Wilson's theorem for economic environments and continuous social preferences

Wilson's generalization of Arrow's impossibility theorem has been proved for the realm of private goods and economic preferences by Border and by Bordes and Le Breton. However, their proofs require the exclusion of the zero vector from the commodity space. This paper assumes continuity of social preference to obtain the impossibility theorem for the entire allocation space, even if the society is infinite. A simple corollary reveals that there is some individual who is assigned the zero consumption vector at every social optimum whenever the social welfare function is nonnull and nonimposed, and satisfies Arrow's independence axiom and continuity and transitivity of social preference.Financial support from the Social Sciences and Humanities Research Council is gratefully acknowledged, as are the suggestions of Charles Plott and an anonymous referee. The author assumes responsibility for any errors.     

Social choice with analytic preferences

Arrow's axioms for social welfare functions are shown to be inconsistent when the set of alternatives is the nonnegative orthant in a multidimensional Euclidean space and preferences are assumed to be either the set of analytic classical economic preferences or the set of Euclidean spatial preferences. When either of these preference domains is combined with an agenda domain consisting of compact sets with nonempty interiors, strengthened versions of the Arrovian social choice correspondence axioms are shown to be consistent. To help establish the economic possibility theorem, an ordinal version of the Analytic Continuation Principle is developed. Received: 4 July 2000/Accepted: 2 April 2001     

Rational budgeters in the theory of social choice

The purpose of this paper is to explore duality in the theory of social choice. As application Arrow's Impossibility Theorem and another impossibility theorem using the notion of positive responsiveness are chosen. It will be seen that we can establish notions and theorems which are symmetric to the original ones. However, if we establish impossibility theorems when rational behaviour is described by budget correspondences and not by choice correspondences, we need not assume that every subset of X (a family of alternatives) with cardinality 2 is a budget set. Therefore the dual theorems also may hold for families of competitive budget sets. It will also be shown that although the underlying preferences on X need not be acyclic, local decisiveness on budget sets may lead to global decisiveness on these sets.     

Independent Decisiveness and the Arrow theorem

I show that the condition of Independence of Irrelevant Alternatives in Arrow's impossibility theorem can be weakened into Relational Independent Decisiveness. The condition of Relational Independent Decisiveness is essentially a translation of Sen's Independent Decisiveness into the traditional Arrovian framework. I also show by example that Relational Independent Decisiveness is indeed weaker than Arrow's Independence of Irrelevant Alternatives. Received: 30 October 1996 / Accepted: 22 May 1997     

Connecting and resolving Sen's and Arrow's theorems

It is shown that the source of Sen's and Arrow's impossibility theorems is that Sen's Liberal condition and Arrow's IIA counter the critical assumption that voters have transitive preferences. But if the procedures are not permitted to treat the transitivity of individual preferences as a valued input, then we cannot expect rational outputs. Once this common cause for these perplexing conclusions is understood, these classical conclusions end up admitting quite benign interpretations where it becomes possible to propose several resolutions. Received: 2 April 1996 / Accepted: 15 October 1996     

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

We prove that Arrow's theorem and, with quasi-transitive social preferences, a version of Mas-Colell and Sonnenschein's 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.We thank Donald Campbell and two anonymous referees for helpfull suggestions.     

The pre-history of Kenneth Arrow's social choice and individual values

The purpose of this article is to give an historical sense of the intellectual developments that determined the form and content of Kenneth Arrow's path-breaking work published in 1951. One aspect deals with personal influences that helped shape Arrow's own thinking. A second aspect is concerned with the early history of the general theory of relations, which is mainly centered in the nineteenth century, and also with the essentially independent modern development of the axiomatic method in the same time period. Arrow's use of general binary relations and of axiomatic methods to ground, in a clear mathematical way, his impossibility theorem marks a turning point in welfare economics, and, more generally, in mathematical economics.     

Arrow's Theorem with a fixed feasible alternative

Arrow's Theorem, in its social choice function formulation, assumes that all nonempty finite subsets of the universal set of alternatives is potentially a feasible set. We demonstrate that the axioms in Arrow's Theorem, with weak Pareto strengthened to strong Pareto, are consistent if it is assumed that there is a prespecified alternative which is in every feasible set. We further show that if the collection of feasible sets consists of all subsets of alternatives containing a prespecified list of alternatives and if there are at least three additional alternatives not on this list, replacing nondictatorship by anonymity results in an impossibility theorem.Most of the research for this article was completed while we were participants in the Public Choice Institute held at Dalhousie University during the summer of 1984. We wish to record here our thanks to the Institute Director, E.F. McClennen, and its sponsors, the Council for Philosophical Studies, the U.S. National Science Foundation, and the Social Science and Humanities Research Council of Canada. We are grateful to our referees for their comments and the Center for Mathematical Studies in Economics and Management Science at Northwestern University, where Weymark was a visitor during 1985–86, for secretarial assistance.     

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.     

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     

Sequential path independence and social choice

Arrow's general impossibility theorem shows that every Paretian social choice function which satisfies independence of irrelevant alternatives and the Axiom of Sequential Path Independence is necessarily dictatorial. It is shown that the existence of a dictator can be established without invoking full path independence. We propose an axiom of weak path independence of a sequential choice procedure. This axiom turns out to be independent of the factor that is critical in obtaining dictatorship or oligarchy results in the choice theoretic framework.     

