Sets of alternatives as Condorcet winners

We characterize sets of alternatives which are Condorcet winners according to preferences over sets of alternatives, in terms of properties defined on preferences over alternatives. We state our results under certain preference extension axioms which, at any preference profile over alternatives, give the list of admissible preference profiles over sets of alternatives. It turns out to be that requiring from a set to be a Condorcet winner at every admissible preference profile is too demanding, even when the set of admissible preference profiles is fairly narrow. However, weakening this requirement to being a Condorcet winner at some admissible preference profile opens the door to more permissive results and we characterize these sets by using various versions of an undomination condition. Although our main results are given for a world where any two sets – whether they are of the same cardinality or not – can be compared, the case for sets of equal cardinality is also considered. Received: 15 March 2001/Accepted: 31 May 2002 This paper was written while Barış Kaymak was a graduate student in Economics at Boğazi?i University. We thank ?ağatay Kayı and İpek ?zkal-Sanver who kindly agreed to be our initial listeners. The paper has been presented at the Economic Theory seminars of Bilkent, Ko? and Sabancı Universities as well as at the Fifth Conference of the Society for the Advancement of Economic Theory, July 2001, Ischia, Italy and at the 24^th Bosphorus Workshop on Economic Design, August 2001, Bodrum, Turkey. We thank Fuad Aleskerov, İzak Atiyas, ?zgür Kıbrıs, Semih Koray, Gilbert Laffond, Bezalel Peleg, Murat Sertel, Tayfun S?nmez, Utku ünver and all the participants. Remzi Sanver acknowledges partial financial support from İstanbul Bilgi University and the Turkish Academy of Sciences and thanks Haluk Sanver and Serem Ltd. for their continuous moral and financial support. Last but not the least, we thank Carmen Herrero and two anonymous referees. Of course we are the sole responsible for all possible errors.     

The Nakamura numbers for computable simple games

The Nakamura number of a simple game plays a critical role in preference aggregation (or multi-criterion ranking): the number of alternatives that the players can always deal with rationally is less than this number. We comprehensively study the restrictions that various properties for a simple game impose on its Nakamura number. We find that a computable game has a finite Nakamura number greater than three only if it is proper, nonstrong, and nonweak, regardless of whether it is monotonic or whether it has a finite carrier. The lack of strongness often results in alternatives that cannot be strictly ranked. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users. We would like to thank an anonymous referee for useful suggestions. The discussion in footnote 3 and Remark 4, among other things, would not have been possible without his/her suggestion.     

The equivalence of the cost share equilibria and the core of a voting game in a public goods economy

We consider a social situation where individuals choose one or some alternative(s) from a set of feasible alternatives. When one alternative is socially chosen, each individual must share the cost for it according to an exogenously given sharing rule. Individuals are allowed to collude among themselves through money transfers. As the mechanism of choosing a social alternative, we consider the cost share equilibrium and a voting game. The cost share equilibrium is an allocation where each individual maximizes his/her utility independently and where unanimity is still maintained. We will show the equivalence between the set of all equilibrium allocations and the core of the voting game with compensation.I would like to thank Professors Mamoru Kaneko, Kotaro Suzumura, William Thomson, Mikio Nakayama, Hiroo Sasaki, Mr. Ken-ichi Shimomura and anonymous referees for their many useful suggestions and comments. I also acknowledge grammatical suggestions of Mr. Ronald M. Siani.     

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin in Rev Econ stud 66: 23–38, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackon and Palfrey in J Econ Theory 98: 1–25, 2001).I thank Toyo Sakai for his comments on a previous draft. I also thank two anonymous referees and an editor of this journal for helpful comments that improved this paper. A previous version circulated as “A note on Maskin monotonicity”. After the results presented here were obtained, I became aware of a new unpublished paper by Benoit and Ok (2004). The result of Theorem 2 and the discussion that follows is partially similar to their Theorem 1.     

Endogenous agenda formation in three-person committees

This paper analyzes a 3-person voting game in which two or three players have the ability to choose alternatives to be considered. Once the set of possible alternatives and the structure of the voting procedure are known, the players can solve for the outcome. Thus, the actual choice over outcomes takes place in the choice of alternatives to be voted on, i.e., the agenda. An equilibrium to this agenda-formation game in shown to exist under different assumptions about the information relative to the order of the players in the voting game. Further, this equilibrium is computed and found to possess certain features which are attractive from a normative point of view.Prepared for delivery at the 1985 Annual Meeting of the American Political Science Association, The New Orleans Hilton, August 29–September 1, 1985. We would like to thank Richard McKelvey, Norman Schofield and two anonymous referees for valuable comments and suggestions. We retain responsibility for remaining errors.     

