Choosing Wisely: A Multibidding Approach

* Perez-CastrUniversitat Autoncelona), Spain;Gurion UniversEconomic ReseHerve Moulin fotuswami who enlibria, and FrancJordi Masso´, andcomments. Pe´rezcial support from

Choosing Wisely: A Multibidding Approach

By DAVID PEREZ-CASTRILLO AND DAVID WETTSTEIN*

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Reaching decisions about the location of noious facilities, such as dump sites, environmtally hazardous plants, nuclear power generaand the like, is a highly contentious issue. Finstance, in February 2000, the U.S. Sendecided that nationwide nuclear waste wouldshipped to the Yucca mountain site in Neva(conditional to it being approved as a high-levnuclear waste repository). Despite the attractcompensation package, the State of Nevvoiced vehement opposition. President Clintvetoed the bill, and in May 2000 the Senafailed to overturn it. As such, the major problefacing the U.S. nuclear industry on where to sa high-level radioactive waste repository rmains virtually unresolved.

Neither are decisions about the locationdesirable events easily solvable. Every tyears, the International Olympic Committee slects hosts for the summer and winter gameprocedure that is constantly under review arevision in order to bring about the “best posible” outcome. Until about a year ago, tdecision was made by a vote based on bidsplans submitted by candidate cities. Howevthe recent controversy surrounding the choiceSalt Lake City to host the 2002 winter gamculminated in the design of a new procedurechoosing the host of the 2006 winter gamThis procedure called for the selection of twfinalist cities (in this case, Turin and Sionimmediately after which the host was selecby secret ballot (in this case, Turin).

Another similar issue is currently the subje
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illo: Department of Economics & CODE,oma de Barcelona, 08193 Bellatera (BarWettstein: Department of Economics, Beity of the Negev, Monaster Center foarch, Beer-Sheva 84105, Israel. We thar his helpful suggestions; Suresh Mu-couraged us to check for strong Nash eqis Bloch, Mark Gradstein, Moshe Justmatwo anonymous referees for their usefu

-Castrillo gratefully acknowledges finan-BEC2000-0172 and 2000SGR-00054.

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of heated debate in Spain. The Spanish government, both on the national and local levelhas recently been discussing various possiblprojects for transferring water from rivers in thenorth of Spain to the arid southern regions. Theparties involved have conflicting interests withregard to several aspects of the projects proposed. Particularly, the water-rich regions arehesitant to give up parts of their water re-sources. The parties involved have still notagreed upon the choice of a transport mode othe size of the transfer payments that wouldcompensate the localities that are relinquishingpart of the water supply.

These and several other cases belong toclass of problems in which a group of agentshas to choose one out of several projects. Aproject is efficient if it maximizes the aggre-gate welfare of the group members. Reachinan efficient outcome is trivial when the de-signer has all the relevant information. How-ever, it is often the case that the partiesconcerned possess much more informatiothan the designer. Hence, the designer facesnontrivial problem if she wishes to optimallychoose one of the projects. Moreover, from anormative point of view, even if the designerhas all the information, it is easier to justifythe use of a fixed procedure to reach a decision than to modify the procedure for variouscases in light of information disclosed or al-ready available to the designer.

Several well-known methods are availablefor the problem of choosing one out of a setof alternatives. One class of methods consistof various voting schemes, such as majorityvoting or the Borda rule. The outcomes generated by these methods need not be efficient duto strategic behavior on the part of the agentsAnother class of methods consists of Vickrey-Clarke-Groves mechanisms. For this class, “truthtelling” is a dominant strategy that alwaysresults in the choice of an efficient alternative.However, these mechanisms are not budgebalanced and the payments collected from th

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1578 THE AMERICAN ECONOMIC REVIEW DECEMBER 2002

agents often generate a surplus that is a lossfrom the agents’ point of view.1

We address the problem of reaching an effi-cient decision by suggesting a simple mecha-nism that is budget-balanced and focusing on itsNash equilibria.2 We construct a multibiddingmechanism in which agents place bids in orderto determine which project will be chosen. Theequilibrium outcomes of the mechanism notonly determine the project, but also generate asystem of transfer payments that serve, in part,to compensate those agents who are not pleasedwith the chosen project.

The one-stage mechanism proceeds as fol-lows. Each agent submits a vector of bids, theqth component of which can be interpreted asthe amount of money an agent is willing to payif project q is chosen. The only restriction im-posed on the bids submitted by an agent is thatthey have to sum up to zero. This introduces ameasure of relative worth whereby more desir-able projects receive a larger bid. The agent alsoselects one of the projects. We define the “ag-gregate bid” for a project as the sum of bidsmade for this project. The project with the high-est aggregate bid is chosen. In case there ismore than one such project, the winning projectis randomly chosen from those with the highestaggregate bid that have also been selected by atleast one agent. If there is no such project thewinning project is randomly chosen from thosewith the highest aggregate bid. Once the projecthas been selected the agents pay the promisedbid corresponding to this project, it is carriedout, and any surplus is shared among the agents.