The informational basis of the theory of fair allocation

The theory of fair allocation is often favourably contrasted with the social choice theory in the search for escape routes from Arrow’s impossibility theorem. Its success is commonly attributed to the fact that it is modest in its goal vis-à-vis social choice theory, since it does not aspire for a full-fledged ordering of options, and settles with a subset of ‘fair’ options. We show that its success may rather be attributable to a broadened informational basis thereof. To substantiate this claim, we compare the informational basis of the theory of fair allocation with the informational requirements of social choice theory.This paper is derived from a part of an earlier draft of our paper entitled ‘Informational requirements for social choice in economic environments’. The authors thank A. Trannoy, an associate editor and three referees for comments, and participants at seminars in University of Cergy-Pontoise, University of Rochester, Hitotsubashi University, and Waseda University, and the 5th International Meeting of the Society for Social Choice and Welfare in Alicante. Financial support from the Ministry of Education of Japan through Grant-in-Aid No. 10045010 (‘Economic Institutions and Social Norms: Evolution and Transformation’) and the 21st Century Center of Excellence Project on the Normative Evaluation and Social Choice of Contemporary Economic Systems is gratefully acknowledged.     

Essentially lexicographic aggregation

In the paper a new approach to lexicography is developed by which in the general framework of ordered blocks with a monotonic basis it is shown that a nontrivial ordering is translation-invariant if and only if it is essentially lexicographic of degree n. Here, the latter means that the ordering can be represented by an ordinary lexicographic ordering in n dimensions. As an application it is shown that a nontrivial social welfare ordering on Euclidean space possesses a useful invariance property (cardinality and non comparability) if and only if the ordering is essentially lexicographic of a strong kind in that it can be obtained from ordinary lexicography by permutation, cutting-off and order reversal with respect to components. This result generalizes the characterization of lexical individual dictatorship obtained by Gevers and d'Aspremont and it provides, within the social welfare approach, a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all.The author thanks K. Arrow, W. Gaertner, L. Gevers, and two anonymous referees for helpful hints and suggestions.     

The free triple assumption

Many impossibility results, like Arrow's Theorem, can be strengthened by using a domain constraint that is substantially weaker than the usual domain condition.     

Positional independence in preference aggregation

If, for strict preferences, a unique choice function (CF) is used to aggregate preferences position-wise then the resulting social welfare function (SWF) is dictatorial. This suggests that the task performed by non-dictatorial SWFs must be “more complex” than just selecting an alternative from a list using a single criterion. This is because the information required by non-dictatorial SWFs to aggregate preferences cannot be compressed into a CF. It is also shown that the attempt to reduce the working of a SWF to the working of a CF involves the adoption of certain positional requirements, whose relationship with the conditions in Arrow's theorem is established. Received: 28 May 2001/Accepted: 25 March 2002 My deepest gratitude to Donald G. Saari, who rescued this paper from the worst fate, and to the referee, who showed the escape route.     

Extended Pareto 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 2 other axioms: “Anonymity” and a weak version of Arrow's Independence of Irrelevant Alternatives axiom: “Weak IIA” it is shown that there is a unique Social Welfare Function called “Relative Utilitarianism” that consists of normalising individual utilities between 0 and 1 and adding them. Received: 7 June 1994 / Accepted: 28 April 1997     

Equity, continuity, and myopia: A generalization of Diamond’s impossibility theorem

 We consider social preferences over infinite horizon intergenerational consumption paths. We use the Mackey topology to define continuity of social preferences. Our main objective is to generalize one of Diamond’s impossibility theorems. First, we show that the trivial preference relation is the only asymmetric social preference relation satisfying equity and continuity. Second, we compare Campbell’s impossibility theorem with ours. Finally we use an order-based notion of myopia and establish another impossibility result. Received: 26 August 1994/Accepted: 5 June 1996     

On the robustness of the impossibility result in the topological approach to social choice

In the first part we make an assessment of the impossibility result, due to Chichilnisky, in the topological approach to social choice theory. We observe that this result depends essentially on the choice of the topology for the set of preferences. In the second part, we present two positive results, obtained using the global approach. The first one deals with the space of continuous and strictly convex preferences, a space which, in the local approach, would produce an impossibility result. The second result deals with a class of preferences which is dense in the space of all continuous preferences. Thus, an approximate solution of the Chichilnisky problem has been obtained on this space.     

Acyclic and continuous social choice in T 1 connected spaces

In this paper, we prove some versions of the Arrovian impossibility theorem in T ₁ connected alternatives spaces, with the collective rationality condition weakened from transitivity to acyclicity, the Pareto condition replaced by some weaker conditions, and a continuity condition of social preferences imposed. Moreover these impossibility theorems are applied to a distributive problem of private goods in economic environments.This is a revised version of my Working Paper No. 111. I would like to thank two anonymous referees of the journal for their insightful comments. This paper was reported at the Western Meeting of Japan Association of Economics and Econometrics, held at Tezukayama University June 1990, and at Second Annual Southeastern Economic Theory Meetings, held at Florida University October 1990. I am grateful to Professors Tetsuya Kishimoto, Makoto Okamura, Koichi Suga, and Donald Campbell for their insightful comments. This paper is supported in part by the Tokyo Center for Economic Research.