Collective choice rules and collective rationality: a unified method of characterizations

The purpose of this paper is to investigate the relationship between collective rationality and permissible collective choice rules using a unified approach inspired by Bossert and Suzumura (J Econ Theory 138:311–320, 2008). We consider collective choice rules satisfying four axioms: unrestricted domain, strong Pareto, anonymity, and neutrality. A number of new classes of collective choice rules as well as the Pareto and Pareto extension rules are characterized under various concepts of collective rationality: acyclicity, transitivity, quasi-transitivity, semi-transitivity, and the interval order property. Further, new concepts of collective rationality, K-term acyclicity and K-term consistency, are proposed and the corresponding characterizations are provided.     

Social choice with infinite populations: construction of a rule and impossibility results

 We provide a simple construction of social choice rules for economies with infinite populations. The rules are continuous, Pareto and non-dictatorial; they are constructed as limits of individual preferences when the limit exists, and otherwise as adequate generalizations. This contrasts with the impossibility results of Arrow (1951) and Chichilnisky (1980), which are valid on economies with finitely many individuals. Our social choice rules are, however, limits of dictatorial rules. This paper was written in 1979. Received: 30 October 1994/Accepted: 22 April 1996     

Quasi-transitive and Suzumura consistent relations

We examine properties of binary relations that complement quasi-transitivity and Suzumura consistency in the sense that they, in conjunction with the original axiom(s), are equivalent to transitivity. In general, the conjunction of quasi-transitivity and Suzumura consistency is weaker than transitivity but in the case of collective choice rules that satisfy further properties, this conjunction implies transitivity of the social relation. We prove this observation by characterizing the Pareto rule as the only collective choice rule such that collective preference relations are quasi-transitive and Suzumura consistent, and standard social choice axioms are satisfied.     

Characterization of social choice sets in terms of individuals' maximal sets: The fixed agenda framework

In this paper we analyze the explicit representation of fixed agenda social choice correspondences under different rationality assumptions (independence, Pareto optimality, etc.). It is well known that, under some of these assumptions, the existence of dictators, oligarchies or individuals with veto power can be proven ([6] and [10]); but only a partial characterization of the social choice set is obtained. We establish a relationship between the social choice set and the individuals' maximal sets, which explicitly describes a fixed agenda social choice correspondence that satisfies these rationality assumptions. Received: 30 December 1997/Accepted: 20 December 1999     

Coalitionally strategy-proof social choice correspondences and the Pareto rule

This paper studies coalitional strategy-proofness of social choice correspondences that map preference profiles into sets of alternatives. In particular, we focus on the Pareto rule, which associates the set of Pareto optimal alternatives with each preference profile, and examine whether or not there is a necessary connection between coalitional strategy-proofness and Pareto optimality. The definition of coalitional strategy-proofness is given on the basis of a max–min criterion. We show that the Pareto rule is coalitionally strategy-proof in this sense. Moreover, we prove that given an arbitrary social choice correspondence satisfying the coalitional strategy-proofness and nonimposition, all alternatives selected by the correspondence are Pareto optimal. These two results imply that the Pareto rule is the maximal correspondence in the class of coalitionally strategy-proof and nonimposed social choice correspondences.     

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 ordinal egalitarian bargaining solution for finite choice sets

Rubinstein et al. (Econometrica 60:1171–1186, 1992) introduced the Ordinal Nash Bargaining Solution. They prove that Pareto optimality, ordinal invariance, ordinal symmetry, and IIA characterize this solution. A feature of their work is that attention is restricted to a domain of social choice problems with an infinite set of basic allocations. We introduce an alternative approach to solving finite social choice problems using a new notion called the Ordinal Egalitarian (OE) bargaining solution. This suggests the middle ranked allocation (or a lottery over the two middle ranked allocations) of the Pareto set as an outcome. We show that the OE solution is characterized by weak credible optimality, ordinal symmetry and independence of redundant alternatives. We conclude by arguing that what allows us to make progress on this problem is that with finite choice sets, the counting metric is a natural and fully ordinal way to measure gains and losses to agents seeking to solve bargaining problems.     

Partially efficient voting by committees

On the separable preference domain, voting by committees is the only class of voting rules that satisfy strategy-proofness and unanimity, and dictatorial rules are the only ones that are strategy-proof and Pareto efficient. To fill the gap, we define a sequence of efficiency conditions. We prove that for strategy-proof rules on the separable preference domain, the various notions of efficiency reduce to three: unanimity, partial efficiency, and Pareto efficiency. We also show that on the domain, strategy-proofness and partial efficiency characterize the class of voting rules represented as simple games which are independent of objects, proper and strong. We call such rules voting by stable committee.The author is deeply indebted to William Thomson for many helpful discussions on an earlier draft. The current version is greatly benefited from detailed comments of an anonymous referee. Thanks are also due to Jeffrey Banks, Salvador Barberà, Marcus Berliant, Ryo-ichi Nagahisa, Takehiko Yamato, and participants in a seminar at Rochester in 1992, the 1992 Midwest Conference at Michigan State, and the 1993 Summer Meeting of Econometric Society at Boston University for conversations and suggestions.     

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.     

