Unequivocal majority and Maskin-monotonicity

The unequivocal majority of a social choice rule is a number of agents such that whenever at least this many agents agree on the top alternative, then this alternative (and only this) is chosen. The smaller the unequivocal majority is, the closer it is to the standard (and accepted) majority concept. The question is how small can the unequivocal majority be and still permit the Nash-implementability of the social choice rule; i.e., its Maskin-monotonicity. We show that the smallest unequivocal majority compatible with Maskin-monotonicity is n- ë \fracn-1m û{n-\left\lfloor \frac{n-1}{m} \right\rfloor} , where n ≥ 3 is the number of agents and m ≥ 3 is the number of alternatives. This value is equal to the minimal number required for a majority to ensure the non-existence of cycles in pairwise comparisons. Our result has a twofold implication: (1) there is no Condorcet consistent social choice rule satisfying Maskin-monotonicity and (2) a social choice rule satisfies k-Condorcet consistency and Maskin-monotonicity if and only if k 3 n- ë \fracn-1m û{k\geq n-\left\lfloor \frac{n-1}{m}\right\rfloor}.     

Choice rules with fuzzy preferences: Some characterizations

Consider an agent with fuzzy preferences. This agent, however, has to make exact choices when faced with different feasible sets of alternatives. What rule does he follow in making such choices? This paper provides an axiomatic characterization of a class of binary choice rules called the α satisfying rule. When α=1, this rule is the Orlovsky choice rule. On the other hand, for α≤1/2, the rule coincides with the M _α rule that has been extensively analyzed in the literature on fuzzy preferences. Received: 3 August 1995/Accepted: 19 November 1997     

Strategy-proof allocation of multiple public goods

This paper characterizes strategy-proof social choice functions (SCFs), the outcome of which are multiple public goods. Feasible alternatives belong to subsets of a product set . The SCFs are not necessarily “onto”, but the weaker requirement, that every element in each category of public goods A _k is attained at some preference profile, is imposed instead. Admissible preferences are arbitrary rankings of the goods in the various categories, while a separability restriction concerning preferences among the various categories is assumed. It is found that the range of the SCF is uniquely decomposed into a product set in general coarser than the original product set, and that the SCF must be dictatorial on each component B _l . If the range cannot be decomposed, a form of the Gibbard–Satterthwaite theorem with a restricted preference domain is obtained.     

Acyclic sets of linear orders

 A set of linear orders on {1,2, ℕ, n} is acyclic if no three of its orders have an embedded permutation 3-cycle {abc, cab, bca}. Let f (n) be the maximum cardinality of an acyclic set of linear orders on {1,2, ℕ, n}. The problem of determining f (n) has interested social choice theorists for many years because it is the greatest number of linear orders on a set of n alternatives that guarantees transitivity of majority preferences when every voter in an arbitrary finite set has any one of those orders as his or her preference order. This paper gives improved lower and upper bounds for f (n). We note that f (5)=20 and that all maximum acyclic sets at n=4, 5 are generated by an “alternating scheme.” This procedure becomes suboptimal at least by n=16, where a “replacement scheme” overtakes it. The presently-best large-n lower bound is approximately f (n)≥(2.1708) ⁿ . Received: 5 April 1995/Accepted: 10 November 1995     

Existence of a multicameral core

When a single group uses majority rule to select a set of policies from an n-dimensional compact and convex set, a core generally exists if and only if n = 1. Finding analogous conditions for core existence when an n-dimensional action requires agreement from m groups has been an open problem. This paper provides a solution to this problem by establishing sufficient conditions for core existence and characterizing the location and dimensionality of the core for settings in which voters have Euclidean preferences. The conditions establish that a core may exist in any number of dimensions whenever n ≤ m as long as there is sufficient preference homogeneity within groups and heterogeneity between groups. With m > 1 the core is however generically empty for . These results provide a generalization of the median voter theorem and of non-existence results for contexts of concern to students of multiparty negotiation, comparative politics and international relations.     

Free triples,large indifference classes and the majority rule

We study classes of voting situations where agents may exhibit a systematic inability to distinguish between the elements of certain sets of alternatives. These sets of alternatives may differ from voter to voter, thus resulting in personalized families of preferences. We study the properties of the majority relation when defined on restricted domains that are the cartesian product of preference families, each one reflecting the corresponding agent’s objective indifferences, and otherwise allowing for all possible (strict) preference relations among alternatives. We present necessary and sufficient conditions on the preference domains of this type, guaranteeing that majority rule is quasi-transitive and thus the existence of Condorcet winners at any profile in the domain, and for any finite subset of alternatives. Finally, we compare our proposed restrictions with others in the literature, to conclude that they are independent of any previously discussed domain restriction.     