The multibidding mechanism is simple andstraightforward. Moreover, it satisfies the ap-pealing property that an agent is at least as welloff when playing the mechanism as he is whenstaying out of the process (an agent can alwayschoose a bids vector equal to the zero vector).We show that all Nash equilibrium outcomes ofthis mechanism are efficient. That is, the project

1 An analysis of the voting methods and Vickrey-Clarke-Groves mechanisms can be found, for example, in Jean-Jacques Laffont (1988).

2 Our mechanism is much simpler and more intuitivethan general mechanisms constructed in the Nash imple-mentation literature, which are designed to handle a largeclass of environments, as in Eric Maskin (1999).

chosen in equilibrium is efficient. Furthermore,we provide a detailed analysis of the structure ofthe equilibria outcomes and their ranking by theparticipating agents. In particular, we show that,at equilibrium, each agent’s payoff is at least theexpected payoff he would obtain in a situationwhere all the projects have the same probabilityof being developed.

We also show that the Nash equilibria of themultibidding mechanism are strong Nash equi-libria. This makes the mechanism even moreattractive since it will achieve efficient out-comes also in environments where agents mightcollude and coordinate their bids, which mayoften be the case in real-world situations.

The mechanism we propose is partly basedon Lester E. Dubins’ work (1977). We adoptthe same environment that Dubins used toanalyze the properties of decision devices andwe construct a bidding mechanism that sharessome common characteristics with his. Hisanalysis mainly focused on the outcomes gen-erated by the choice of max-min strategies,leaving open the issues related to agents’ stra-tegic behavior.

William Samuelson (1980) studied a class ofprocedures suggested by H. Steinhaus (1949)designed to divide indivisible objects. However,the analysis concentrated only on normativeissues and ignored problems caused by strategicbehavior.

Herve Moulin (1984) proposed a two-stagegame in which each agent first bids for the rightto choose a project, states his utility function,and chooses a number. The agent with the high-est bid (where ties are broken by the numberschosen) selects the project to be carried out inthe second stage. Then transfer payments, de-pending on the project chosen, are made. Thismechanism implements in subgame-perfectequilibrium the equal sharing of the surplusabove the “average” utility level. In a morespecific environment, Matthew Jackson andMoulin (1992) considered a set of agents whodecide whether to carry out a particular publicproject that yields a benefit to each agent. Atevery undominated Nash equilibrium of themechanism they design, the public project isundertaken exactly when its total benefit out-weighs its cost. This mechanism also allows thedesigner to realize a variety of cost distribution

1579VOL. 92 NO. 5 PEREZ-CASTRILLO AND WETTSTEIN: A MULTIBIDDING APPROACH

rules.3 Parimal Kanti Bag (1997) adapts theprevious mechanism to a framework with divis-ible public goods when agents’ benefits frompublic good consumption are linear.

Our mechanism would also realize efficientoutcomes in Nash equilibria for discrete eco-nomic environments with public goods and ex-ternalities, continuous versions of which wereconsidered by Hal R. Varian (1994a, b). Varian(1994a) analyzed the equilibrium outcomes ofseveral games entailing private contributions toa public good. In one of the games he had theagents bid for the right to move first. This is aspecial case of our general setup where thereare two agents and the two alternatives arewhich one of them will move first. This gamedid not realize Pareto-efficient outcomes whentaking into account all possible levels of pub-lic good provision. Another game, where agents“subsidized” the contributions made by otheragents, did realize Pareto-efficient outcomesin subgame-perfect equilibria. In another paper,Varian (1994b) considered complete-informationeconomic environments with externalities, andconstructed two-stage mechanisms that re-alize efficient allocations in subgame-perfectequilibria.

R. Preston McAfee (1992) considered theproblem of dissolving a partnership, and ana-lyzed the equilibrium properties of four simplemechanisms in environments with risk-averseagents and asymmetric information. He showedthat bidding mechanisms similar to first-priceand second-price sealed-bid auctions attain expost efficient outcomes in suitably restrictedenvironments. These bidding mechanisms bearsome similarities to our multibidding mecha-nism for two agents. The mechanism con-structed in our paper realizes efficient outcomesfor complete-information equivalents of the en-vironments in McAfee.

An interesting subclass of the environmentscovered by our analysis is one in which a set ofagents has to choose who from among them willdevelop a project. In this environment, the set ofprojects corresponds to the set of agents. Sev-

3 Jackson and Moulin (1992) also describe a simplemodification of the mechanism so that the unique subgame-perfect equilibrium coincides with the desired outcome.

eral papers have addressed strategic consider-ations for this class of problems includingHoward Kunreuther and Paul R. Kleindorfer(1986), Jacob Glazer and Ching-to Albert Ma(1989), Rafael Rob (1989), Arthur O’Sullivan(1993), Daniel E. Ingberman (1995), PhilippeJehiel et al. (1996), and Motty Perry and PhilipJ. Reny (1999). We discuss their findings andour own results regarding this environment inSection III.

