition as less

unhappy as possible [9], and given that the nucleolus isalways in a non-empty core, it is the utility vector thatretains the most number of bidders if the core is emptyand is the most stable one retaining all the bidders if thecore is non-empty. We compute the nucleolus by solving aseries of linear programs [4], [5] and obtain the utility vector(x1, x2, . . . , xn) for then players and the search engine (x0).

C. Mapping the Utility Vector to a Correlated Bid Profile

1) Obtaining a locally envy-free bid profile for eachvalid coalition: To satisfy the stability criterion in the non-cooperative sense, and ensure truthful participation of all thebidders in the proposed bid optimizer, we aim to find outlocally envy-free bids for each of the

(

nk

)

possible sets ofwinning bidders. For finding these bids, consider a subsetof (k + 1) bidders and allocate slots to the bidders inthis subset in the sorted order of their willingness-to-payvalues to satisfy the requirement for the locally envy-freeequilibrium. Now, the bids can be calculated as follows. The(k + 1)th bidder bids the reserve price (assumed to be0here without loss of generality). The bid of thekth bidder(who payssk+1) is now calculated by solving forbk inβk(sk−bk+1) = βk−1(sk−bk) to satisfy the envy-freeness.Once we obtainbk, we proceed recursively by replacing thebk+1 by bk andk by (k − 1) in the above equation to getbk−1 and so on till we get the bids of all thek players. Notethat the bid of the first player does not have a role here aslong as it is greater than the next highest bid. Thus we obtaina set of bids which are in locally envy-free equilibrium.

2) Obtaining a correlated bid profile:The solution givenby the nucleolus provides a utility for each bidder. This can-not be used directly in the GSP auction of the search engine.Towards this end, we map the nucleolus to a correlated bidprofile which defines the required rotation among the biddersfor occupying the slots.

The characterization of a correlated bid profile corre-sponds to assigning the probabilities associated with eachof the bid profiles associated with the bidders. There existseveral algorithms in general, for finding the correlated bidprofile. But we would like to exploit the structure of theproblem and obtain a simpler solution without going intothe complex details about modifying the ellipsoid algorithmas done in most of the work in this area. See [7] for example.Any correlated strategy we consider here has a subset of sizek bidders bidding their corresponding LEF (locally envy-free) bids (obtained in the previous section) and all otherbidders bidding the reserve price. Considering only these(

nk

)

strategy profiles corresponding to each subset of sizek

bidders winning the slots would suffice since they exhaustall the possible outcomes of the underlying GSP auction.

The probability distribution which yields the utilitiessuggested by the nucleolus to the players is any distributionwhich satisfies the constraints that it is a probability distri-bution, it is individually rational for each player and it must

yield the payoffs suggested by the nucleolus to the bidderssubject to their budget constraints. This can be obtained bysolving a linear program as follows.

min z0+∑

i∈N

sizi

subject to

∀i ∈ N zi ≥

∑

C⊆N, |C|=ki∈C, j=yiC

pCβj(si − bj)

− xi

∀i ∈ N z0 ≥

∑

C⊆N, |C|=k

pC∑

i∈C, j=yiC

βjbj+1

− x0

∀i ∈ N z0 ≥ x0 −

∑

C⊆N, |C|=k

pC∑

i∈C, j=yiC

βjbj+1

∀i ∈ N zi ≥ xi −

∑

C⊆N, |C|=ki∈C, j=yiC

pCβj(si − bj)

pCβj(si − bj) ≥ 0 ∀C ⊆ N ∀i ∈ C ∀j ∈ K∑

C⊂N

pC = 1

∀C ⊂ N pC ≥ 0

whereyiC denotes the slot that playeri wins in a locallyenvy-free allocation if only the setC of players were to winall the slots.

The linear program maps the utility vector suggested bythe nucleolus into a correlated bid profile. The objectivefunction minimizes the difference between the utility sug-gested by nucleolus and the expected utility in the correlatedbid profile for each player. The minimization of differenceleads to two constraints for each player. This is because forany two variablesx andy,

min | x− y |

is the same asmin z

subject to

z ≥ x− y

z ≥ y − x

In the minimization, the higher valuation bidders are givena preference over the lower valuation bidders. This is doneby weighting each player’s difference from the nucleolus inthe objective function by their valuation. This is a heuristicto ensure that the error in the inverse mapping of the utility

vector to a correlated bid profile is biased towards thehigher valuation bidders so that they voluntarily participatein the bid optimizer. Since the only problem to Individualrationality is when the higher valuation bidders shade theirwillingness-to-pay, this weighing gives the incentive forthem to reveal their true valuations.

