Majority vote following a debate

Voters determine their preferences over alternatives based on cases (or arguments) that are raised in the public debate. Each voter is characterized by a matrix, measuring how much support each case lends to each alternative, and her ranking is additive in cases. We show that the majority vote in such a society can be any function from sets of cases to binary relations over alternatives. A similar result holds for voting with quota in the case of two alternatives.We wish to thank Enriqueta Aragones, Jean-Francois Laslier, Andrew Postlewaite, and David Schmeidler for the comments and discussions. We thank Lada Burde for her help in proofreading.     

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.     

Condorcet choice and the Ostrogorski paradox

The Ostrogorski paradox refers to the fact that, facing finitely many dichotomous issues, choosing issue-wise according to the majority rule may lead to a majority defeated overall outcome. This paper investigates the possibility for a similar paradox to occur under alternative specifications of the collective preference relation. The generalized Ostrogorski paradox occurs when the issue-wise majority rule leads to an outcome which is not maximal according to some binary relation φ defined over pairs of alternatives. We focus on three possible definitions of φ, whose sets of maximal elements are respectively the Uncovered Set, the Top-Cycle, and the Pareto Set. We prove that a generalized paradox may prevail for the Uncovered Set. Moreover, it may be avoided for the same issue-wise majority margins as for the Ostrogorski paradox. However, the issue-wise majority rule always selects a Pareto-optimal alternative in the Top-Cycle. Gilbert Laffond and Jean Lainé are grateful to two anonymous referees for their helpful comments and suggestions.     

Hansson’s theorem for generalized social welfare functions: an extension

 Hansson (1969) sets forth four conditions satisfied by no generalized social welfare function (GSWF), a mapping from profiles of individual preferences to arbitrary social preference relations. Though transitivity is not imposed on social preferences, one of Hansson’s conditions requires that socially maximal alternatives always exist. Of course, this condition is not satisfied by the majority GSWF. We prove a generalization of Hansson’s theorem that requires the existence of maximal alternatives only in very special cases. Our result applies to the majority GSWF and a large class of other GSWFs that sometimes produce no maximal alternatives. Received: 10 July 1995/Accepted: 4 March 1996     

A characterization of the single-peaked domain

We provide in this paper two properties that are both necessary and sufficient to characterize the domain of single-peaked preference profiles. This characterization allows for a definition of single-peaked preference profiles without using an ad hoc underlying order of the alternatives and also sheds light on the structure of single-peaked profiles. Considering the larger domain of value-restricted preference profiles (Sen, Econometrica 34:491–499, 1966) we also provide necessary and sufficient conditions for a preference profile to be single-caved or group-separable. Our results show that for single-peaked, single-caved and group-separable profiles it is sufficient to restrict to profiles containing of either three individuals and three alternatives or two individuals and four alternatives.     

Choosing from a large tournament

A tournament can be viewed as a majority preference relation without ties on a set of alternatives. In this way, voting rules based on majority comparisons are equivalent to methods of choosing from a tournament. We consider the size of several of these tournament solutions in tournaments with a large but finite number of alternatives. Our main result is that with probability approaching one, the top cycle set, the uncovered set, and the Banks set are equal to the entire set of alternatives in a randomly chosen large tournament. That is to say, each of these tournament solutions almost never rules out any of the alternatives under consideration. We also discuss some implications and limitations of this result.     

A Social Choice Lemma on Voting Over Lotteries with Applications to a Class of Dynamic Games

We prove a lemma characterizing majority preferences over lotteries on a subset of Euclidean space. Assuming voters have quadratic von Neumann–Morgenstern utility representations, and assuming existence of a majority undominated (or “core”) point, the core voter is decisive: one lottery is majority-preferred to another if and only if this is the preference of the core voter. Several applications of this result to dynamic voting games are discussed.This paper was completed after Jeff Banks’s death. John Duggan is deeply indebted to him for his friendship and his collaboration on this and many other projects.     

