Single-Crossing, Strategic Voting and the Median Choice Rule

This paper studies the strategic foundation of the Representative Voter Theorem (Rothstein in: Pub Choice 72:193–212, 1991), also called the “second version” of the Median Voter Theorem. As a by-product, it also considers the existence of strategy-proof social choice functions over the domain of single-crossing preferences. The main result shows that single-crossing constitutes a domain restriction over the real line that allows not only majority voting equilibria, but also non-manipulable choice rules. In particular, this is true for the median rule, which is found to be group strategic-proof over the full set of alternatives and over every nonempty subset. In addition, the paper also examines the relation between single-crossing and order-restriction. And it uses this relation together with the strategy-proofness of the median rule to prove that the outcome predicted by the Representative Voter Theorem can be implemented in dominant strategies through a simple mechanism. This mechanism is a two-stage voting procedure in which, first, individuals select a representative among themselves, and then the winner chooses a policy to be implemented by the planner.     

Manipulability measures of common social choice functions

All social choice functions are manipulable when more than two alternatives are available. I evaluate the manipulability of the Borda count, plurality rule, minimax set, and uncovered set. Four measures of manipulability are defined and computed stochastically for small numbers of agents and alternatives.  Social choice rules derived from the minimax and uncovered sets are found to be relatively immune to manipulation whether a sole manipulating agent has complete knowledge or absolutely no knowledge of the preferences of the others. The Borda rule is especially manipulable if the manipulating agent has complete knowledge of the others. Received: 5 January 1996/Accepted: 31 July 1998     

The majority rule with a chairman

For the case of two alternatives and a given finite set of at least three individuals, seven axioms are shown to characterize the rules that are either the relative majority rule or the relative majority in which a given individual, the chairman, can always break ties. An axiomatization of the relative majority rules with a chairman is suggested that holds for an even number of individuals and that, for an odd number of individuals, characterizes the rules that are either the relative majority rule or a relative majority rule with a chairman.     

How to choose a non-controversial list with k names

Barberà and Coelho (WP 264, CREA-Barcelona Economics, 2007) documented six screening rules associated with the rule of k names that are used by diferent institutions around the world. Here, we study whether these screening rules satisfy stability. A set is said to be a weak Condorcet set à la Gehrlein (Math Soc Sci 10:199–209) if no candidate in this set can be defeated by any candidate from outside the set on the basis of simple majority rule. We say that a screening rule is stable if it always selects a weak Condorcet set whenever such set exists. We show that all of the six procedures which are used in reality do violate stability if the voters do not act strategically. We then show that there are screening rules which satisfy stability. Finally, we provide two results that can explain the widespread use of unstable screening rules.     

Bargaining over a finite set of alternatives

We analyze bilateral bargaining over a finite set of alternatives. We look for "good" ordinal solutions to such problems and show that Unanimity Compromise and Rational Compromise are the only bargaining rules that satisfy a basic set of properties. We then extend our analysis to admit problems with countably infinite alternatives. We show that, on this class, no bargaining rule choosing finite subsets of alternatives can be neutral. When rephrased in the utility framework of Nash (1950), this implies that there is no ordinal bargaining rule that is finite-valued. Professor Sertel passed away on January 25, 2003.     

Limiting distributions for continuous state Markov voting models

This paper proves the existence of a stationary distribution for a class of Markov voting models. We assume that alternatives to replace the current status quo arise probabilistically, with the probability distribution at time t+1 having support set equal to the set of alternatives that defeat, according to some voting rule, the current status quo at time t. When preferences are based on Euclidean distance, it is shown that for a wide class of voting rules, a limiting distribution exists. For the special case of majority rule, not only does a limiting distribution always exist, but we obtain bounds for the concentration of the limiting distribution around a centrally located set. The implications are that under Markov voting models, small deviations from the conditions for a core point will still leave the limiting distribution quite concentrated around a generalized median point. Even though the majority relation is totally cyclic in such situations, our results show that such chaos is not probabilistically significant.We acknowledge the support of NSF Grants #SOC79-21588, SES-8106215 and SES-8106212.     

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     

Opportunity sets and individual well-being

 An opportunity set ranking rule assigns an ordering of opportunity sets to each individual utility function (defined on the universal set of alternatives) within the domain of this rule. Using an axiomatic approach, this paper characterizes a general class of opportunity set ranking rules which are based on the utilities associated with the elements of an opportunity set. It is argued that the addition of an alternative to a given opportunity set is not necessarily desirable in terms of overall well-being, and this position is reflected in replacing a commonly used monotonicity axiom with an alternative condition. Received: 15 May 1995/Accepted: 14 December 1995     

Self-selective social choice functions

It is not uncommon that a society facing a choice problem has also to choose the choice rule itself. In such situations, when information about voters’ preferences is complete, the voters’ preferences on alternatives induce voters’ preferences over the set of available voting rules. Such a setting immediately gives rise to a natural question concerning consistency between these two levels of choice. If a choice rule employed to resolve the society’s original choice problem does not choose itself, when it is also used for choosing the choice rule, then this phenomenon can be regarded as inconsistency of this choice rule as it rejects itself according to its own rationale. Koray (Econometrica 68: 981–995, 2000) proved that the only neutral, unanimous universally self-selective social choice functions are the dictatorial ones. Here we introduce to our society a constitution, which rules out inefficient social choice rules. When inefficient social choice rules become unavailable for comparison, the property of self-selectivity becomes more interesting and we show that some non-trivial self-selective social choice functions do exist. Under certain assumptions on the constitution we describe all of them.     

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     

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.     