In the objective function, we minimize thedifference(thisis done by the first4 constraints of the linear program)between the utility vector and the obtained expected utilityin the above linear program since the restriction of the bidprofiles to the set of locally envy-free equilibria may nothave a feasible correlated bid profile. The minimization isdone in such a way that the higher valuation bidders obtainrelatively higher utility (due to the weights given to thedifference in the objective function) than the lower valuationbidders in case the optimal value of the objective functionis non-zero. This is a heuristic to ensure that the error inthe inverse mapping of the utility vector to a correlated bidprofile is biased towards the higher valuation bidders so theyvoluntarily participate in the bid optimizer.

D. Properties of the Proposed Solution

The properties of the proposed solution are as follows:

• The proposed solution has the bidders participating vol-untarily in the bid optimizer for the following reasons.(i) The auctioneer is benefited since he has a guaranteedrevenue of at least what is suggested by the nucleolus.(ii) The high valuation bidders are benefited sincethey are offered the same slots at a relatively lowerprice. Also, since the nucleolus tries to retain the grandcoalition intact, it will be individually rational for thehigh valuation bidders to participate in the bid optimizerrather than to deviate and bid higher. (iii) The lowervaluation bidders are benefited because they get moreslots and hence more clicks and their campaign is moreeffective. Thus, the utility of every player increases andthe individual rationality (IR) condition is satisfied.

• The bids suggested are in a locally envy-free equilib-rium of the game and also are in the core since thenucleolus is always in core if the core is non-empty.This indicates that the proposed solution is strategicallystable. In other words, no one can profitably deviateunilaterally from the solution proposed by the bidoptimizer.

• Retention and efficient division of the worth of thegrand coalition are guaranteed by the nucleolus since itis the allocation which tries to retain the grand coalition.

• Truthfulness is difficult to satisfy, given that the GSPmechanism is non-truthful. But note that the lowervaluation bidders have no incentive to shade their bids.If they do so, they may lose their slots or run intonegative utilities. Hence there is a problem only whenthe higher valuation bidders do not participate in the bidoptimizer or they understate their willingness-to-pay.

The higher valuation bidders cannot understate theirvaluations by a large amount since they have a threatof losing their slots to lower valuation bidders whoare retained in the system. Also, the higher valuationbidders are given more benefits to participate in thebid optimizer and it is individually rational for them toparticipate in the bid optimizer.

Hence this solution satisfies all the properties which werementioned in Section II-B.

III. E XPERIMENTAL RESULTS

This section presents simulation based experimental re-sults to explore the effectiveness of the approach presentedin the paper. First we start with a model for the drop outsof bidders.

A. Bidder drop out model

The bidders drop out if they do not get enough slots (oralternatively clicks) consistently over a period of time. Theconditional probability that a bidder drops out given that hedid not get a slot in a round may vary from bidder to bidder.Also, the positions of slots occupied by the bidders in theprevious few rounds of auction could play an important rolein the dropping out of a bidder.

To model this behavior of the bidder dropping out basedon the outcomes of the previous auctions and giving moreimportance to recent outcomes, we propose a discountedweighting of the outcomes of the previous auctions tocompute the probability that a bidder will continue in thenext round of the auction. This model also captures themyopic human behavior that the bidder’s choice is dependenton only the recent outcomes. That is, the history the bidderlooks into, before taking a decision to continue or not forthe next round is limited. The amount of history howeverdepends on the bidder in the form of his discounting factor.Let x−i ∈ {0, 1} denote the outcome of theith previousround. A “1“ denotes that the bidder received a click in theith previous auction with a zero indicating otherwise. Wepropose that the probability that the bidder will participate in

the next round is given byP

∞

i=1γix−i

P

∞

i=1γi = (1−γ)

∑∞i=1 γ

ix−i

whereγ is the discount factor of the bidder. To see how themyopic nature of the bidders is captured, suppose that thebidder’s discount factorγ = 0.95. The discount factor forthe 101st round will be 0.006 which is negligible. Hencethe bidder’s decision is dependent on at most100 previousauctions. Thus, the discount factor decides the nature of thebidder.

B. Experimental Setup

Given a fixed set of CTRs, the valuations of the biddersare chosen close enough to each other, to analyse ourmodel in competitive environment. The retention problem,is fundamental in competitive environment, as search engineneeds to retain the bidders and allow them compete in further

auctions. The results indicate that the proposed bid optimizernot only retains a higher number of bidders than the normalGSP but also yields better cumulative revenue to the searchengine in the long run.

0 10 20 30 40 50 60 70 80 90 1000

100

200

300

400

500

600

700

800

900

1000

Rounds of Auction

Searc

h E

ngin

e’s

Reve

nue

GSPOur Approach

Figure 2. Cumulative revenue of the search engine

C. Cumulative Revenue of the Search Engine

First we consider the cumulative revenue of the searchengine for comparing the non-cooperative bidding and usingcooperative bid optimizer. We consider10 advertisers with5slots to be allocated.We run successive auctions and find thecumulative revenue of the search engine after each auctionusing the two approaches. We run this experiment until thechange in the average cumulative revenue after each auctionbecomes acceptably small. Figure 2 shows a comparisonof the cumulative revenue of the search engine under theproposed approach with that of a standard GSP auction.

