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     

Strategy-proofness on Euclidean spaces

 In this paper we characterize strategy-proof voting schemes on Euclidean spaces. A voting scheme is strategy-proof whenever it is optimal for every agent to report his best alternative. Here the individual preferences underlying these best choices are separable and quadratic. It turns out that a voting scheme is strategy-proof if and only if (α) its range is a closed Cartesian subset of Euclidean space, (β) the outcomes are at a minimal distance to the outcome under a specific coordinatewise veto voting scheme, and (γ) it satisfies some monotonicity properties. Neither continuity nor decomposability is implied by strategy-proofness, but these are satisfied if we additionally impose Pareto-optimality or unanimity. Received: 18 October 1993/Accepted: 2 February 1996     

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.     

Weighting the transitivity of undirected weighted social networks with triadic edge dissimilarity scores

When an individual is socially connected to two others, the resulting triplet can be closed (if the two social partners are themselves connected) or open (if they are not connected). The proportion of closed triplets, referred to as the binary network transitivity, is a classic measure of the level of interconnectedness of a social network. However, in any given triplet, if the closing link is weak, or indeed if any of the links in the triplet is weak, then the triplet should not contribute as much to network transitivity as if all three links were equally strong. I propose two ways to weight the contribution of each triplet according to the dissimilarity between the three links in the triplet. Empirically, the resulting new metrics conveyed information not picked up by any other network-level metric. I envision that this approach could prove useful in studies of triadic mechanisms, i.e., situations where pre-existing social ties influence the interactions with third parties. These metrics could also serve as repeatable synthetic variables that summarize information about the variability of the strength of social connections.     

On strategy-proof social choice under categorization

I consider social choice problems such that (i) the set of alternatives can be partitioned into categories based on a prominent and objective feature and (ii) agents have strict preferences over the alternatives. Main results are characterizations of the structure of the strategy-proof social choice functions. I prove that each social choice function is strategy-proof if and only if it is decomposable into “small” strategy-proof social choice functions; one of them chooses one category and each of the others chooses one alternative from a category.     

Special majority rules and the existence of voting equilibria

This paper is concerned with voting rules in which one alternative defeats a second alternative if and only if it is preferred to the second alternative by a pre-specified proportion of the individuals who have strict preferences on the pair. In particular, the paper focuses on the potential of two alternative lower bounds (for the proportion used) as tools for studying the existence of voting equilibria (i.e., for studying the existence of alternatives which cannot be defeated). It establishes that there are certain important contexts where one of the two bounds does not directly reveal whether any voting equilibria exist, but the other one does.     

General concepts of value restriction and preference majority

Regenwetter and Grofman [17] offer a probabilistic generalization of Sen's [25, 27] classic value restriction condition when individual preferences are linear orders. They provide necessary and sufficient conditions for transitive majority preferences on linear orders. They call these conditions net value restriction and net preference majority. We study parallel generalizations for general binary relations. In general, neither net value restriction nor net preference majority is necessary for transitive majority preferences. Net value restriction is sufficient for transitive strict majority preferences, but not sufficient for transitive weak majority preferences. Net majority is sufficient for transitive majorities only if the preference relation with a net majority is a weak order. An application of our results to four U.S. National Election Study data sets reveals, in each case, transitive majorities despite a violation of Sen's original value restriction condition. We thank the National Science Foundation for funding this collaborative research through NSF grants SBR 97-30076 to Regenwetter and SBR 97-30578 to Grofman and Marley. We are indebted to the Interuniversity Consortium for Political and Social Research (ICPSR) for access to the 1968, 1980, 1992 and 1996 U.S. National Election Study (NES) data. We thank Mark Berger for helping us with the necessary data extraction. We are grateful to the action editor and the referees for extensive and helpful comments. Most of this work was carried out while the first author was a faculty member at the Fuqua School of Business, Duke University, which has generously supported our collaboration. Marley was a fellow at the Hanse-Wissenschaftskolleg, Germany, during the paper's completion.     