The paper proceeds as follows: In Section I,we present the basic environment and the multi-bidding mechanism, and in Section II we studythe outcomes generated by the mechanism andprovide a full characterization of all the equi-librium payoffs. Section III studies the environ-ment in which a set of agents decides who fromamong them will develop a project that gener-ates (positive or negative) externalities. SectionIV presents conclusions and suggests furtherdirections of research.

I. The Environment and theMultibidding Mechanism

We consider a set of agents N � {1, ... , n}and a set of possible projects K � {1, ... , k}.The agents’ utilities depend on the project cho-sen and the agents have to choose which projectwill be carried out. The utility (payoff) of agenti if project q is carried out is given by vq

i .Project q is efficient if:

�i � N

vqi � �

i � Nvp

i for all p � K.

We denote by E the set of efficient projects andby Ve the maximum total value that can beattained by developing the project, that is,

Ve � �i � N

vqi for some q � E.

Although each agent has complete information,that is, he knows all the values of vq

i , the plan-ner does not. Alternatively, we can assume thatthe planner, while possibly having some partialinformation, wants to design a system that willbe applicable to any similar situation.

We construct a multibidding mechanismthrough which agents influence the choice of the


project. The winning project will be the onehaving received the highest bid. Informally, themechanism is described as follows: Each agenti announces k bids, one for each possibleproject, that are constrained to sum up to zero.Furthermore, he chooses one of the projects.The project with the highest aggregate bid ischosen as the winner. We define the aggregatebid for a project as the sum of bids made to thatproject. In case of a tie, the winning project israndomly chosen among those with the highestaggregate bid that have been selected by at leastone agent. Otherwise, the winning project israndomly chosen among those with the highestaggregate bid. Once the winning project is iden-tified, the bids corresponding to it are paid (orreceived). The surplus, if any, is shared amongthe agents, and the project is developed.

A key feature of our mechanism is that weallow for more than one bid by each agent.Forcing the agents to submit a single bid makesit impossible to “fi ne tune” the strategy andleads to inefficient equilibria.

We now describe the multibidding mech-anism more formally: Each agent i � Nmakes bids bq

i in R; one for every q in K with¥q�K bq

i � 0, and announces mi � K. Hence, astrategy for agent i is a vector ((bq

i )q � K, mi) inH k � K (mi is “ interpreted” to mean that theagent would like to see project mi developed),where

H k��z � Rk� �q � K

zq � 0� .

For each q � K, we let Bq � ¥i � N bqi denote

the aggregate bid for project q.4 The set ofprojects with the highest aggregate bid is de-noted by �(b) � {q � K �Bq � Bp for all p �K}. Finally, �1(b, m) � {q � �(b)�mi � qfor some i � N } denotes the set of projectswith the highest aggregate bid that have beenselected by some agent. The winning project �is randomly chosen among the members of�1(b, m) if �1(b, m) is not empty. Otherwise,it is randomly chosen in �(b).

4 Since the bids of every agent sum up to zero, the sumof the aggregate bids is also zero.

The final payment to any agent i is given by(�b�

i � B�/n).5

II. Equilibrium Outcomes of theMultibidding Mechanism

We are interested in the equilibrium out-comes generated by the multibidding mecha-nism; that is, which project will be finallyimplemented and what will be the final utilitylevels of the agents? As this is a game of com-plete information, we analyze the resultingNash equilibria (NE). We start by proving sev-eral important properties satisfied by all equi-librium strategies. In Lemma 1 we show that inequilibrium the aggregate bid for any projectmust be zero. Using Lemma 1, we derive inLemma 2 a set of inequalities satisfied by allequilibrium strategies.

LEMMA 1: In any NE of the multibiddingmechanism, the aggregate bid of any project iszero, i.e., Bq � 0 for any q.

PROOF:If �(b) � K the claim is satisfied since the

sum of aggregate bids is zero. If �(b) � K,that is, the winning project(s) received a posi-tive aggregate bid, there must be another projectthat received a negative aggregate bid. In thiscase, any agent could, by slightly reducing thebid he made for the winning projects and in-creasing the bids (so as to satisfy the summingup to zero) for the losing projects, keep thesame set of winning projects and increase hispayoff.

