Arrow's Theorem, Weglorz' Models and the Axiom of Choice (original) (raw)
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Infinite Populations, Choice and Determinacy
Studia Logica
This paper criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow's theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the Hildenbrand criterion and argue that results based on unrestricted Axiom of Choice (AC) do not meet this criterion. The technically novel part of this paper is a proposal to use a set-theoretic principle known as the Axiom of Determinacy (AD), not as a replacement for Choice, but simply to eliminate applications of set theory violating the Hildenbrand criterion. A particularly appealing aspect of AD from the point of view of the research area in question is its game-theoretic character.
Non-dictatorial extensive social choice
Economic Theory, 2005
Different social planners may have different opinions on the wellbeing of individuals under different social options . If utilities are translation-or ratio-scale measurable, or if the social ranking might be incomplete, or if interplanner comparability is allowed; then there exists non-dictatorial aggregation rules. We propose extensions, intersections, and mixtures of the Pareto, utilitarian, leximin, Kolm-Pollak, and iso-elastic rules. * We are extremely grateful to the referee who was willing to review this paper four times. Her/his 'extensive' and in-depth comments had a strong impact. Further thanks are due to Bart Capéau, Marc Fleurbaey, Maurice Salles, Erik Schokkaert, and Alain Trannoy. The first author gratefully acknowledges the financial support by the TMR network Living Standards, Inequality and Taxation (ERBFMRXCT 980248) of the European Communities.
The famous Choice Axiom by Luce is now more than half a century old. We investigate the consequences of going back to Luce's original formulation of the axiom, which had a more succinct formulation. Using this, what we term Strong Choice Axiom, we prove a sort of "lexicographic" version of the "Luce form" choice probabilities. Further, we study whether these extended choice probabilities are representable by Random Utility models.
The separability axiom and equal-sharing methods
Journal of Economic Theory, 1985
A quasi-linear social choice problem is defined as selecting one (among finitely many) indivisible public decision and a vector of monetary transfers among agents to cover the cost of this decision. This decision is based upon individual preferences, which are assumed to be additively separable and linear in money. The Separability axiom is a consistency property for choice methods on societies with variable size: the decision is not affected if we remove an arbitrary agent under the condition that he be guaranteed his original utility level and the cost to the remaining agents is modified accordingly. Thus the utility level assigned by the social choice function to agent i is the price at which the other agents are unanimously willing to buy agent l*s share of the decision power. A general characterization of choice methods satisfying this axiom is provided. Three subclasses of particular interest are characterized by additional milder axioms. Those are: (i) equal sharing of the surplus left over some reference utility (e.g., the utility at a status quo decision), (ii) utilitarian methods that merely select the efficient public decision and perform no monetary transfers, and (iii) equal allocation of nonseparable costs, which divides equally the surplus left over from the utility derived from the pivotal mechanism (also known as the Vickrey-Clarke-Groves mechanism). Journal of Economic Literature Classification Number: 025. 0 1985 Academic press, hc.