Justifiable preferences over opportunity sets

In this paper we say that a preference over opportunity sets is justifiable if there exists a reflexive and complete binary relation on the set of alternatives, such that one opportunity set is at least as good as a second, if and only if the there is at least one alternative from the first set which is no worse than any alternative of the two sets combined together, with respect to the binary relation on the alternatives. In keeping with the revered tradition set by von Neumann and Morgenstern we call a reflexive and complete binary relation, an abstract game (note: strictly speaking von Neumann and Morgenstern refer to the asymmetric part of a reflexive and complete binary relation as an abstract game; hence our terminology though analytically equivalent, leads to a harmless corruption of the original meaning). In this paper we obtain a necessary and sufficient condition for the justifiability of transitive and quasi transitive preferences over opportunity sets. I would like to thank Prasanta Pattanaik for motivation in this research. I would also like to thank Maurice Salles and an anonymous referee of this journal for useful suggestions leading to considerable improvement in the presentation. An earlier version of this paper was presented at the Sixth Annual Conference on Econometric Modeling for Africa, held at Pretoria from July 4–6, 2001. I would like to thank all the conference participants for their observations on my paper. However, the sole responsibility for errors that do remain is my own.     

Assent-maximizing social choice

We take a decision theoretic approach to the classic social choice problem, using data on the frequency of choice problems to compute social choice functions. We define a family of social choice rules that depend on the population’s preferences and on the probability distribution over the sets of feasible alternatives that the society will face. Our methods generalize the well-known Kemeny Rule. In the Kemeny Rule, it is known a priori that the subset of feasible alternatives will be a pair. We define a distinct social choice function for each distribution over the feasible subsets. Our rules can be interpreted as distance minimization—selecting the order closest to the population’s preferences, using a metric on the orders that reflects the distribution over the possible feasible sets. The distance is the probability that two orders will disagree about the optimal choice from a randomly selected available set. We provide an algorithmic method to compute these metrics in the case where the probability of a given feasible set is a function only of its cardinality.     

Nonverbal communication: Toward syntax,by way of semantics

The identification of syntactical rules that govern nonverbal communication in humans and the determination of its semantic rules has often been viewed by researchers as two separate issues. In the present paper we depart from this traditional approach by studying the relationship among various nonverbal channels through their meaning. We propose that the functions which relate behavior to meaning are undulatory, and each is characterized by a different number of bending points. Supportive results of preliminary empirical testing are reported. Implications for cross-cultural research, development and psychopathology are discussed.The authors thank Sandra Tunis for her suggestions and Clara Mayo for her comments, as well as anonymous reviewers of this journal.     

Intermediate preferences and Rawlsian arbitration rules

Given the preferences of two agents over a finite set of alternatives, an arbitration rule selects some fair compromise. We study the idea that more consensus should not be harmful: the closer your preferences are to mine (in the sense of Grandmont's (1978) intermediate preferences), the better I like the selected alternative. We describe several Pareto optimal rules satisfying this principle. If, in addition, a condition akin to Suppes' (1966) grading principle is imposed, the rule must always choose an alternative maximizing the welfare of the worst-off agent, measured by the number of alternatives that he finds worse than the chosen one.Stimulating discussions with H. Moulin and helpful comments from J. Crémer are gratefully acknowledged. The author also wishes to thank a referee and an associate editor for challenging remarks. This research was partly supported by a CAFIR grant from the Université de Montréal.     

Freedom of choice and diversity of options: Some difficulties

This paper explores the possibility for a (non-preference-based) freedom ranking of opportunity sets that is sensitive to the diversity of the options. It turns out that how distances between sets and alternatives are measured is crucial to the derivation of such a ranking. Several proposals are examined, each of which is shown to lead to impossibility results. Earlier versions of this paper were presented at conferences in Kortrijk, Osnabrück, Oisterwijk, Caen and Pavia. I thank the participants at these conferences for their comments. I would also like to thank Steven Hartkamp, Theo Kuipers, Clemens Puppe and especially Kotaro Suzumura for their helpful comments. Furthermore, I am very grateful to Somdeb Lahiri for pointing out a mistake in a previous version of this paper.     

