Utility function of fuzzy preferences on a countable set under max-*-transitivity

We determine, by means of max-*-transitivity, necessary and sufficient conditions for a fuzzy binary relation R defined on a countable (finite or denumerable) set A to be representable by a utility function. We display one example of its application. The first author thanks AUF (Agence Universitaire de la Francophonie). This paper was revised when he was Visiting Researcher at CREM-University of Caen under the Research grant “Bourse Post-doctorale de la Francophonie 2005–2006”.     

Reconsidering two-agent Nash implementation

In this paper, we reconsider the full characterization of two-agent Nash implementation provided in the celebrated papers by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991), since we are able to show that the characterizing conditions are not logically independent. We prove that an amended version of the conditions proposed in these papers is still necessary and sufficient for Nash implementability. Then, by using our necessary and sufficient condition, we show that Maskin’s impossibility result can be avoided under restrictions on the outcomes and the domain of preferences much weaker than those previously imposed by Moore and Repullo (Econometrica 58:1083–1099, 1990) and Dutta and Sen (Rev Econ Stud 58:121–128, 1991).     

Market arbitrage, social choice and the core

 This paper establishes a clear connection between equilibrium theory, game theory and social choice theory by showing that, for a well defined social choice problem, a condition which is necessary and sufficient to solve this problem – limited arbitrage – is the same as the condition which is necessary and sufficient to establish the existence of an equilibrium and the core. The connection is strengthened by establishing that a market allocation, which is in the core, can always be realized as a social allocation, i.e. an allocation which is optimal according to an ordering chosen by a social choice rule. Limited arbitrage characterizes those economies without Condorcet triples, and those for which Arrow’s paradox can be resolved on choices of large utility values. Received: 30 December 1994/Accepted: 12 August 1996     

Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

“Subset voting” denotes a choice situation where one fixed set of choice alternatives (candidates, products) is offered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, “approval voting“ from political science, the “weak utility model” from mathematical psychology, and “social welfare orderings” from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sufficient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988). Received: 18 August 1995 / Accepted: 13 February 1997     

Bayesian utilitarianism and probability homogeneity

This paper investigates, within the axiomatic framework of Jeffrey-Bolker decision theory, two kinds of conditions and the relation between them: (1) The Utilitarian condition that social rankings of prospects be representable by an expected utility function that is a weighted sum of the expected utility functions representing individual rankings; and (2) Homogeneity conditions on the probabilities and preferences of individuals. In particular, we show that identity of individuals’ probabilities is necessary and sufficient for the Utilitarian condition to hold and that the homogeneity of individuals’ probabilities can be derived from a Pareto condition on the relation between individual and social rankings, provided that these rankings are separable in a particular sense.This paper has considerably benefited from comments by John Broome, Isaac Levi and an anonymous referee. Special thanks to Philippe Mongin who provided encouragement, help and careful criticism throughout the development of this paper.     

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.     

Nash Implementation with Lottery Mechanisms

Consider the problem of exact Nash Implementation of social choice correspondences. Define a lottery mechanism as a mechanism in which the planner can randomize on alternatives out of equilibrium while pure alternatives are always chosen in equilibrium. When preferences over alternatives are strict, we show that Maskin monotonicity (Maskin in Rev Econ stud 66: 23–38, 1999) is both necessary and sufficient for a social choice correspondence to be Nash implementable. We discuss how to relax the assumption of strict preferences. Next, we examine social choice correspondences with private components. Finally, we apply our method to the issue of voluntary implementation (Jackon and Palfrey in J Econ Theory 98: 1–25, 2001).I thank Toyo Sakai for his comments on a previous draft. I also thank two anonymous referees and an editor of this journal for helpful comments that improved this paper. A previous version circulated as “A note on Maskin monotonicity”. After the results presented here were obtained, I became aware of a new unpublished paper by Benoit and Ok (2004). The result of Theorem 2 and the discussion that follows is partially similar to their Theorem 1.     