Modeling large electorates with Fourier series, with applications to Nash equilibria in proximity and directional models of spatial competition

 In this paper we introduce harmonic analysis (Fourier series) as a tool for characterizing the existence of Nash equilibria in two-dimensional spatial majority rule voting games with large electorates. We apply our methods both to traditional proximity models and to directional models. In the latter voters exhibit preferences over directions rather than over alternatives, per se. A directional equilibrium can be characterized as a Condorcet direction, in analogy to the Condorcet (majority) winner in the usual voting models, i.e., a direction which is preferred by a majority to (or at least is not beaten by) any other direction. We provide a parallel treatment of the total median condition for equilibrium under proximity voting and equilibrium conditions for directional voting that shows that the former result is in terms of a strict equality (a knife-edge result very unlikely to hold) while the latter is in terms of an inequality which is relatively easy to satisfy. For the Matthews [3] directional model and a variant of the Rabinowitz and Macdonald [7] directional model, we present a sufficiency condition for the existence of a Condorcet directional vector in terms of the odd-numbered components of the Fourier series representing the density distribution of the voter points. We interpret our theoretical results by looking at real-world voter distributions and direction fields among voter points derived from U.S. and Norwegian survey data. Received: 7 July 1995 / Accepted: 14 May 1996     

Maximal domains of preferences preserving strategy-proofness for generalized median voter schemes

We characterize the maximal sets of preferences under which generalized median voter schemes are strategy-proof. Those domains are defined by a qualified version of single-peakedness, which depends on the distribution of power among agents implied by each generalized median voter scheme. Received: 28 April 1997/Accepted: 30 January 1998     

Aggregation of coarse preferences

We consider weak preference orderings over a set A _n of n alternatives. An individual preference is of refinement?≤n if it first partitions A _n into ? subsets of `tied' alternatives, and then ranks these subsets within a linear ordering. When ?<n, preferences are coarse. It is shown that, if the refinement of preferences does not exceed ?, a super majority rule (within non-abstaining voters) with rate 1− 1/? is necessary and sufficient to rule out Condorcet cycles of any length. It is argued moreover how the coarser the individual preferences, (1) the smaller the rate of super majority necessary to rule out cycles `in probability'; (2) the more probable the pairwise comparisons of alternatives, for any given super majority rule. Received: 29 June 1999/Accepted: 25 February 2000     

On the likelihood of Condorcet's profiles

Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, w _j, to the alternative she ranks in j ^th position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P ^* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P ^* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases. Received: 30 April 1999/Accepted: 14 September 2000     

Social choice with analytic preferences

Arrow's axioms for social welfare functions are shown to be inconsistent when the set of alternatives is the nonnegative orthant in a multidimensional Euclidean space and preferences are assumed to be either the set of analytic classical economic preferences or the set of Euclidean spatial preferences. When either of these preference domains is combined with an agenda domain consisting of compact sets with nonempty interiors, strengthened versions of the Arrovian social choice correspondence axioms are shown to be consistent. To help establish the economic possibility theorem, an ordinal version of the Analytic Continuation Principle is developed. Received: 4 July 2000/Accepted: 2 April 2001     

Extended preferences and freedom of choice

The common choice theory in economics is based on the assumption that an individual is defined in terms of a binary preference relation. This preference relation is defined over alternatives without taking into account menu dependence and, in particular, freedom of choice or, more generally, the set that contains the alternatives. In this study we clarify the nature and the significance of freedom of choice which may positively or negatively affect the individual's welfare. Our proposed extended preference relation of the individual takes into account both the particular alternative and the opportunity set that he faces. This extended relation does not induce ranking of opportunity sets. Its restriction to a particular opportunity set is the paradigmatic preference relation and it can capture the dependence of preferences on freedom of choice. Our main result establishes the inconsistency between dependence of extended preferences on freedom of choice and the existence of a utility that represents the paradigmatic preference relation and any of its restrictions. Received: 30 December 1997/Accepted: 6 September 1999     