Order restricted preferences and majority rule

This paper develops the social choice foundations of a restriction that, in different guises, is utilized in a number of economic models; illuminates the key features of these models; and provides a specific class of applications. Order Restriction (on triples) is strictly weaker than Single Peakedness (or Single Cavedness) but strictly stronger than Sen's Value Restriction. It therefore guarantees quasi-transitivity of majority rule. This condition is most useful in models where there is a natural ordering of the individuals, not the alternatives. We show for a class of applications that1) it may be imposed through conditions with meaningful economic interpretations;2) imposing Single Peakedness weakens the usefulness of the models; 3) directly establishing Value Restriction is difficult and has not been done.I thank Bob Anderson for his help over a long period of time. I also thank the participants in many seminars at U.C. Berkeley. All remaining errors are my own.     

Distribution of coalitional power in randomized multi-valued social choice

This paper considers the distribution of coalitional influence under probabilistic social choice functions which are randomized social choice rules that allow social indifference by mapping each combination of a preference profile and a feasible set to a social choice lottery over all possible choice sets from the feasible set. When there are at least four alternatives in the universal set and ex-post Pareto optimality, independence of irrelevant alternatives and regularity are imposed, we show that: (i) there is a system of additive coalitional weights such that the weight of each coalition is its power to be decisive in every two-alternative feasble set; and (ii) for each combination of a feasible proper subset of the universal set and a preference profile, the society can be partioned in such a way that for each coalition in this partition, the probability of society's choice set being contained in the union of the best sets of its members is equal to the coalition's power or weight. It is further shown that, for feasible proper subsets of the universal set, the probability of society's choice set containing a pair of alternatives that are not jointly present in anyone's best set is zero. Our results remain valid even when the universal set itself becomes feasible provided some additional conditions hold. Received: 10 May 1999/Accepted: 18 June 2000 I would like to thank Professor Prasanta Pattanaik for suggesting to me the line of investigation carried out in this paper. I am solely responsible for any remaining errors and omissions.     

An arbitration game and the egalitarian bargaining solution

We analyze a simple arbitration procedure which is a multi-stage variant of Nash's demand game. In the absence of discounting, all Nash equilibria of the game yield the egalitarian solution in the first stage. The crucial feature of our arbitration procedure is that, in the case of incompatible demands, the game is allowed to continue and the player who demands the higher gain over the disagreement point is penalized by restricting her or his feasible demands in the following stage. Suitable modifications of the arbitration game yield the lexicographic extension of the egalitarian solution, resp. the proportional solutions. Journal of Economic Literature Classification Numbers: C72, C78.Financial support through grants from the Social Sciences and Humanities Research Council of Canada and the University of Waterloo is gratefully acknowledged. Earlier versions of the paper were presented at McMaster University and Brock University. The authors thank Nejat Anbarci, Michele Piccione, Venkatraman Sadanand, an editor, and the referees for their useful comments.     

A note on the core and von Neumann-Morgenstern solutions of simple games

Following the results of Nakamura (1979) and Muto (1984), we derive, for a given proper voting game G, the upper bound on the size of the space of alternatives, which guarantees that the core constitutes a von Neumann-Morgenstern solution for any profile of voter's preferences. In particular, we show that if the space of alternatives consists of more than two elements, then, in general, the core is not a von Neumann-Morgenstern solution.We wish to thank Bezalel Peleg for pointing us out that one of the results in the previous version of this paper has already been proved by Muto (1984) and two anonymous referees for their useful remarks.     

Sequential entry in many-to-one matching markets

We study sequential bargaining in many-to-one matching markets. We show that there is an advantage to entering late in the market, and that the last agent to enter the market will receive his or her best partner in a stable matching, extending the results of Blum and Rothblum (J Econ Theory 103(2):429–443, 2002) and Cechlárová (Randomized matching mechanism revisited. Mimeo, Safarik University, 2002) for the marriage model. We also discuss the relation between sequential bargaining and a possible alternative formulation based on the NTU Shapley value. We thank Peter Biró and Utku ünver for their advice on the related literature. We also thank an anonymous referee for comments.     