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.     

Stochastic Dominance and Absolute Risk Aversion

In this paper, we propose the infimum of the Arrow–Pratt index of absolute risk aversion as a measure of global risk aversion of a utility function. We show that, for any given arbitrary pair of distributions, there exists a threshold level of global risk aversion such that all increasing concave utility functions with at least as much global risk aversion would rank the two distributions in the same way. Furthermore, this threshold level is sharp in the sense that, for any lower level of global risk aversion, we can find two utility functions in this class yielding opposite preference relations for the two distributions.This paper has benefited from insightful comments made by James Mirrless, two anonymous referees, and by seminar participants at IAE and Simposio de Análisis Económico in Salamanca. They should not bear any responsibility for the remaining errors. Financial support from the Spanish Ministry of Education and Science and FEDER through grants SEC2003-306 and SEC2003-1961, from the Generalitat of Catalonia through the Barcelona Economics program (CREA) and grants 2005SGR00447 and 2005SGR00626 is gratefully acknowledged. This paper is part of the “Polarization and Conflict” project, contract 3CIT2-CT-2004-506084 funded by the European Commission.     

Representation of effectivity functions in coalition proof Nash equilibrium: A complete characterization

The concept of coalition proof Nash equilibrium was introduced by Bernheim et al. [5]. In the present paper, we consider the representation problem for coalition proof Nash equilibrium: For a given effectivity function, describing the power structure or the system of rights of coalitions in society, it is investigated whether there is a game form which gives rise to this effectivity function and which is such that for any preference assignment, there is a coalition proof Nash equilibrium.  It is shown that the effectivity functions which can be represented in coalition proof Nash equilibrium are exactly those which satisfy the well-known properties of maximality and superadditivity. As a corollary of the result, we obtain necessary conditions for implementation of a social choice correspondence in coalition proof Nash equilibrium which can be formulated in terms of the associated effectivity function. Received: 24 June 1999/Accepted: 20 September 2000     

Utility-gap Dominances and Inequality Orderings

The justification for using Lorenz dominance as an inequality ranking condition has been based on the aggregate social welfare comparison and the Pigou–Dalton principle of transfers. Since both the aggregating aspect of the social welfare function and certain implications of the principle of transfers are debatable, ordering conditions stronger than Lorenz dominance are worth exploring. A particularly interesting direction to pursue is to follow the frequently invoked notion that inequality is the “gap” between the rich and the poor. This paper follows this notion to formally propose a unified utility-gap concept and characterizes several utility-gap based conditions as general stronger-than-Lorenz-dominance ranking criteria. Specifically, we propose utility-gap dominance which requires all pair-wise utility-gaps in one distribution to be uniformly smaller than those of the other distribution. We then explore a conceptually weaker dominance concept – quasi dominance – which imposes conditions only on the gap between each person’s utility and some reference utility point of the distribution. I am grateful to two anonymous referees and Peter Lambert for their very constructive comments and suggestions on an earlier version of the paper. The usual caveat applies.     

Games of capacity allocation in many-to-one matching with an aftermarket

In this paper, we study many-to-one matching (hospital–intern markets) with an aftermarket. We first show that every stable matching system is manipulable via aftermarket. We then analyze the Nash equilibria of capacity allocation games, in which preferences of hospitals and interns are common knowledge and every hospital determines a quota for the regular market given its total capacity for the two matching periods. Under the intern-optimal stable matching system, we show that a pure-strategy Nash equilibrium may not exist. Common preferences for hospitals ensure the existence of equilibrium in weakly dominant strategies whereas unlike in games of capacity manipulation strong monotonicity of population is not a sufficient restriction on preferences to avoid the non-existence problem. Besides, in games of capacity allocation, it is not true either that every hospital weakly prefers a mixed-strategy Nash equilibrium to any larger regular market quota profiles.     