‘One and a Half Dimensional’ Preferences and Majority Rule

I examine a model of majority rule in which alternatives are described by two characteristics: (1) their position in a standard, left-right dimension, and (2) their position in a good-bad dimension, over which voters have identical preferences. I show that when voters’ preferences are single-peaked and concave over the first dimension, majority rule is transitive, and the majority’s preferences are identical to the median voter’s. Thus, Black’s (The theory of committees and elections, 1958) theorem extends to such a “one and a half” dimensional framework. Meanwhile, another well-known result of majority rule, Downs’ (An economic theory of democracy, 1957) electoral competition model, does not extend to the framework. The condition that preferences can be represented in a one-and-a-half-dimensional framework is strictly weaker than the condition that preferences be single-peaked and symmetric. The condition is strictly stronger than the condition that preferences be order-restricted, as defined by Rothstein (Soc Choice Welf 7:331–342;1990).     

Topological aggregation of preferences: the case of a continuum of agents

 This paper studies the topological approach to social choice theory initiated by G. Chichilnisky (1980), extending it to the case of a continuum of agents. The social choice rules are continuous anonymous maps defined on preference spaces which respect unanimity. We establish that a social choice rule exists for a continuum of agents if and only if the space of preferences is contractible. We provide also a topological characterization of such rules as generalized means or mathematical expectations of individual preferences. Received: 30 November 1994/Accepted: 22 April 1996     

The C1 topology on the space of smooth preference profiles

This paper defines a fine C ¹-topology for smooth preferences on a “policy space”, W, and shows that the set of convex preference profiles contains open sets in this topology.  It follows that if the dimension(W)≤v(?)−2 (where v(?) is the Nakamura number of the voting rule, ?), then the core of ? cannot be generically empty. For higher dimensions, an “extension” of the voting core, called the heart of ?, is proposed. The heart is a generalization of the “uncovered set”. It is shown to be non-empty and closed in general. On the C ¹-space of convex preference profiles, the heart is Paretian. Moreover, the heart correspondence is lower hemi-continuous and admits a continuous selection. Thus the heart converges to the core when the latter exists. Using this, an aggregator, compatible with ?, can be defined and shown to be continuous on the C ¹-space of smooth convex preference profiles. Received: 3 April 1995/Accepted: 8 April 1998     

The replacement principle for the provision of multiple public goods on tree networks

This article considers the provision of two public goods on tree networks where each agent has a single-peaked preference. We show that if there are at least four agents, then no social choice rule exists that satisfies efficiency and replacement-domination. In fact, these properties are incompatible, even if agents’ preferences are restricted to a smaller domain of symmetric single-peaked preferences. However, for rules on an interval, we prove that Miyagawa’s (Soc Choice Welf 18:527–541, 2001) characterization that only the left-peaks rule and the right-peaks rule satisfy both of these properties also holds on the domain of symmetric single-peaked preferences. Moreover, if agents’ peak locations are restricted to either the nodes or the endpoints of trees, rules exist on a subclass of trees. We provide a characterization of a family of such rules for this tree subclass.     

Ranking opportunity sets on the basis of their freedom of choice and their ability to satisfy preferences: A difficulty

This paper examines a possibility of enlarging the domain of definition of individual preferences suggested by the recent literature on freedom of choice. More specifically, the possibility for an individual to have preferences that depend upon both the opportunity set that she faces and the particular alternative that she chooses from that set is considered. Even more specifically, the possibility for these preferences to value freedom of choice, as defined by the set theoretic relation of inclusion, while being consistent, in a certain sense, with the existence of a preference ordering over the options contained in opportunity sets is investigated. It is shown in the paper that a necessary condition for the existence of any transitive extended preferences of this type is for freedom of choice to be given no intrinsic importance. Received: 22 November 1995 / Accepted: 11 January 1997     

Special majority rules necessary and sufficient condition for quasi-transitivity with quasi-transitive individual preferences

A condition on preferences called strict Latin Square partial agreement is introduced and is shown to be necessary and sufficient for quasi-transitivity of the social weak preference relation generated by any special majority rule, under the assumption that individual preferences themselves are quasi-transitive.     

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     

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.     

Identification of domain restrictions over which acyclic, continuous-valued, and positive responsive social choice rules operate

We consider a social choice problem in various economic environments consisting of n individuals, 4≤n<+∞, each of which is supposed to have classical preferences. A social choice rule is a function associating with each profile of individual preferences a social preference that is assumed to be complete, continuous and acyclic over the alternatives set. The class of social choice rules we deal with is supposed to satisfy the two conditions; binary independence and positive responsiveness. A new domain restriction for the social choice rules is proposed and called the classical domain that is weaker than the free triple domain and holds for almost all economic environments such as economies with private and/or public goods. In this paper we explore what type of classical domain that admits at least one social choice rule satisfying the mentioned conditions to well operate over the domain. The results we obtained are very negative: For any classical domain admitting at least one social choice rule to well operate, the domain consists only of just one profile.     

