Georges Bordes

Consistency, Rationality and Collective Choice

Though they are related, consistency and rationality are different concepts. Consistency is concerned with what happens to choices when the set of available alternatives expands or contracts. Rationality is concerned with how the choices are related to a binary relation on the set of all alternatives. In this paper we define a weakened consistency by a strengthening of Sens ,B condition (Sen [9], [10], [11]). We also define a weakened rationality by two conditions, related to (but different from) Schwartzs rationality conditions [8]. We study the relations between consistency and rationality thus defined, and apply the results to collective choice theory by proving that there exists one and only one Collective Choice Rule that is fully democratic and yields consistent, rational choice functions. Also, the revealed preference relations of these choice functions are transitive.

On the possibility of reasonable consistent majoritarian choice: Some positive results

Abstract Strictly Majoritarian Social Choice Functions (SMSCFs) are such that the choice on an agenda can be defined with the knowledge of the simple majority relation on the agenda as the sole information. The possibility for SMSCFs to satisfy both the General Pareto condition and choice consistency conditions strong enough to be meaningful has been doubted. Here we exhibit three reasonable SMSCFs that do both. One of them can be interpreted as eliminating from the agenda all alternatives one can suspect of being Pareto dominated by some other. We compare some of their properties with those of the SMSCFs already known in the literature.

Independence of irrelevant alternatives in the theory of voting

In social choice theory there has been, and for some authors there still is, a confusion between ArrowsIndependence of Irrelevant Alternatives (IIA) and somechoice consistency conditions. In this paper we analyze this confusion. It is often thought that Arrow himself was confused, but we show that this is not so. What happened was that Arrow had in mind a condition we callregularity, which implies IIA, but which he could not state formally in his model because his model was not rich enough to permit certain distinctions that would have been necessary. It is the combination of regularity and IIA that he discusses, and the origin of the confusion lies in the fact that if one uses a model that does not permit a distinction between regularity and IIA, regularity looks like a consistency condition, which it is not. We also show that the famous example that ‘proves’ that Arrow was confused does not prove this at all if it is correctly interpreted.

Arrovian theorems for economic domains. The case where there are simultaneously private and public goods

We prove that Arrows theorem and, with quasi-transitive social preferences, a version of Mas-Colell and Sonnenscheins theorem, hold when there are simultaneously private and public goods, and the individuals are supposed to have selfish, continuous, convex and strictly increasing preferences. We first prove the results in an abstract general setting, and show that the above-mentioned economic domain is a model for this setting.

Covering relations, closest orderings and hamiltonian bypaths in tournaments

We show that the Slaters set of a tournament, i.e. the set of the top elements of the closest orderings, is a subset of the top cycle of the uncovered set of the tournament. We also show that the covering relation is related to the hamiltonian bypaths of a strong tournament in that if x covers y, then there exists an hamiltonian bypath from x to y.

Social Welfare Functionals on Restricted Domains and in Economic Environments

March 1997 Arrows ``impossibility and similar classical theorems are usually proved for an unrestricted domain of preference profiles. Recent work extends Arrows theorem to various restricted but ``saturating domains of privately oriented, continuous, (strictly) convex, and (strictly) monotone ``economic preferences for private and/or public goods. For strongly saturating domains of more general utility profiles, this paper provides similar extensions of Wilsons theorem and of the strong and weak ``welfarism results due to dAspremont and Gevers and to Roberts. Hence, for social welfare functionals with or without interpersonal comparisons of utility, most previous classification results in social choice theory apply equally to strongly saturating economic domains. Journal of Economic Literature classification: D71. Keywords: social welfare functionals, Arrows theorem, Wilsons theorem, welfarism, neutrality, restricted domains, economic domains, economic environments.

Arrovian theorems for economic domains: Assignments, matchings and pairings

In this paper, we show that Arrows and Wilsons theorems hold for permutation, symmetric permutation and symmetric domains. Permutation domains concern assignments: n indivisible private goods are distributed betweenn selfish individuals. Symmetric permutation domains concern matchings: two sets ofn selfish individuals being given, pairs with an individual from each set are to be made. Symmetric domains concern pairings: a set of 2n selfish individuals being given,n pairs of individuals are to be made.

Arrow's Theorem for Economic Domains and Edgeworth Hyperboxes

Kenneth J. Arrows theorem holds when the set of alternatives is an Edgeworth hyperbox and the individuals have classical economic preferences over their consumption sets. (Free disposability is not assumed.) By classical individual preferences the authors mean preorders satisfying continuity, strict convexity, strict monotonicity, and selfishness. A minor, but noteworthy, accomplishment is the development of a general technique for extending two-commodity impossibility theorems to the general m-commodity counterpart. Copyright 1995 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.