Lemma 1 implies that in any NE, players’utilities from a given project q are given by(vq

i � bqi ), and not (vq

i � bqi � Bq/n), since in

equilibrium Bq � 0 for any q. Moreover, inequilibrium, the bids are such that each agentcan by a slight change in his bid decide upon theidentity of the winning project. Indeed, since allaggregate bids are equal, agent i can turnproject q into a certain winner by slightly in-

5 We chose the equal sharing rule to allocate the possiblesurplus B� � 0 resulting from the bids made to the winningproject. Any other sharing rule would generate the sameresults.


creasing bqi . Similarly, agent i could turn project p

into a sure loser by slightly decreasing bpi . There-

fore, the project finally chosen must be the mostpreferred one for every agent given the equilib-rium bids. This result is stated as Lemma 2.

LEMMA 2: If project q is chosen as a NEoutcome of the multibidding mechanism, thefollowing condition must be satisfied:

(1) vqi � bq

i � vpi � bp

i

for all i � N and p � K.

The inequalities derived in Lemma 2 are nowused to show that all Nash equilibrium out-comes are efficient.

THEOREM 1: In every NE of the multibiddingmechanism, the project chosen is efficient.

PROOF:Assume that project q was chosen. Take any

project p different from q. Consider condition(1) corresponding to this particular project p.Summing up over the set of agents, we get:

�i � N

vqi � �

i � Nbq

i � �i � N

vpi � �

i � Nbp

i .

Since aggregate bids for projects q and p arezero, we obtain

�i � N

vqi � �

i � Nvp

i .

Therefore project q is efficient.

Hence, the multibidding mechanism leadsto an efficient project even though the author-ities in charge lack precise information on thebenefits associated with each project. The in-tuition for this result follows from Lemma 2.If the project chosen in equilibrium was notefficient, then it would be possible to increasethe aggregate utility of the agents by thechoice of an alternate project. This wouldcertainly improve the situation of at least oneperson and (by Lemma 2) that person could,by slightly modifying his strategy, attain abetter outcome, in contradiction to the fact

that the choice of the project was an equilib-rium outcome.

Theorem 1 states the main property of theNE. The (constructive) proof of the existence ofsuch NE is postponed until Theorem 3. We nowprovide some characteristics of the utilities ob-tained by the agents in equilibrium.

Denote by ui the final level of utility thatagent i obtains by participating in the multibid-ding mechanism. Also, denote by vi the ex-pected utility agent i obtains if there is aprobability of 1/k that each project is carriedout. Formally,

v i �1

k �q � K

vqi .

Note that the payoff vi is associated with therandom assignment mechanism, which is a pop-ular benchmark for evaluating performances.The following theorem shows that the vi utilitylevels provide a lower bound on the payoffsreceived by the agents at the equilibria of themultibidding mechanism.

THEOREM 2: In every NE of the multibiddingmechanism, the final utility of any agent i isgreater or equal to his average valuation, thatis, ui � vi for all i in N.

PROOF:Consider an equilibrium outcome in which a

project q is chosen. Summing up condition (1)for all p in K, we obtain:

k�vqi � bq

i � � �p � K

�vpi � bp

i � for all i � N.

Given that the sum of bpi ’s over the set K

equals zero by construction, we get:

ui � vqi � bq

i �1

k �p � K

vpi � vi

for all i � N.

The intuition behind Theorem 2 is also sim-ple. Agent i can always choose a vector of bidsso that he guarantees himself a certain final


utility irrespective of the choices made by otheragents. Agent i can choose bids in H k satisfy-ing vq

i � bqi � vp

i � bpi for all q, p in K,

namely bqi � vq

i � vi for all q. It is easy tocheck that by using this “safe” strategy, agent i’spayoff always equals at least v i. Indeed, agent iobtains precisely v i if Bq � 0 for all q (as happensat equilibrium); he obtains strictly more than v i

if Bq 0 for some q. Hence, the utility that agenti obtains in any NE must be, at least, v i.

Theorems 1 and 2 highlight the main propertiesof the equilibrium outcomes. All equilibrium out-comes are efficient and guarantee for each agent atleast his expected utility when the probability ofcarrying out the project is uniformly distributedamong all the projects. Hence, all equilibriumoutcomes belong to the set P, where

P �u � Rn� �i � N

ui � Ve and ui � vi

for all i � N � .

We now derive further properties of the equi-librium strategies using the efficiency of anyequilibrium outcome.

LEMMA 3: At any NE of the multibiddingmechanism, mi � E for every agent i.

PROOF:Since in equilibrium all aggregate bids are

equal, if an agent i sent mi � E, there would bea positive probability that an inefficient projectwill be chosen at equilibrium. This is in contra-diction to Theorem 1.

LEMMA 4: In any NE of the multibidding mech-anism, all agents are indifferent to the identity ofthe project chosen as long as it is efficient.

PROOF:Assume we are at a NE where an efficient

project q is chosen. Furthermore, there is an-other efficient project p. Since

�i � N

vqi � �

i � Nvp

i ,

it is also the case that:

�i � N

vqi � �

i � Nbq

i � �i � N

vpi � �

i � Nbp

i .