On relative egalitarianism

We reconsider the problem of aggregating individual preference orderings into a single social ordering when alternatives are lotteries and individual preferences are of the von Neumann–Morgenstern type. Relative egalitarianism ranks alternatives by applying the leximin ordering to the distributions of 0–1 normalized utilities they generate. We propose an axiomatic characterization of this aggregation rule.     

The structure of fuzzy preferences: Social choice implications

It has been shown that, with an alternative factorization of fuzzy weak preferences into symmetric and antisymmetric components, one can prove a fuzzy analogue of Arrow's Impossibility Theorem even when the transitivity requirements on individual and social preferences are very weak. It is demonstrated here that the use of this specification of strict preference, however, requires preferences to also be strongly connected. In the absence of strong connectedness, another factorization of fuzzy weak preferences is indicated, for which nondictatorial fuzzy aggregation rules satisfying the weak transitivity requirement can still be found. On the other hand, if strong connectedness is assumed, the fuzzy version of Arrow's Theorem still holds for a variety of weak preference factorizations, even if the transitivity condition is weakened to its absolute minimum. Since Arrow's Impossibility Theorem appeared nearly half a century ago, researchers have been attempting to avoid Arrow's negative result by relaxing various of his original assumptions. One approach has been to allow preferences – those of individuals and society or just those of society alone – to be “fuzzy.” In particular, Dutta [4] has shown that, to a limited extent, one can avoid the impossibility result (or, more precisely, the dictatorship result) by using fuzzy preferences, employing a particularly weak version of transitivity among the many plausible (but still distinct) definitions of transitivity that are available for fuzzy preferences. Another aspect of exact preferences for which the extension to the more general realm of fuzzy preferences is ambiguous is the factorization of a weak preference relation into a symmetric component (indifference) and an antisymmetric component (strict preference). There are several ways to do this for fuzzy weak preferences, all of them equivalent to the traditional factorization in the special case when preferences are exact, but quite different from each other when preferences are fuzzy (see, for example, [3]). A recent paper in this journal [1], by A. Banerjee, argues that the choice of definitions for indifference and strict preference, given a fuzzy weak preference, can also have “Arrovian” implications. In particular, [1] claims that Dutta's version of strict preference presents certain intuitive difficulties and recommends a different version, with its own axiomatic derivation, for which the dictatorship results reappear even with Dutta's weak version of transitivity. However, the conditions used to derive [1]'s version of strict preference imply a restriction on how fuzzy the original weak preference can be, namely, that the fuzzy weak preference relation must be strongly connected. Without this restriction, I will show that the rest of [1]'s conditions imply yet a third version of strict preference, for which Dutta's possibility result under weak transitivity still holds. On the other hand, if one accepts the strong connectedness required in order for it to be valid, I show that [1]'s dictatorship theorem can in fact be strengthened to cover any version of transitivity for fuzzy preferences, no matter how weak, and further, that this dictatorship result holds for any “regular” formulation of strict preference, including the one originally used by Dutta. Received: 13 May 1996 / Accepted: 13 January 1997     

The geometry of implementation: a necessary and sufficient condition for straightforward games

 We characterize games which induce truthful revelation of the players’ preferences, either as dominant strategies (straightforward games) or in Nash equilibria. Strategies are statements of individual preferences on R ⁿ . Outcomes are social preferences. Preferences over outcomes are defined by a distance from a bliss point. We prove that g is straightforward if and only if g is locally constant or dictatorial (LCD), i.e., coordinate-wise either a constant or a projection map locally for almost all strategy profiles. We also establish that: (i) If a game is straightforward and respects unanimity, then the map g must be continuous, (ii) Straightforwardness is a nowhere dense property, (iii) There exist differentiable straightforward games which are non-dictatorial. (iv) If a social choice rule is Nash implementable, then it is straightforward and locally constant or dictatorial. Received: 30 December 1994/Accepted: 22 April 1996     

A pedagogical proof of Arrow's Impossibility Theorem

In this note I consider a simple proof of Arrow's Impossibility Theorem (Arrow 1963). I start with the case of three individuals who have preferences on three alternatives. In this special case there are 13³=2197 possible combinations of the three individuals' rational preferences. However, by considering the subset of linear preferences, and employing the full strength of the IIA axiom, I reduce the number of cases necessary to completely describe the SWF to a small number, allowing an elementary proof suitable for most undergraduate students.  This special case conveys the nature of Arrow's result. It is well known that the restriction to three options is not really limiting (any larger set of alternatives can be broken down into triplets, and any inconsistency within a triplet implies an inconsistency on the larger set). However, the general case of n≥3 individuals can be easily considered in this framework, by building on the proof of the simpler case. I hope that a motivated student, having mastered the simple case of three individuals, will find this extension approachable and rewarding.  This approach can be compared with the traditional simple proofs of Barberà (1980); Blau (1972); Denicolò (1996); Fishburn (1970); Kelly (1988); Mueller (1989); Riker and Ordeshook (1973); Sen (1979, 1986); Suzumura (1988), and Taylor (1995). Received: 5 January 1999/Accepted: 10 December 1999