Characterization of self-selective social choice functions on the tops-only domain

Self-selectivity is a new kind of consistency pertaining to social choice rules. It deals with the problem of whether a social choice rule selects itself from among other rival such rules when a society is also to choose the choice rule that it will employ in making its choice from a given set of alternatives. Koray [3] shows that a neutral and unanimous social choice function is universally self-selective if and only if it is dictatorial. In this paper, we confine the available social choice functions to the tops-only domain and examine whether such restriction allow us to escape the dictatoriality result. A neutral, unanimous, and tops-only social choice function, however, turns out to be self-selective relative to the tops-only domain if and only if it is top-monotonic, and thus again dictatorial. Received: 8 October 2001/Accepted: 4 June 2002     

Arrow's Theorem with a fixed feasible alternative

Arrow's Theorem, in its social choice function formulation, assumes that all nonempty finite subsets of the universal set of alternatives is potentially a feasible set. We demonstrate that the axioms in Arrow's Theorem, with weak Pareto strengthened to strong Pareto, are consistent if it is assumed that there is a prespecified alternative which is in every feasible set. We further show that if the collection of feasible sets consists of all subsets of alternatives containing a prespecified list of alternatives and if there are at least three additional alternatives not on this list, replacing nondictatorship by anonymity results in an impossibility theorem.Most of the research for this article was completed while we were participants in the Public Choice Institute held at Dalhousie University during the summer of 1984. We wish to record here our thanks to the Institute Director, E.F. McClennen, and its sponsors, the Council for Philosophical Studies, the U.S. National Science Foundation, and the Social Science and Humanities Research Council of Canada. We are grateful to our referees for their comments and the Center for Mathematical Studies in Economics and Management Science at Northwestern University, where Weymark was a visitor during 1985–86, for secretarial assistance.     

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 Average Minimum Size of a Manipulating Coalition

We study the asymptotic average minimum manipulating coalition size as a characteristic of quality of a voting rule and show its serious drawback. We suggest using the asymptotic average threshold coalition size instead. We prove that, in large electorates, the asymptotic average threshold coalition size is maximised among all scoring rules by the Borda rule when the number m of alternatives is 3 or 4, and by -approval voting when m ≥ 5.     

SYMMETRIC SCORING RULES AND A NEW CHARACTERIZATION OF THE BORDA COUNT

Young developed a classic axiomatization of the Borda rule almost 50 years ago. He proved it is the only voting rule satisfying the normative properties of decisiveness, neutrality, reinforcement, faithfulness and cancellation. Often overlooked is that the uniqueness of Borda applies only to variable populations. We present a different set of properties which only Borda satisfies when both the set of voters and the set of alternatives can vary. It is also shown Borda is the only scoring rule which will satisfy all of the new properties when the number of voters stays fixed. (JEL D71, D02, H00)     

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.     

On the social stability of coalitional property rights regimes

We present a model of coalitional property rights (CPR) regimes– regimes in which ownership of a good is attributable to coalitions of various sizes. Specifically, for each good, we define a legal structure that specifies the legal coalitions of individuals that share a communal claim to that good. Generally, each legal coalition may use exclusionary rules to allocate its holdings internally. These rules allow eligible subcoalitions to recontract by expropriating some fraction of the legal coalition's endowment. We then ask: what types of CPR regimes are socially stable in the sense of having a nonempty core? We give conditions on the legal structure and the primitives of the economy that achieve social stability in this sense. We emphasize two cases of particular interest. ( I ) Unanimity. Unanimity is required for a legal coalition to recontract against (block) the status quo. In this case, the core is nonempty under standard assumptions. Each agent's ability to veto an alternative allocation allows a partial characterization in terms of the economies that are privatized by dividing up the communal endowment among the members of each legal coalition. We show that in some economies' collective vs private ownership matters in terms of social stability. ( II ) Exclusion. Many eligible subcoalitions can expropriate the legal coalition's entire endowment. An example is the collection of simple majorities. The presence of cycles can easily lead to social instability. We show that if endowment holdings are sufficiently “specialized” and each agent's “veto power” sufficiently large, then stability can be achieved despite the presence of cycles in some goods. Received: 30 June 1993/Accepted: 28 February 1998     

Topological aggregation, the case of an infinite population

 The literature on infinite Chichilnisky rules considers two forms of anonymity: a weak and a strong. This note introduces a third form: bounded anonymity. It allows us to prove an infinite analogue of the “Chichilnisky– Heal-resolution” close to the original theorem: a compact parafinite CW-complex X admits a bounded anonymous infinite rule if and only if X is contractible. Furthermore, bounded anonymity is shown to be compatible with the finite and the [0, 1]-continuum version of anonymity and allows the construction of convex means in infinite populations. With X=[0, 1], the set of linear bounded anonymous rules coincides with the set of medial limits. Received: 30 October 1993/Accepted: 22 April 1996     

Stability of voting games

This paper generalizes the result of Le Breton and Salles (1990) about stable set (far-sighted core of order 1) for voting games to far-sighted core of arbitrary order. Let m be the number of alternatives, n be the number of voters and G(n,k) be a proper symmetric simple game in which the size of a winning coalition is greater or equal to k. It is shown that the far-sighted core of order d for G(n,k) is nonempty for all preference profiles and for all n and k with n/(n–k)=v ¹ iff m(d+1)(v–1).This paper is part of my dissertation. I am grateful to my thesis advisor Leonid Hurwicz for his guidance and encouragement. I would like to thank Edward Green, Lu Hong, James Jordan, Andrew McLennan, Herve Moulin and Marcel Richter for their very helpful suggestions. Especially a referee and Maurice Salles made many good comments. Of course, any errors that remain are the sole responsibility of the author.