Therefore, all the inequalities involving projectsq and p in equation (1) must be satisfied asequalities. Hence, at this NE, any agent i getsthe same payoff if project q or p is chosen.

We can now use the above properties to pro-vide a full characterization of all the equilibriumstrategies.

LEMMA 5: A set of strategies constitutes a NEof the multibidding mechanism if and only if thefollowing three properties hold:

(a) The aggregate bid for every project is equalto zero.

(b) Given the equilibrium bids, an agent’s pay-off is maximal if an efficient project is cho-sen, and it is the same regardless of whichefficient project is chosen.

(c) The message mi � E for every agent i.

PROOF:The “only if ” part has been shown in the

previous lemmata. We now prove that the con-ditions are also sufficient. Suppose that thestrategies of the agents satisfy conditions (a)–(c). We prove that no agent i in N profits bydeviating from these strategies.

Note first that conditions (a) and (c) implythat �1(b, m) � E and hence by (b) everyagent is indifferent to the identity of theproject chosen in �1(b, m). Therefore, giventhe bids, no agent i can gain by a change inhis message mi since his utility is maximizedwhen an efficient project is chosen. If anagent decides to increase his bid for a partic-ular project, it will be chosen with certainty[because of (a)]. However, since the projectsin �1(b, m) yielded a maximal payoff, hispayoff will decrease.

Theorem 3 constructively shows that anyvector in P can indeed be realized as an equi-librium outcome of our mechanism. Thus weprovide a full characterization of the set of NEutilities.


THEOREM 3: Any u � P can be realized asa Nash equilibrium outcome of the multibiddingmechanism.

PROOF:See the Appendix.

The following corollary sets our results in theframework of implementation theory.

COROLLARY 1: The multibidding mecha-nism implements in NE the set of utility vectorsgiven by:

� �u1, ... , un� � Rn� �i � N

ui � Ve and ui � vi

for all i � N � .

PROOF:It follows directly from Theorems 1, 2, and 3.

One concern regarding the realization of ef-ficient outcomes via NE is that the results areusually not immune to coalitional manipula-tions. A subset of the agents can often improvetheir situation by a joint deviation from theequilibrium strategies. A NE is a strong NE iffor any subset of agents the strategies played byits members are a best response to the strategiesadopted by the remaining agents. We now showthat the NE of the multibidding mechanism areimmune to coalitional deviations and constitute,in fact, strong NE.6 This adds further credibilityto our efficiency results in that they remainintact even when one allows for collusionamong agents.

THEOREM 4: Any NE of the multibiddingmechanism is a strong NE.

PROOF:Take any NE of the multibidding mechanism

6 Game forms for which the set of NE coincides with theset of strong NE have also been constructed for implement-ing competitive outcomes (Bezalel Peleg, 1996) and taxa-tion methods (Nir Dagan et al., 1999).

and let (bqi )i � N,q � K be the bids announced in

that equilibrium. Assume by way of contradic-tion that there exists a subset of agents C � Nwhich could achieve a better outcome for all ofits members by deviating to a new set of strat-egies (while agents outside of C keep their NEstrategies). Let (dq

i )i � C,q � K denote the bidsused in the new set of strategies.

Let q be a project that is chosen as an out-come of the mechanism given the new bids andmessages. The sum of utilities obtained by theagents in C is:

�i � C

�vqi � dq

i � � �i � C

vqi � �

i � Cdq

i .

Since the (new) aggregate bid for project qmust be nonnegative, and the bids by the agentsoutside of C did not change, it must be the casethat the sum of the new bids of the agents in Cis at least as large as previously, hence:

�i � C

vqi � �

i � Cdq

i � �i � C

vqi � �

i � Cbq

i

� �i � C

�vqi � bq

i � � �i � C

�vpi � bp

i �,

for any efficient p.Given that the last expression is the aggregate

utility of the members in C at the NE, thedeviation by C could not have increased theaggregate utility of the members of C. Hence, itis not possible for the agents in C to propose ajoint set of strategies in which they will bebetter off than under the NE strategies.

A strong NE is immune to any coalitionaldeviation and as such is also a coalition-proofequilibrium (coalition-proofness looks only at asubset of all the possible deviations; see B.Douglas Bernheim et al. [1987] for a precisedefinition). Hence, the mechanism implementsthe efficient outcomes in coalition-proof equi-libria as well.