A necessary and sufficient condition for stable matching rules to be strategy-proof

We study one-to-one matching problems and analyze conditions on preference domains that admit the existence of stable and strategy-proof rules. In this context, when a preference domain is unrestricted, it is known that no stable rule is strategy-proof. We introduce the notion of the no-detour condition, and show that under this condition, there is a stable and group strategy-proof rule. In addition, we show that when the men’s preference domain is unrestricted, the no-detour condition is also a necessary condition for the existence of stable and strategy-proof rules. As a result, under the assumption that the men’s preference domain is unrestricted, the following three statements are equivalent: (i) a preference domain satisfies the no-detour condition, (ii) there is a stable and group strategy-proof rule, (iii) there is a stable and strategy-proof rule.     

Nash implementation via hyperfunctions

Hyperfunctions are social choice rules which assign sets of alternatives to preference profiles over sets. Therefore, they are more general objects compared to standard (social choice) correspondences. In fact, every correspondence can be expressed in terms of an equivalent hyperfunction. We give a partial characterization of Nash-implementable hyperfunctions and explore the conditions under which correspondences have Nash-implementable equivalent hyperfunctions. While the strength of these conditions depends on the axioms used to extend preferences over alternatives to sets, they are at most as strong as the conjunction of Maskin monotonicity with the no veto power condition. Thus, our approach expands the set of Nash-implementable social choice rules. In fact, social choice rules such as the majority rule and the top cycle are Nash-implementable through their equivalent hyperfunctions, while they are not Maskin-monotonic, and thus, not Nash-implementable in the standard framework.

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Social choice by majority rule with rational participation

This paper explores the social choice properties of majority rule when voters rationally participate in elections. Employing the basic model of voter behavior developed originally by Ledyard (1984) and Palfrey and Rosenthal (1985), we show that when the electorate is sufficiently large, a participation equilibrium exists and is unique. The main result of the paper shows that under these conditions the social preference ordering induced by majority rule is identical to that given by the expected utility of a randomly selected voter, implying the existence of a Condorcet point in the proposal space. A final section provides intuition for the main theorems and relates the equilibrium of this majority rule game to the median voter result.     

Condorcet efficiencies under the maximal culture condition

The Condorcet winner in an election is a candidate that could defeat each other candidate in a series of pairwise majority rule elections. The Condorcet efficiency of a voting rule is the conditional probability that the voting rule will elect the Condorcet winner, given that such a winner exists. The study considers the Condorcet efficiency of basic voting rules under various assumptions about how voter preference rankings are obtained. Particular attention is given to situations in which the maximal culture condition is used as a basis for obtaining voter preferences. Received: 4 February 1998/Accepted: 13 April 1998     

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     

Dictatorial domains in preference aggregation

We call a domain of preference orderings “dictatorial” if there exists no Arrovian (Pareto optimal, IIA and non-dictatorial) social welfare function defined over that domain. In a finite world of alternatives where indifferences are ruled out, we identify a condition which implies the dictatoriality of a domain. This condition, to which we refer as “being essentially saturated”, is fairly weak. In fact, independent of the number of alternatives, there exists an essentially saturated (hence dictatorial) domain which consists of precisely six orderings. Moreover, this domain exhibits the superdictatoriality property, i.e., every superdomain of it is also dictatorial. Thus, given m alternatives, the ratio of the size of a superdictatorial domain to the size of the full domain may be as small as 6/m!, converging to zero as m increases.     

Sophisticated voting rules: the case of two tournaments

A voting rule maps voter preferences into outcomes, and is called sophisticated if there exists a voting tree whose sophisticated outcomes coincide with the voting rule for every voter preference. As yet, no complete characterization of such rules is available. In this paper, we provide an important step toward this characterization by completely solving the problem when there are two possible sets of voter preferences.The second author was supported by the Office of Naval Research, Grant N00014-92-J-1387.