Complete characterization of functions satisfying the conditions of Arrow’s theorem

Arrow??s theorem implies that a social welfare function satisfying Transitivity, the Weak Pareto Principle (Unanimity), and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are also allowed, a dictatorial social welfare function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions, since non-strict preferences of the dictator are not necessarily followed. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow??s theorem, in the case of non-strict preferences, does not provide a complete characterization of all social welfare functions satisfying Transitivity, the Weak Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow??s theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow??s and Wilson??s result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Weak Pareto Principle). Additionally, we derive formulae for the number of functions satisfying these conditions.     

A systematic approach to the construction of non-empty choice sets

Suppose a strict preference relation fails to possess maximal elements, so that a choice is not clearly defined. I propose to delete particular instances of strict preferences until the resulting relation satisfies one of a number of known regularity properties (transitivity, acyclicity, or negative transitivity), and to unify the choices generated by different orders of deletion. Removal of strict preferences until the subrelation is transitive yields a new solution with close connections to the “uncovered set” from the political science literature and the literature on tournaments. Weakening transitivity to acyclicity yields a new solution nested between the strong and weak top cycle sets. When the original preference relation admits no indifferences, this solution coincides with the familiar top cycle set. The set of alternatives generated by the restriction of negative transitivity is equivalent to the weak top cycle set.     

Positional Independence in Preference Aggregation: A Remark

In a recent article, Quesada (Soc Choice Welfare 20 363–370, 2003) showed that if, for strict preferences, a unique strongly unanimous Choice Function (CF) is used to aggregate preferences position-wise then the resulting Social Welfare Function (SWF) is dictatorial. We show that this result is still true if one allows the use of different choice functions, one for each position. We also show that allowing choice functions not to be unanimous leads to SCW that are perm-dictatorial or imposed. We also show a way to set a dual problem to the one exposed in Que03, namely only perm-dictatorial and imposed SWFs can be decomposed into several Dual Choice Functions.An anonymous referee is thanked for his useful remarks.     

Bayesian implementation in exchange economies with state dependent preferences and feasible sets

This paper considers Bayesian and Nash implementation in exchange economic environments with state dependent preferences and feasible sets. We fully characterize Bayesian implementability for both diffuse and non-diffuse information structures. We show that, in exchange economic environments with three or more individuals, a social choice set is Bayesian implementable if and only if closure, non-confiscatority, Bayesian monotonicity, and Bayesian incentive compatibility are satisfied. As such, it improves upon and contains as special cases previously known results about Nash and Bayesian implementation in exchange economic environments. We show that the individual rationality and continuity conditions, imposed in Hurwicz et al. [12], can be weakened to the non-confiscatority and can be dropped, respectively, for Nash implementation. Thus we also give a full characterization for Nash implementation when endowments and preferences are both unknown to the designer. Received: 4 March 1996 / Accepted: 8 September 1997     

Endogenous Voting Agendas

Existence of a “simple” pure strategy subgame perfect equilibrium is established in a model of endogenous agenda formation and sophisticated voting; upper hemicontinuity of simple equilibrium outcomes is demonstrated; and connections to the set of undominated, or “core,” alternatives are examined. In one dimension with single-peaked preferences, the simple equilibrium outcome is essentially unique and lies in the core, providing a game-theoretic foundation for the median voter theorem in terms of endogenous agenda setting. Existence of equilibrium relies on a general characterization of sophisticated voting outcomes in the presence of “majority-ties,” rather than the standard tie-breaking convention in voting subgames in favor of the alternative proposed later. The model is illustrated in a three-agent distributive politics setting, and it is shown there that the standard tie-breaking convention leads to non-existence of equilibrium.     