III. In Whose Backyard?

One subclass of the environments we haveconsidered in the previous sections has beenextensively discussed in the literature, namely


the case where each project is “associated” withone agent. Consider the case where one of kcities is to host a project. The project can eitherbe a “good,” for example a museum or a largesports event, or it can be a “bad,” such as adump site or a nuclear waste disposal facility. Inthis setting vq

i and bqi can be interpreted as

follows: vqi is the utility of city i if city q ends

up with the project; bqi is the amount of money

city i is ready to pay in case city q gets theproject. If the project is a “good,” then vq

i istypically a negative number if q � i, and apositive number if q � i. The opposite holds ifthe project is a “bad.” Clearly, the mechanismwe have constructed applies to this environmentand realizes efficient outcomes in this settingwith (positive or negative) externalities. In thiscase, the lower bound vi for the equilibriumpayoffs has a natural interpretation. Given thatthe environment is one in which every agent hasthe same a priori rights, a natural and fair (al-though not efficient) procedure for allocatingthe project is to randomly choose one of theagents with probability 1/n. The value vi is theexpected payoff of agent i under this random-assignment mechanism.

Problems of this class have been analyzed inseveral papers. In particular, sealed-bid mecha-nisms have been suggested for siting noxiousfacilities, for incomplete-information settingswhere each agent knows his own preferencesbut does not know the preferences of the otheragents. Kunreuther and Kleindorfer (1986)showed that the outcomes realized by max-minstrategies are efficient in those environmentswhere each agent is indifferent as to all theoutcomes, as long as he is not the host. For thecase of two cities, O’Sullivan (1993) provedthat symmetric Bayes-Nash equilibria of thebidding game yield an efficient outcome whenthe cost parameters of the two cities are inde-pendently drawn.

Ingberman (1995) added a further dimensionto the problem of siting a noxious facility byrelating the cost to the agents to their distancefrom the noxious facility. He analyzed the pro-cess of majority agreement by using an auctionapproach and concluded that decisions reachedin this manner would not be efficient.

Rob (1989) approached the siting problemfrom a mechanism design point of view. The

designer’s role is assumed by a firm that has todecide where to build its plant. To each costvector reported by the locations, the optimalincentive-compatible and individually rationalmechanism associates a randomized decisionrule and an expected compensation for eachlocation. As is customary in such environments,the resulting mechanism could lead to ineffi-cient outcomes.

The environment characterizing King Solo-mon’s Dilemma is another special case of ourgeneral setup. In this environment there areseveral agents and a single prize that must beawarded to one of them. The designer’s prob-lem is to award the prize to the agent that valuesit the most without imposing any cost on thatagent. Glazer and Ma (1989) assumed theagents have complete information and con-structed a multistage mechanism whose uniquesubgame-perfect equilibrium realizes that out-come. Perry and Reny (1999) relaxed the infor-mational assumptions, requiring only that it iscommon knowledge that the agent who mostvalues the object knows who he is. They pro-vided a mechanism that realizes the desiredoutcome in iteratively weakly undominatedstrategies. The mechanism we have constructedwould realize an efficient outcome in Nashequilibria for the complete-information KingSolomon’s Dilemma. Moreover, it would alsowork in a more general framework in whicheach agent’s welfare depended not only onwhether or not he received the prize, but also onwhich of the other agents is the final owner.However, in most cases our mechanism wouldimpose a monetary cost on the winner.

Jehiel et al. (1996) analyzed a more complexenvironment where the benefit (or cost) to anagent depends on the identity of the agent car-rying out the project. In their setting, one sellerhas an object he wants to sell to one of n agents.First, the authors assume the seller is aware ofall the benefits and show how he could constructa revenue-maximizing selling mechanism. Thenin the incomplete-information setting they char-acterize the incentive-compatible and individu-ally rational mechanisms that maximize theseller’s revenue. An advantage of our simplermechanism is that it can be applied without anyknowledge of the particular environment inwhich it is being used. In contrast, the outcome


function of the mechanism proposed by Jehiel etal. (1996) is very sensitive to the particularsof the environment to which it is applied.However, Jehiel et al. (1996) consider bothcomplete- and incomplete-information settingswhile we only perform our analysis when agentshave complete information about each other.

For the environments considered in this sec-tion, it is in fact possible to adopt another mech-anism that is different than (although similar to)the multibidding mechanism that achieves iden-tical outcomes, but in which strategies have adifferent interpretation.7 The mechanism canbe informally described as follows: Each agenti announces (n � 1) bids, one for each ofthe other agents. Furthermore, he announceswhether he “ really” wants to have the project.The agent with the highest net bid is chosen asthe winner. We define the “net bid” of an agentas the difference between the sum of bids hemakes to the others and the sum of bids theothers make to him. In case of a tie, the winneris randomly chosen among those agents with thehighest net bid who have announced that they“ really” want the project. If there are no suchagents, the winner is randomly chosen from allthe agents with the highest net bid. Once thewinner is determined, he pays the bids to theother agents and proceeds with the project. Inthis mechanism, the bids are naturally inter-preted as transfer payments (positive or nega-tive) offered for the right to carry out aparticular project.