Pareto models for top incomes and wealth

Top incomes are often related to Pareto distribution. To date, economists have mostly used Pareto Type I distribution to model the upper tail of income and wealth distribution. It is a parametric distribution, with interesting properties, that can be easily linked to economic theory. In this paper, we first show that modeling top incomes with Pareto Type I distribution can lead to biased estimation of inequality, even with millions of observations. Then, we show that the Generalized Pareto distribution and, even more, the Extended Pareto distribution, are much less sensitive to the choice of the threshold. Thus, they can provide more reliable results. We discuss different types of bias that could be encountered in empirical studies and, we provide some guidance for practice. To illustrate, two applications are investigated, on the distribution of income in South Africa in 2012 and on the distribution of wealth in the United States in 2013.

     

Public information and social choice

We examine the value of public information when a society uses a social choice rule to decide among a set of outcomes. We require that a social choice function satisfies unrestricted domain, non-decisiveness and the Pareto principle. We show that there exist payoff structures for every social choice function, such that an arbitrary subset of voters is worse off by public information. We apply the proposition to collective information acquisition and to irreversible investments. Received: 2 June 1997/Accepted: 30 September 1998     

Reference functions and possibility theorems for cardinal social choice problems

 In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions. Received: 8 August 1994/Accepted: 12 February 1996     

A characterization of the Walras rule

This paper has three purposes. First, we refine the characterization of the Walras rule proposed by Nagahisa (JET 1991) over a more natural and simple domain than the one he employed. We show that the Walras rule is the only social choice rule defined over the domain and satisfying Individual Rationality, Pareto Efficiency, and Local Independence. Second, assuming endowments to be collectively owned, we show that the Walras rule operated from equal division is the only social choice rule satisfying No Envy, Pareto Efficiency, and Local Independence. Third, we show that for every social choice rule satisfying Individual Rationality and Pareto Efficiency, Local Independence is equivalent to a condition of Nash implementation with a game form satisfying convexity.This article is a revised version of Toyama University Working Paper No. 141. We are grateful to Professors William Thomson, Shinsuke Nakamura, Tomoichi Shinotsuka and two anonymous referees for their detailed comments. Nagahisa is grateful for hospitality of the economics department of the University of Rochester.     

Minimal manipulability: anonymity and unanimity

This paper is concerned with the minimal number of profiles at which a unanimous and anonymous social choice function for three alternatives is manipulable. The lower bound is derived and examples of social choice functions attaining the lower bound are given. It is conjectured that these social choice functions are in fact all minimally manipulable social choice functions. Since some of these social choice functions are Pareto optimal, it follows that the lower bound also holds for Pareto optimal and anonymous social choice functions. Some of the minimally manipulable Pareto optimal and anonymous social choice functions can be interpreted as status quo voting.     

Constrained allocation problems with single-peaked preferences: An axiomatic analysis

We introduce a new class of problems that contains two existing classes: allocation problems with single-peaked preferences and bankruptcy problems. On this class, we analyze the implications of well-known properties such as Pareto optimality, strategy-proofness, resource-monotonicity, no-envy, equal treatment of equals, and two new properties we introduce, hierarchical no-envy and independence of nonbinding constraints. Unlike earlier literature, we consider rules that allow free-disposability. We present characterizations of a rule we introduce on this domain. We relate this rule to well-known rules on the aforementioned subdomains. Based on this relation, we present a characterization of a well-known bankruptcy rule called the constrained equal awards rule. Received: 22 June 2000/Accepted: 21 March 2002 This paper is based on the first chapter of my Ph.D. thesis submitted to the University of Rochester. I wish to thank my advisor, William Thomson, for helpful comments and suggestions.     

Multi-profile intergenerational social choice

In an infinite-horizon setting, Ferejohn and Page showed that any social welfare function satisfying Arrow’s axioms and stationarity must be a dictatorship of the first generation. Packel strengthened this result by proving that no collective choice rule generating complete social preferences can satisfy unlimited domain, weak Pareto and stationarity. We prove that this impossibility survives under a domain restriction and without completeness. We propose an alternative stationarity axiom and show that a social welfare function on a specific domain satisfies this modified version and some standard social choice axioms if and only if it is a chronological dictatorship.     

A note on the structure of stochastic social choice functions

This note sharpens the result of Nandeibam (J Econ Theory 68:212–233, 1996). We show that a stochastic social choice function which satisfies regularity, independence of irrelevant alternatives and weak ex-post Pareto optimality is essentially a weak random dictatorship.     

A note on restricted majority rules: invariance to rule selection and outcome distinctiveness

Recently, Karotkin (1993) has shown that in the symmetric uncertain dichotomous choice model the set of restricted majority rules (RMRs) is special in the sense that a member of this family of rules is always the worst rule among the potentially optimal weighted majority rules (WMRs). In the current paper we establish two additional special properties of RMRs. First, given a particular configuration of the group members' decisions, the collective choice is invariant to the selection of WMRs if it is invariant to the selection of RMRs. Second, given a particular decision profile, a potentially optimal WMR can result in a distinctive collective choice which is different from the choice of any other potentially optimal WMR, if and only if it is a RMR.     

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.     

The proximity condition

We investigate the social choice implications of what we call “the proximity condition”. Loosely speaking, this condition says that whenever a profile moves “closer” to some individual’s point of view, then the social choice cannot move “further away” from this individual’s point of view. We apply this idea in two settings: merging functions and preference aggregation. The precise formulation of the proximity condition depends on the setting. First, restricting attention to merging functions that are interval scale invariant, we prove that the only functions that satisfy proximity are dictatorships. Second, we prove that the only social welfare functions that satisfy proximity and a version of the Pareto criterion are dictatorships. We conclude that either proximity is not an attractive normative requirement after all, or we must give up some other social choice condition. Another possibility is that our normative intuition about proximity needs to be codified using different axioms.