Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem

Arrow's theorem implies that a social choice function satisfying Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are allowed, a dictatorial social choice 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. 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 choice functions satisfying Transitivity, the 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 Pareto Principle). Additionally, we derive formulas for the number of functions satisfying these conditions.Comment: 11 pages, 1 figur

Computability of simple games: A characterization and application to the core

It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable simple games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted

Arrow's Theorem in Spatial Environments

In spatial environments, we consider social welfare functions satisfying Arrow's requirements. i.e., weak Pareto and independence of irrelevant alternatives. When the policy space os a one-dimensional continuum, such a welfare function is determined by a collection of 2n strictly quasi-concave preferences and a tie-breaking rule. As a corrollary, we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voter's preference is strictly quasi-concave. When the policy space is multi-dimensional, we establish Arrow's impossibility theorem. Among others, we show that weak Pareto, independence of irrelevant alternatives, and non-dictatorship are inconsistent if the set of alternatives has a non-empty interior and it is compact and convex.Dans des environnements spatiaux, nous considérons des fonctions de bien-être social satisfaisant les hypothèses d’Arrow, i.e. la faiblesse au sens de Pareto et l’indépendance des alternatives non pertinentes. Lorsque l’espace des politiques est un continuum unidimensionnel, une telle fonction de bien-être est déterminée par une collection de 2 N préférences strictement quasi-concaves et une règle de bris d’égalité. Comme corollaire, nous obtenons que, lorsque le nombre d’électeurs est impair, le vote à la majorité simple est transitif si et seulement si la préférence de chaque électeur est strictement quasi concave. Lorsque l’espace des politiques est multidimensionnel, nous établissons le théorème d’impossible d’Arrow. Nous montrons, entre autres, que la faiblesse au sens de Pareto, l’indépendance des alternatives non pertinentes et la non-dictature sont incompatibles si l’ensemble des alternatives possède un intérieur non vide et est compact et convexe

Path Independence, Rationality, and Social Choice

The paper provides several axiomatizations of the concept of "path independence" as applied to choice functions defined over finite sets. The axioms are discussed in terms of their relationship to "rationality" postulates and their meaning with respect to social choice models

Extended paretian rules and relative utilitarianism

This paper introduces the 'Extended Pareto' axiom on Social welfare functions and gives a characterization of the axiom when it is assumed that the Social Welfare Functions that satisfy it in a framework of preferences over lotteries also satisfy the restrictions (on the domain and range of preferences) implied by the von-Neumann-Morgenstern axioms. With the addition of two other axioms: Anonymity and Weak IIA* it is shown that there is a unique Social Welfare Function called Relative Utilitarianism that consists of normalizing individual utilities between zero and one and then adding them

Social choice and game theory: recent results with a topological approach

This chapter presents a summary of recent results obtained in game and social choice theories, and highlights the application and the development of tools in algebraic topology. The purpose is expository: no attempt is made to provide complete proofs, for which references are given, nor to review the previous work in this area, which covers a significant subset of the economic literature. The aim is to provide an oriented guide to recent results, through economic examples with geometric interpretations, and to indicate possible fruitful avenues of research.social choice; game theory; algebraic topology; topological; pareto condition; preferences; Nash equilibrium; transversality

Extended paretian rules and relative utilitarianism.

This paper introduces the 'Extended Pareto' axiom on Social welfare functions and gives a characterization of the axiom when it is assumed that the Social Welfare Functions that satisfy it in a framework of preferences over lotteries also satisfy the restrictions (on the domain and range of preferences) implied by the von-Neumann-Morgenstern axioms. With the addition of two other axioms: Anonymity and Weak IIA* it is shown that there is a unique Social Welfare Function called Relative Utilitarianism that consists of normalizing individual utilities between zero and one and then adding them.Group Preferences; Multi-profile;

Are interpersonal comparisons of utility indeterminate?

On the orthodox view in economics, interpersonal comparisons of utility are not empirically meaningful and "hence" impossible. To reassess this view, this paper draws on the parallels between the problem of interpersonal comparisons of utility and the problem of translation of linguistic meaning, as explored by Quine. The paper discusses several cases of what the empirical evidence for interpersonal comparisons of utility might be and shows that, even on the strongest of these, interpersonal comparisons are empirically underdetermined and, if we also deny any appropriate truth of the matter, indeterminate. However, the underdetermination can be broken non-arbitrarily (though not purely empirically) if (i) we assign normative significance to certain states of affairs or (ii) we posit a fixed connection between certain empirically observable proxies and utility. It is concluded that, even if interpersonal comparisons are not empirically meaningful, they are not in principle impossible

Social Choice Theory and the Informational Basis Approach

For over a quarter of a century, the use of utility information based upon interpersonal comparisons has been seen as an escape route from the Arrow Impossibility Theorem. This paper critically examines this informational basis approach to social choice. Even with comparability of differences and levels, feasible social choice rules must be insensitive to a range of distributional issues. Also, the Pareto principle is not solely to blame for the inability to adopt rules combining utility and non-utility information: if the Pareto principle is not invoked then there is no way of combining utility and non-utility information in a ranking of states unless levels of utility are comparable; with only level comparability, information must be combined in restrictive ways and the notion of giving different independent weight to different considerations is ruled out. If informational bases are viewed as the restriction on information that is available, rather than a theoretical limit on information, then there exist methods to estimate richer informational structures and overcome some of these difficulties.

Complete Characterization of Functions Satisfying the Conditions of Arrow\u27s 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