An economic approach to social choice ? II
Public Choice, 1978
There is a formal equivalence between games, societies, and economies. Lindahl equilibrium for a game or society corresponds to competitive equilibrium for the equivalent economy. Results on existence and optimality of competitive equilibrium thus apply to the theory of games and societies. The "core" for a game or society as derived by extension from the core of an economy is "too large" to be interesting. An example illustrates that the a-core may be disjoint from the set of Lindahl equilibria. However, if the power of coalitions to inflict negative externalities is suitably restricted, Lindahl equilibria must be in the a-coreo I. Introduction The fagt that an economy with public goods may be treated as a private goods economy in a commodity space of higher dimension was noted by Arrow (1969) and later exploited by others, including Starrett (1973), Rader (1972) and Bergstrom (1976). Here we develop this notion systematically by demonstrating the equivalence of the notions of game, society, and economy. This enables us to define Lindahl ecLuilibrium for a game or society in such a way as to correspond to competitive equilibrium for the equivalent economy. In view of this equivalence, theorems on the existence of Lindahl equilibrium become straightforward corollaries of theorems on the existence of competitive equilibria for the corresponding economies. We show that the notion of a core for a game or society derived by extension from the core of the equivalent economy is relatively uninteresting since it is "too large." We then study the a-core of Scarf (1971) for a game or society. We show by example that the a-core and the set of Lindaht equilibria may be disjoint. This extends and sharpens the observation of Shapley and Shubik (1969) and Bergstrom (1975) that some
Anonymity and Neutrality in Arrow's Theorem with Restricted Coalition Algebras
1994
In the very general setting of for Arrow's Theorem, I show two results. First, in an infinite society, Anonymity is inconsistent with Unanimity and Independence if and only if a domain for social welfare functions satisfies a modest condition of richness. While Arrow's axioms can be satisfied, unequal treatment of individuals thus persists. Second, Neutrality is consistent with Unanimity (and Independence). However, there are both dictatorial and nondictatorial social welfare functions satisfying Unanimity and Independence but not Neutrality. In Armstrong's setting, one can naturally view Neutrality as a stronger condition of informational simplicity than Independence.
From preference to utility: a problem of descriptive set theory
Notre Dame Journal of Formal Logic, 1985
Some years ago J. H. Silver proved that a co-analytic equivalence relation on a Polish space has either countably many or continuum many equivalence classes. Later L. Harrington greatly simplified the complicated original proof. The present paper is a sort of footnote to Harrington's lectures on these matters. It will be shown that information developed in his proof settles a problem of (hyper-)theoretical mathematical economics first investigated by Wesley [13] and Mauldin [8]. Namely, it will be shown that any family of closed preference orders that is parametrized in a Borel fashion can be represented by a family of continuous utility functions parametrized in an absolutely measurable fashion. Though the author is greatly indebted to Mauldin's work [8], the treatment of the problem here will be self-contained. Background and motivation for problems of this kind can be found in [6], Section 2.1. Terminology and notation pertaining to descriptive set theory will be as in [9]. 2 Definitions Throughout let ψ be a topological space. A preference order on 'ψ is any transitive, connected binary relation <*. Associated are the strict preference and indifference relations given by: x <* y <-> x <* y & ~y <* x X =* y +-> x <* y & y <* χ m Note that Ξ* i s an equivalence relation, and that <* induces a linear order on its equivalence classes, [x]* will denote the equivalence class of x. <* will be *Research in part supported by USA National Science Foundation Grant MCS 8003254.
Arrovian impossibilities in aggregating preferences over non-resolute outcomes
Social Choice and Welfare, 2008
Given a society confronting a set of alternatives A, we consider the aggregation of individual preferences over the power set A of A into a social preference over A. In case we allow individuals to have any complete and transitive preference over A, Arrow's impossibility theorem naturally applies. However, the Arrovian impossibility prevails, even when the set of admissible preferences over A is severely restricted by strong axioms that relate preferences over A to preferences over A. In fact, we identify a very narrow domain of lexicographic orderings over A which exhibits the Arrovian impossibility in all of its superdomains. As the lexicographic extension we use is compatible with almost all standard extension axioms, we interpret our results as the strong prevalence of Arrow's impossibility theorem in aggregating preferences over sets.
Robertsʼ Theorem with neutrality: A social welfare ordering approach
Games and Economic Behavior, 2012
We consider dominant strategy implementation in private values settings, when agents have multi-dimensional types, the set of alternatives is finite, monetary transfers are allowed, and agents have quasi-linear utilities. We focus on private-value environments. We show that any implementable and neutral social choice function must be a weighted welfare maximizer if the type space of every agent is an m-dimensional open interval, where m is the number of alternatives. When the type space of every agent is unrestricted, Roberts' Theorem with neutrality (Roberts, 1979) becomes a corollary to our result. Our proof technique uses a social welfare ordering approach, commonly used in aggregation literature in social choice theory. We also prove the general (affine maximizer) version of Roberts' Theorem for unrestricted type spaces of agents using this approach.