Money-egalitarian-equivalent and gain-maximin allocations of indivisible items with monetary compensation

Suppose that a certain quantity M of money and a finite number of indivisible items are to be distributed among n people, all of whom have equal claims on the whole. Different allocations are presented using various criteria of fairness in the special case where each player's utility function is additively separable. An allocation is “money-egalitarian-equivalent” (MEE) if each player's monetary valuation of his or her bundle is a fixed constant. We show that there is an essentially unique allocation that is MEE and Pareto-optimal; it is also envy-free. Alternatively, the “gain” of a player may be defined as the difference between how the player evaluates his bundle and an exact nth part of the whole according to his numerical evaluation of the whole. A “gain-maximin” criterion would maximize the minimum gain obtained by any player. We show that Knaster's procedure finds an allocation which is optimal under the gain-maximin criterion. That allocation is not necessarily envy-free, so we also find the envy-free allocation that is optimal under the gain-maximin criterion among all envy-free allocations. It turns out that, even though there exist allocations that are simultaneously envy-free and Pareto-optimal, this optimal allocation may fail to be Pareto-optimal, and it may also violate monotonicity criteria. Received: 30 September 1996/Accepted: 6 March 2002 The author would like to thank Professor William Thomson for a discussion on this subject; and he would like to thank the anonymous referees, who made many substantive suggestions for improving this paper – shortening it, streamlining the arguments, improving the terminology, making further ties with the literature, and improving the exposition.     

Strategic requirements with indifference: single-peaked versus single-plateaued preferences

We concentrate on the problem of the provision of one pure public good whenever agents that form the society have either single-plateaued preferences or single-peaked preferences over the set of alternatives. We are interested in comparing the relationships between different nonmanipulability notions under these two domains. On the single-peaked domain, under strategy-proofness, non-bossiness is equivalent to convexity of the range. Thus, minmax rules are the only strategy-proof non-bossy rules. On the single-plateaued domain, only constant rules are non-bossy or Maskin monotonic; but strategy-proofness and weak non-bossiness are equivalent to weak Maskin monotonicity. Moreover, strategy-proofness and plateau-invariance guarantee convexity of the range. We thank Salvador Barberà, Matthew Jackson, Bettina Klaus, Jordi Massó, John Weymark, and two anonymous referees and the Associate Editor for helpful comments and suggestions. We also thank the participants in the 3rd Workshop on Social Decisions that took place in Málaga in November 2007. Dolors Berga acknowledges the financial support by the Spanish Ministry of Education and Science through Research Grants SEJ2004-03276 and SEJ2007-60671 and also by the Generalitat de Catalunya through Research Grant 2005SGR-213 and the Barcelona Economics Program (CREA). Bernardo Moreno gratefully acknowledges financial support from Junta de Andalucía through grant SEJ522 and the Spanish Ministry of Science and Technology through grant SEC2005-04805.     

Costly voting when both information and preferences differ: is turnout too high or too low?

We study a model of costly voting over two alternatives, where agents’ preferences are determined by both (i) a private preference in favour of one alternative e.g. candidates’ policies, and (ii) heterogeneous information in the form of noisy signals about a commonly valued state of the world e.g. candidate competence. We show that depending on the level of the personal bias (weight on private preference), voting is either according to private preferences or according to signals. When voting takes place according to private preferences, there is an unique equilibrium with inefficiently high turnout. In contrast, when voting takes place according to signals, turnout is locally too low. Multiple Pareto-ranked voting equilibria may exist and in particular, compulsory voting may Pareto dominate voluntary voting. Moreover, an increase in personal bias can cause turnout to rise or fall, and an increase in the accuracy of information may cause a switch to voting on the basis of signals and thus lower turnout, even though it increases welfare. This is a substantially revised version of Department of Economics University of Warwick Working Paper 670, “Information Aggregation, Costly Voting and Common Values”, January 2003. We would like to thank B. Dutta, M. Morelli, C. Perrroni, V. Bhaskar and seminar participants at Warwick, Nottingham and the ESRC Workshop in Game Theory for their comments. We would also like to thank the editor and an anonymous referee for their comments.     

Welfarism,IIA and arrovian constitutional rules

Two contradictory views on the relationship between independence of irrelevant alternatives and welfarism have appeared in the social choice literature. I show that the contradictory views result from the use of two different forms of independence of irrelevant alternatives. It is shown that one form implies welfarism while the other does not.I would like to thank Peter Murrell, Yew-Kwang Ng, Amartya Sen and an anonymous referee for helpful comments on earlier drafts of this paper.     

Ranking opportunity sets: An approach based on the preference for flexibility

We describe a criterion to evaluate subsets of a finite set of alternatives which are considered as opportunity sets. The axioms for set comparison are motivated within the preference for flexibility framework. We assume the preference over the universal set of alternatives to be made of two disjoint binary relations. The result is the axiomatic characterization of a procedure which is formally similar to the leximax ordering, but in our case it incorporates the presence of some uncertainty about the decision-maker final tastes. Received: 20 January 1999/Accepted: 20 October 1999