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Strategy-Proofness and Essentially Single-Valued Cores Author(S): Tayfun Sonmez Source: Econometrica, Vol

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Strategy-Proofness and Essentially Single-Valued Cores Author(S): Tayfun Sonmez Source: Econometrica, Vol Strategy-Proofness and Essentially Single-Valued Cores Author(s): Tayfun Sonmez Source: Econometrica, Vol. 67, No. 3 (May, 1999), pp. 677-689 Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/2999552 Accessed: 10/08/2009 18:29 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=econosoc. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. The Econometric Society is collaborating with JSTOR to digitize, preserve and extend access to Econometrica. http://www.jstor.org Economnteica, Vol. 67, No. 3 (May,1999), 677-689 STRATEGY-PROOFNESS AND ESSENTIALLY SINGLE-VALUED CORES BY TAYFUNS6NMEZ' 1. INTRODUCTION IN THIS PAPER WE SEARCHfor solutionis to various classes of allocatioll problemlls.We require them to be Pareto efficient and inldiliduIally ratioia (in the senIse that I10 agent is ever worse off than he would be on his own). In addition to these minimualrequirements we also would like agents not to be able to profitably misrepresenit their prefereinces. This property is known as strategy-proofiIess. Two extensively analyzed classes of allocation problems are inarriage pr-oblemls (Gale and Shapley (1962)), and hlolusinig mnarkets(Shapley and Scarf (1974)). Results concerning strategy-proofness in marriage problems are quite disappointing. Roth (1982a) shows that in the context of marriage problems there is no selection from the core correspondence that is strategy-proof. Moreover, Alcalde and Barbera (1994) shiow tlhat th1ere is no solution that is Pareto efficient, individually rational, ancl strategy-proof.3 OIn the other hand results pertaining to housing markets are muchi more encouraging. Roth (1982b) shows that in the context of housing markets the core correspondenlce, whiclh is shown to be single-valued by Roth and Postlewaite (1977), is strategy-proof. Moreover Ma (1994) shows that it is the only solution that is Pareto efficient, individually rational, and str ategy-proof. We search for foundations of the differences in these results. We do this by studying strategy-proofness on a general class of allocation problems that includes both models as subclasses. In addition to marriage problems and housing markets this class also includles This paper is a substantial revision of my discuission paper 'Strategy-Proofiness and Singleton Cores in Generalized Matching Problems." The revision is inspired by the suggestions of Joseplh Greenberg to whom I am most grateful. I also thank Steve Ching, Matthew Jacksoni,Hideo Konislhi, John Ledyard, Alvin Roth, Ismail Saglam, James Schumiimer,Tomoichi Shiniotsuka,Enniio Stacclhetti, Koichi Tadenuma, Robert Wilsoni, and in particular to an eclitor, Tanlk Kara, William Tlhomsoni,anicI three anonymous referees whose comments and suggestions significantly improvecl the paper. I also thank seminar participants at Bilkent University (Tturkey),Caltech, New York University, Ohio State University, Stanford University (Engiineerinlg-EconiomicSystems), University of Michigan, University of Pittsburgh, University of Rochester, University of Western Ontario, WashinigtoniUniversity at St. Louis, SITE 1995, the conference New Directions in the Theory of Markets and Gamiies(Toronto, 1995), 1996 North American Meeting for the EconomiietricSociety for their comlllenlts. All errors are my own responsibility. Strategy-proofness was first anialyzedin abstract social choice models wlhere there are few or no restrictions on preferences. Gibbard (1973) and Satterthwaite (1975) show that, unicer minlor conditions strategy-proofness is equivalent to dictatorship. In moclels with more structure (such as economic models) some positive results are available. See, for example, Barbera, Giil, and Stacclhetti (1993), Barberaiand Jackson (1994), Ching (1994), Moulini (1980, f994), Moulini and Shelnker (1992), Roth (1982b), Sonmez (1996), Sprumonit(1991). Kara ancdSonmez (1996) weaken the incentive requiiremiienitancl searclh for Nash implemiienitable allocation rules. They show that any solutioni that is Pareto efficient, inidividuallyratioial, and Nash implementable is a supersoltution of the core correspondenice. 