The NE outcomes of this mechanism possessall the properties satisfied by the NE outcomesof the multibidding mechanism except for thefact that they are not strong NE.8

IV. Conclusion

We have addressed the problem of choosingan efficient alternative by a group of agents withconflicting interests. The underlying problemmight be the siting of a major sports event, theelection of a department chairman, or the loca-

7 A similar bidding procedure was used by Perez-Castrillo and Wettstein (2001) as part of a noncooperativeimplementation of the Shapley value.

8 For more details on this mechanism, the reader isreferred to Perez-Castrillo and Wettstein (2000).

tion of a nuclear reactor. We assumed that theagents are fully informed and that an unin-formed designer has to reach a decision that willaffect all of them. We then constructed a simpleand straightforward mechanism whose Nashequilibria realize any efficient outcome that sat-isfies intuitively appealing lower bounds on thepayoffs attained in equilibrium by the partici-pating agents. Furthermore, these equilibriawere shown to be strong Nash equilibria andhence are immune to coalitional deviations.

The mechanism proposed here improves pre-vious constructions in that it can handle a largeset of environments. It involves a single stageof play with each agent being able to expresshis preferences by sending a multidimensionalmessage.

The complete-information assumption, whilerestrictive, can approximate realistic environ-ments where the agents taking part in the mech-anism are much better informed than thedesigner. For example, the designer could be auniversity president and the agents members ofa department, one of whom should be desig-nated as a chairman. The complete-informationframework allows the derivation of a powerfulefficiency result and can serve as a benchmarkfor the study of environments with incompleteinformation.

The mechanism could operate in environ-ments with asymmetric information as well.Determining the properties of the resultingequilibria in such environments is a topic forfurther research.

APPENDIX

PROOF OF THEOREM 3:The proof proceeds as follows: First we show

that both the set of equilibrium bidding strate-gies and of equilibrium payoff vectors are con-vex. Second, we show by construction that forany agent i � N there exists an equilibriumwhere the payoff for any agent j other than i isv j. We conclude by noting that these payoffvectors are the extreme points of the set P andas such any point in P can be expressed as aconvex combination of these vectors.

Let S1 � (�1, m1) and S2 � (�2, m2) be twostrategy tuples that are NE for the biddingmechanism where �1 � (bq

1i)q � K,i � N and


�2 � (bq2i)q � K,i � N are the bidding strategies

used and m1, m2 are the n-tuples of messagessent in these equilibria. We now prove that thebidding strategies �� (bq

i )q � K,i � N givenby:

bqi � �bq

1i � �1 � ��bq2i,

together with m1 (or m2) are a NE for any 0 �� 1. We prove this by showing that theconditions of Lemma 5 are satisfied.

(a) The aggregate bid for any project in �� iszero, since

Bq� � �

i � N��bq

1i � �1 � ��bq2i�

� � �i � N

bq1i � �1 � ��

i � Nbq

2i

� 0 � 0 � 0.

(c) Moreover, since m1 was part of a NE, itsatisfies condition (c).

(b) Now we consider any efficient project, sayq � E. If q is chosen, agent i’s payoff ishigher than his payoff if any other projectp � q is chosen, since:

vqi � �bq

1i � �1 � ��bq2i

� ��vqi � bq

1i� � �1 � ��vqi � bq

2i�

� ��vpi � bp

1i� � �1 � ��vpi � bp

2i�

� vpi � �bp

1i � �1 � ��bp2i for all p q.

Hence, agent i’s payoff is maximized if theefficient project q is chosen. Also note that inthe two previous expressions the inequalitiesbecome equalities if project p is efficient aswell.

Therefore �� and m1 constitute a NE. Fromthe above calculations we also see that the pay-off vector associated with the �� equilibrium isthe convex combination with weights � and(1 � �) of the payoffs corresponding to the S1

and S2 equilibria. Thus, we proved that the sets

of equilibrium-bidding strategies and of equi-librium payoffs are convex.

We now proceed to construct the “extremal”equilibrium points. Take any agent j � N. Weclaim that the following bidding strategies con-stitute the best equilibrium for agent j:

bqi � vq

i � v i for i j, q � K

bqj � �

i � j

v i � �i � j

vqi for q � K.

Also, the players’ strategy include a vector ofmessages such that only efficient projects arechosen (for example, mi � q for all i � N,where q � E ).

It is easy to check that the previous bid vectorof any agent belongs to Hk. To show this is anequilibrium we prove that all the conditions ofLemma 5 are satisfied. Condition (a) holds sincewith these bids the aggregate bid for any projectis indeed zero. Condition (c) is satisfied byconstruction. It remains to show that condition(b) of Lemma 5 holds; that is, given thesestrategies, each agent’s payoff is maximized ifany efficient project is carried out. The payofffor any agent i � j is vi and hence does notdepend on the project chosen and is maximizedfor any choice of an efficient project. Agent jreceives the difference between the total surplusand the previous payoffs. Therefore, agent jprefers that an efficient project be carried out.Hence, all the conditions of Lemma 5 are sat-isfied and the previous strategies are part of aNE that yields a payoff vector that agent jprefers the most.