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     

Fuzzy choice functions

The R-greatest and maximal sets of standard choice theory are extended to fuzzy R-greatest and fuzzy maximal sets. Unlike the precise counterparts of these concepts, these two sets do not in general coincide when preferences are reflexive and connected. A stronger than usual version of connectedness under which the two sets are equal is provided. The concept of a fuzzy choice function is introduced and conditions under which a fuzzy choice function may be rationalized as a fuzzy R-greatest or a fuzzy maximal set are discussed. Rationalizability with transitive and weakly transitive fuzzy preference relations is also considered.We are indebted to Professor P. K. Pattanaik for his comments on an earlier version of this paper. We also wish to acknowledge comments made by an anonymous referee from which this paper has benefited greatly. The usual caveat about errors applies.     

Arbitrage and global cones: Another counterexample

Chichilnisky (1997) claims that another variant of her condition limiting arbitrage is necessary and sufficient for existence of equilibrium and nonemptiness of the core in an economy with short sales allowing half lines in indifference surfaces. Her proof, however, is based on a proposition purporting to relate her notion of “global cone” (see Chichilnisky (1997) for references) to the Page-Wooders “increasing cone.” In this paper, we present a counterexample showing that parts (i) and (ii) of Chichilnisky's proposition are false. Thus, Chichilnisky's claimed result is without proof. Received: 18 August 1997/Accepted: 30 January 1998     

Congruence indicators for fuzzy choice functions

We introduce the congruence indicators WFCA(·) and SFCA(·) corresponding to fuzzy congruence axioms WFCA and SFCA. These indicators measure the degree to which a fuzzy choice function verifies the axioms WFCA and SFCA, respectively. The main result of the paper establishes for a given choice function the relationship between its congruence indicators and some rationality conditions. One obtains a fuzzy counterpart of the well-known Arrow–Sen theorem in crisp choice functions theory.     

Aggregation of binary evaluations for truth-functional agendas

In the problem of judgment aggregation, a panel of judges has to evaluate each proposition in a given agenda as true or false, based on their individual evaluations and subject to the constraint of logical consistency. We elaborate on the relation between this and the problem of aggregating abstract binary evaluations. For the special case of truth-functional agendas we have the following main contributions: (1) a syntactical characterization of agendas for which the analogs of Arrow’s aggregation conditions force dictatorship; (2) a complete classification of all aggregators that satisfy those conditions; (3) an analysis of the effect of weakening the Pareto condition to surjectivity. This is a sequel to the paper “Aggregation of binary evaluations.” The contents of both papers were presented, under the title “An Arrovian impossibility theorem for social truth functions,” at the Second World Congress of the Game Theory Society, Marseille, July 2004. The first version of “Aggregation of binary evaluations” was completed in June 2005. That working paper was subsequently split into two parts, of which this is the second. The comments of an anonymous referee are gratefully acknowledged. Part of R. Holzman’s work was done while he was a Fellow of the Institute for Advanced Studies at the Hebrew University of Jerusalem.     

Aggregation of individuals' preference intensities into social preference intensity

A result of John Harsanyi concerns the aggregation of individuals' preferences into social preferences. The result states that if the individuals in a society and the society as a whole have preference relations that compare probability distributions on a set of outcomes, and the preference relations satisfy expected-utility conditions and Pareto conditions, then a utility function for the social preference relation is a positive affine function of utility functions for the individuals' preference relations. This paper presents an analogous result for preference relations that denote intensity of preference, i.e., preference relations that compare exchanges of outcomes. This approach avoids the difficulties of requiring that the individuals in the society have common beliefs regarding uncertainty. Received: 14 October 1996 / Accepted: 4 September 1997     

Uncovered set choice rules

I study necessary and sufficient conditions for a choice function to be rationalized in the following sense: there exists a total asymmetric relation T (a tournament) such that, for each feasible (finite) set, the choice set coincides with the uncovered set of T restricted to that feasible set. This notion of ‘maximization’ offers testable restrictions on observable choice behavior.