677 678 TAYFUN SONMEZ roommateproblems (Gale and Shapley (1962)), indivisiblegoods exchangeeconomies, coalitionifoimation problems (Banerjee, Konishi, and Sonmez (1997)), and networks (Jacksonand Wolinsky(1996)). We show that for any model in this class, there exists a Pareto efficient,individually rational, and strategy-proofsolution only if all allocationsin the core are Pareto indifferentfor all problems.In fact, any such solution selects an allocationin the core wheneverthe core is nonempty.Furthermore any selection from the core correspondenceis strategy-proofas long as the core correspondenceis essen- tially single-valuedand the core of each problem is externallystable. We obtain the positiveresults of Roth (1982b),Ma (1994) for the housingmarkets, the negativeresults of Roth (1982a) and Alcalde and Barber'a(1994), for marriage problems, and new negativeresults for indivisiblegoods exchangeeconomies, coalition formation problems, and networksas direct applicationsof our general results. An implicationof our results is that, for a wide class of problems,the core is the key concept when one searches for strategy-proofsolutions that are Pareto efficient and individuallyrational. We believe this conclusion provides important noncooperative supportfor the core correspondence,a cooperativesolution. 2. THE GENERAL MODEL A generalizedindivisible goocls allocation problem, or simply a problem,is a 4-tuple (N, , Vf,R) where N = {,..., n} is a finite set of agents, w = (w(l),..., w(n)) is an initial endowment,cq/f is a set of feasible allocations,and R = (R1,...,R,,) is a list of preferencerelations. Each agent i is endowedwith a set of indivisiblegoods Cl(i)that we refer to as agent i's initial endowment.For all T c N, let cl(T) = U i I T o(li) denote the set of indivisiblegoods ownedby agents in coalitionT. An allocationa is a mappingfrom N to c(N) such that each good is assignedto one agent. Let w/ denote the set of all allocations.Formally, >V= {a: N -- w(N) IVx E w(N), la-l(x)l = 1}. We exogenouslyspecify a subset sV/f of the set of allocations as the set of feasible allocations.We requirethat CoFC/J. The preferencerelation Ri of each agent i E N is a binaryrelation on V-fthat is complete(for all a, b c&if we either have aRib or bRia) and tWansitive(for all a, b, c Ec f if aRib and bRic then aRic). Let Pi denote the strict prefer-encer-elation and Ii denote the indifferencerelation induced by Ri. Let Mi be the class of all preferencerelations for agent i. We require Mi to satisfytwo conditions. ASSUMPTIONA: For all Ri ECi and for all a E=-, aIiw a(i) = (i) That is, an agent is indifferent between an allocation and the initial endowment if and only if he keeps his initial endowment. ASSUMPTIONB: For all Ri EcAi and for all a E/Vf with aRiw, these exists R1iEci stuch that 1. Vb E Jf\{a}, bRia =bRja, 2. V b Ecf \ {a}, aRib aRlb, 3. Vb E-.f\{a}, aPib= aPib, and aRiwRib. STRATEGY-PROOFNESS 679 That is, for any preferencerelation Ri and any allocation a that is at least as good as the initial endowment,there exists a preferencerelation Ri such that, all allocationsthat are better than a under Ri are better than a under Ri, all that are worse remainworse, and the initial endowmentranks right after (or indifferentto) a. An importantclass that satisfiesthese conditionsis the class of preferenceswhere an agent has strict preferencesover own assignmentand does not care for others' assign- ments (i.e., no consumptionexternalities). Let M' be the class of all such preferencesfor agent i. Let R = Hi seN Mi and ?S = Hi seN M.'. For all R E=, and for all T c N, we denote the restrictionof R to T by RT, and the set N\ T by - T. For all i E N, we denote the set N\{i} by -i. Throughoutthe paper we fix N, w,X f - and hence each preference profile R E= defines a problem. An allocationa cEf is individuallyrational under R if aRiw for all i E N. We denote the set of all individuallyrational allocations under R by J(R). An allocation a EJVfPar-eto dominates an allocation b EcXf under R if aRib for all i E N and aPjb for some j E N. An allocationa Ecc. is Paretoefficient if it is not Pareto dominatedby any allocation b ECJ. We denote the set of all Pareto efficientallocations under R by 9(R). An allocation a EVf strictlyPasreto donlinates an allocation b Ecf under R if aPib for all i E N. Two allocationsa, b EcVfare Par-etoindiffer-ent under R if aIib for all i E N. An allocation a EccVfweakly dorninates the allocation b EcVf via the coalition T c N under R if 1. ViE TT a(i) c(T) 2. ViE T aRib, 3. 3jE T, aPjb. In that case we say that the coalition T blocksb underR via a. An allocationa EcWf is in the core4 of the problem R Ec if it is not weaklydominated by any feasible allocation.
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