Iterative elimination of weakly dominated strategies in binary voting agendas with sequential voting

This paper describes the correspondences between the results given by backward induction (BI) and iterative elimination of weakly dominated strategies (IEWDS) in binary voting agendas with sequential voting. When the voters have strict preferences over all candidates, the strategies that survive IEWDS all select the unique candidate selected by the BI strategy profiles. But if some voters are indifferent, this result no longer holds. However, when there are only two candidates, it is possible to demonstrate strong relationships between the results given by BI and IEWDS, even when some voters have indifferences.     

The core of social choice problems with monotone preferences and a feasibility constraint

We consider collective choice with agents possessing strictly monotone, strictly convex and continuous preferences over a compact and convex constraint set contained in ₊^k . If it is non-empty the core will lie on the efficient boundary of the constraint set and any policy not in the core is beaten by some policy on the efficient boundary. It is possible to translate the collective choice problem on this efficient boundary to another social choice problem on a compact and convex subset of ₊^c (c<k) with strictly convex and continuous preferences. In this setting the dimensionality results in Banks (1995) and Saari (1997) apply to the dimensionality of the boundary of the constraint set (which is lower than the dimensionality of the choice space by at least one). If the constraint set is not convex then the translated lower dimensional problem does not necessarily involve strict convexity of preferences but the dimensionality of the problem is still lower. Broadly, the results show that the homogeneity afforded by strict monotonicity of preferences and a compact constraint set makes generic core emptyness slightly less common. One example of the results is that if preferences are strictly monotone and convex on ² then choice on a compact and convex constraint exhibits a version of the median voter theorem.I thank Donald Saari for helpful comments on an earlier version of this paper.     

Measurement of inequality: An attempt at unification and generalization

This paper reviews and extends the theory of ethical inequality indices. It presents a novel axiom (strict separability of social welfare orderings in rank-ordered subspaces). This axiom allows to provide joint characterizations of the most important inequality measures (Atkinson family, Kolm-Pollak family and Generalized Ginis) and of some new more general classes of indices. The whole derivation is based on weak assumptions. In an ordinal framework only continuity of the underlying ordering is required and no cardinal properties are employed.I thank two anonymous referees for helpful comments and suggestions.     

On the continuity of ethical social welfare orders on infinite utility streams

In this paper we study the extent to which ethical social welfare orders on infinite utility streams can be continuous. For a class of metrics, we show that ethical preferences can be continuous if and only if the continuity requirement is in terms of a metric which satisfies a simplex condition. This condition requires that the distance from the origin to the unit simplex in the space of utility streams be positive. We use this characterization result to establish that the metric used by Svensson (Econometrica 48:1251–1256, 1980) induces the smallest topology for which there exist continuous ethical preferences. We thank an associate editor and two referees of this journal for their comments and suggestions.     

Public goods and Arrovian social choice

A social welfare function satisfying Arrow's independence axiom is constant or authoritarian if it generates continuous and transitive social preferences over the space of allocations of public and private goods, and individual preferences have the classical economic properties. The social welfare function will be oligarchial if it generates continuous and quasitransitive social preferences and satisfies a weak version of the Pareto criterion in addition to the independence axiom.This work was supported by the National Sciences Foundation grant no. SES 9007953. I am grateful to anonymous referees whose suggestions led to several substantial improvements.     

A characterization of the uniform rule with several commodities and agents

We consider a problem of allocating infinitely divisible commodities among a group of agents. More specifically, there are several commodities to be allocated and agents have continuous, strictly convex, and separable preferences. We establish that a rule satisfies strategy-proofness, unanimity, weak symmetry, and nonbossiness if and only if it is the uniform rule. This result extends to the class of continuous, strictly convex, and multidimensional single-peaked preferences.     

Equity and the Vickrey allocation rule on general preference domains

We consider the problem of allocating multiple units of an indivisible good among a group of agents in which each agent demands at most one unit of the good and money payment or receipt is required. Under general preference domains that may contain non quasi-linear preferences, the Vickrey allocation rule is characterized by axioms for equity and continuity without use of efficiency: namely, the Vickrey rule is the only rule that satisfies strategy-proofness, weak envy-freeness for equals, non-imposition, and continuity of welfare.