It can be seen in Figure 2 that though initially the non-cooperative approach (GSP) yields more revenue, after a fewruns, the cooperative bid optimizer, with all the solutionvectors, starts outperforming. Initially the GSP outcomeis better, as the advertisers are bidding their maximumwillingness to pay, and hence the search engine gets highlevels of revenue. However, as the utilities of the advertisersare less in the case of the non-cooperative approach, theystart dropping out of the auction and hence in a long run,the cumulative revenue starts declining compared to thecooperative bid optimizer. This is the adverse effect ofthe dropping out of the advertisers leading to the retentionproblem.

D. Number of Bidders Retained in the System

After each auction, we compute the average number ofadvertisers retained, to analyze the retention dynamics in

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

Rounds of Auction

Nu

mb

er

of

Bid

de

rs R

eta

ine

d

Our ApproachGSP

Figure 3. Number of bidders retained in the system

the system. Figure 3 gives a comparison between using thecooperative bid optimizer and the GSP approach.

In Figure 3, it can be observed that there are considerablymore number of advertisers retained in the system whenusing the solution vector suggested by the cooperative bidoptimizer in comparison to the GSP approach. This explainsthe reason for the dip in the non-cooperative cumulativerevenue of the search engine.

Through experimentation, we are able to demonstraterevenue increase for the search engine in the long runand also reduction of the retention problem considerably. Itshould be noted that even though the raise is not substantial,its impact on retaining and attracting the advertisers andthereby other indirect advantages are immense.

IV. SUMMARY AND FUTURE WORK

We have proposed a bid optimizer for sponsored keywordsearch auctions which leads to better retention of advertisersby yielding attractive utilities to the advertisers without de-creasing the long-run revenue to the search engine. The bidoptimizer is a value added tool the search engine provides tothe advertisers which transforms the reported values of theadvertisers for a keyword into a correlated bid profile. Thecorrelated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of theunderlying game. Through detailed simulation experiments,we have shown that the proposed bid optimizer retainsmore customers than a plain GSP mechanism and alsoyields better long-run utilities to the search engine and theadvertisers.

The experiments were carried out with a model thatcaptures the phenomenon of customer drop outs and showed

that our approach produces a better long run utility to thesearch engine and all the advertisers. The proposed bidoptimizer is beneficial for both the bidders and the searchengine in the long run.

We considered GSP auction, which is popularly run inmost of the search engines, in our analysis. However, itwould be interesting to look at the effects of other auctionmechanisms like VCG auctions on the overall process.

The other important components like budget optimizationand ad scheduling are involved in sponsored search auctions.We would further like to combine these components with ourcooperative bid optimizer.

REFERENCES

[1] M Cary, A Das, B Edelman and I Giotis, K Heimerl, A RKarlin, C Mathieu, and M Schwarz. Greedy bidding strategiesfor keyword auctions. InProceedings of the Eighth ACMConference on Electronic Commerce, pages 57–58, 2007.

[2] B Edelman and M Ostrovsky. Strategic bidder behaviorin sponsored search auctions.Decision Support Systems,43(1):192–198, 2007.

[3] J Feng and X Zhang. Dynamic price competition on theInternet: advertising auctions. InProceedings of the EighthACM Conference on Electronic Commerce, pages 57–58,2007.

[4] A Mas-Colell, M D Whinston, and J R Green.MicroeconomicTheory. Oxford University Press, Oxford, 1995.

[5] R B Myerson. Game Theory: Analysis of Conflict. HarvardUniversity Press, Cambridge, Massachusetts, 1997.

[6] N Nisan, T Roughgarden, E Tardos and V V Vazirani.Algorithmic Game Theory. Cambridge University Press,2007.

[7] C H Papadimitriou. Computing correlated equilibria in multi-player games. InProceedings of the thirty-seventh annualACM symposium on Theory of computing, Baltimore, MD,USA, May 22-24 2005.

[8] K Ramakrishnan, D Garg, K Subbian, and Y Narahari. ANash bargaining approach to retention enhancing bid opti-mization in sponsored search auctions with discrete bids. InFourth Annual IEEE Conference on Automation Science andEngineering (IEEE CASE), Arizona, USA, 2008.

[9] P D Straffin. Game Theory and Strategy. MathematicalAssociation of America, New York, 1993.

[10] Y Vorobeychik and D M Reeves. Equilibrium analysis of dy-namic bidding in sponsored search auctions. InProceedingsof Workshop on Internet and Network Economics (WINE),2007.

[11] Y Narahari, D Garg, R Narayanam, H Prakash.GameTheoretic Problems in Network Economics and MechanismDesign Solutions. Springer Series in Advanced Informationand Knowledge Processing, 2009.