The payoff vectors corresponding to the equi-libria we have constructed are the extremalpoints of the set P and as such any point p in Pcan be expressed as a convex combination ofthese vectors. Applying the same convex com-bination to the bidding strategies yields a NE ofthe game with the associated payoff vector co-inciding with p.

REFERENCES

Bag, Parimal Kanti. “Public Goods Provision:Applying Jackson-Moulin Mechanism forRestricted Agent Characteristics.” Journal of


Economic Theory, April 1997, 73(2), pp. 460–72.

Bernheim, B. Douglas; Peleg, Bezalel and Whin-ston, Michael D. “Coalition-Proof Nash Equi-libria. I. Concepts.” Journal of EconomicTheory, June 1987, 42(1), pp. 1–12.

Dagan, Nir; Serrano, Roberto and Volij, Oscar.“Feasible Implementation of Taxation Meth-ods.” Review of Economic Design, February1999, 4(1), pp. 57–72.

Dubins, Lester E. “Group Decision Devices.”American Mathematical Monthly, May 1977,84(5), pp. 350–56.

Glazer, Jacob and Ma, Ching-to Albert. “Effi-cient Allocation of a ‘Prize’—King Solo-mon’ s Dilemma.” Games and EconomicBehavior, September 1989, 1(3), pp. 222–33.

Ingberman, Daniel E. “Siting Noxious Facilities:Are Markets Efficient?” Journal of Environ-mental Economics and Management, No-vember 1995, 29(3), pp. 20–33.

Jackson, Matthew and Moulin, Herve. “ Imple-menting a Public Project and Distributing itsCost.” Journal of Economic Theory, June1992, 57(1), pp. 125–40.

Jehiel, Philippe; Moldovanu, Benny and Stac-chetti, Ennio. “How (Not) to Sell NuclearWeapons.” American Economic Review, Sep-tember 1996, 86(4), pp. 814–29.

Kunreuther, Howard and Kleindorfer, Paul R. “ASealed-Bid Auction Mechanism for SitingNoxious Facilities.” American Economic Re-view, May 1986 (Papers and Proceedings),76(2), pp. 295–99.

Laffont, Jean-Jacques. Fundamentals of publiceconomics. Cambridge, MA: MIT Press,1988.

Maskin, Eric. “Nash Equilibrium and WelfareOptimality.” Review of Economic Studies,January 1999, 66(1), pp. 23–38.

McAfee, R. Preston. “Amicable Divorce: Dis-solving a Partnership with Simple Mecha-

nisms.” Journal of Economic Theory, April1992, 56(2), pp. 266–93.

Moulin, Herve. “The Conditional AuctionMechanism for Sharing a Surplus.” Review ofEconomic Studies, January 1984, 51(1), pp.157–70.

O’Sullivan, Arthur. “Voluntary Auctions forNoxious Facilities: Incentives to Participateand the Efficiency of Siting Decisions.” Jour-nal of Environmental Economics and Man-agement, July 1993, 25(1), pp. 12–26.

Peleg, Bezalel. “A Continuous Double Imple-mentation of the Constrained Walras Equilib-rium.” Economic Design, August 1996, 2(1),pp. 89–97.

Perez-Castrillo, David and Wettstein, David. “ InWhose Backyard? A Generalized BiddingApproach.” Working Paper No. 463.00, Uni-versitat Autonoma de Barcelona, 2000.

. “Bidding for the Surplus: A Non-cooperative Approach to the Shapley Value.”Journal of Economic Theory, October 2001,100(2), pp. 274–94.

Perry, Motty and Reny, Philip J. “A GeneralSolution to King Solomon’s Dilemma.”Games and Economic Behavior, February1999, 26(2), pp. 279–85.

Rob, Rafael. “Pollution Claim Settlements underPrivate Information.” Journal of EconomicTheory, April 1989, 47(2), pp. 307–33.

Samuelson, William. “The Object DistributionProblem Revisited.” Quarterly Journal ofEconomics, February 1980, 94(1), pp. 85–98.

Steinhaus, H. “Sur la Vision Pragmatique.”Econometrica, July 1949, Supp., 17, pp. 315–19.

Varian, Hal R. “Sequential Contributions toPublic Goods.” Journal of Public Economics,February 1994a, 53(2), pp. 165–86.

. “A Solution to the Problem of Exter-nalities When Agents Are Well Informed.”American Economic Review, December1994b, 84(5), pp. 1278–93.

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Choosing Wisely: A Multibidding Approach

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