Deprivation,welfare and inequality

We provide a characterization of the generalised satisfaction—in our terminology non-deprivation—quasi-ordering introduced by S.R. Chakravarty (Keio Econ Stud 34:17–32, (1997)) for making welfare comparisons. The non-deprivation quasi-ordering obeys a weaker version of the principle of transfers: welfare improves only for specific combinations of progressive transfers, which impose that the same amount be taken from richer individuals and allocated to one arbitrary poorer individual. We identify the extended Gini social welfare functions that are consistent with this principle and we show that the unanimity of value judgements among this class is identical to the ranking of distributions implied by the non-deprivation quasi-ordering. We extend the approach to the measurement of inequality by considering the corresponding relative and absolute ethical inequality indices. This is a shortened version of Magdalou and Moyes (2008), which contains the details of the proofs as well as an empirical illustration.     

Is proximity preservation rational in social choice theory?

We establish a strong impossibility theorem of a rational social choice that the proximity preservation (also called weak proximorphism WPX) and the diagonal surjectivity are logically inconsistent. The result is valid for finite or infinite alternatives, discrete or continuous. It generalizes the Baigent theorem, largely weakening his antecedent. For continuum set of alternatives, we clarify the notion of WPX by showing (1) WPX almost implies the continuity, (2) WPX is almost rigid. These observations raise the issue whether WPX is a natural condition for a social welfare function. A splitting reformulation of the proximity preservation which is weaker but rational is suggested.     

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.     

Reformulation of the Nash social welfare function for a continuum of individuals

This paper provides a new formulation of the Nash social welfare function for a continuum of individuals. The new formulation removes redundant assumptions of Kaneko's original formulation and in particular shows that the continuity axiom on social orderings is unnecessary. The derivation of the Nash social welfare function is also much directer and shorter than Kaneko's original derivation.The author thanks a referee of this journal for valuable comments on an eralier draft     

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.     

Generalized median social welfare functions

The article considers the construction of social welfare functions when the set of alternatives is the two-dimensional nonnegative orthant, as would be the case if there are two divisible public goods which can be consumed in any nonnegative quantities. With individual and social preferences required to be linear and strictly monotonic, but otherwise unrestricted, we characterize all of the social welfare functions which satisfy binary independence of irrelevant alternatives and anonymity and which satisfy binary independence of irrelevant alternatives, anonymity, and weak Pareto. These classes of social welfare functions are shown to be formally equivalent to the classes of social choice functions characterized by Moulin in his study of strategy-proof social choice with single-peaked preferences.We are grateful to Charles Blackorby, Hervé Moulin, and an anonymous referee for their comments.     

A remark on a social choice problem

If preferences are continuous monotone complete preorders, then there is a continuous social choice aggregation rule which respects unanimity and is anonymous. The simple proof of this result involves a straightforward application of well-known properties of the closed convergence topology.     

Complete characterization of functions satisfying the conditions of Arrow’s 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.     

Strategy-proofness of continuous aggregation maps

 We consider continuous aggregation maps (e.g., social welfare functions). By assuming that the voters have preferences over social outcomes, we regard the social decision procedure as a noncooperative game, with the aggregation map as a game form. The map is called strategy-proof if it is a Nash equilibrium that the voters report their most preferred outcomes. We give sufficient topological conditions on the space of outcomes so that only dictatorial maps are strategy-proof. Received: 28 February 1994/Accepted: April 22, 1996     

Is a continuous rational social aggregation impossible on continuum spaces?

This paper provides a global topological setting for the social choice theory on continuum spaces of alternatives, in contrast to the local differentiable setting of Chichilnisky. Chichilnisky proved that a rational continuous social choice must be discontinuous in her setting. Our paper revisits her theorem to trace the source of this discontinuity. We find that the discontinuity is irrelevant to social aggregation, per se. The main theorem states that there exist a number of continuous social utility maps which are anonymous and satisfy the Pareto condition. As a corollary, we show that there exist corresponding continuous social welfare functions, if singularity is not separated from regular preferences in social preference topology. This extends the possibility result of Jonnes-Zhang-Simpson on linear preferences, to the general ones. The notion of singularity of preferences, relative to the given mathematical structure of an alternative space, is carefully studied.     

Topological aggregation of preferences: the case of a continuum of agents

 This paper studies the topological approach to social choice theory initiated by G. Chichilnisky (1980), extending it to the case of a continuum of agents. The social choice rules are continuous anonymous maps defined on preference spaces which respect unanimity. We establish that a social choice rule exists for a continuum of agents if and only if the space of preferences is contractible. We provide also a topological characterization of such rules as generalized means or mathematical expectations of individual preferences. Received: 30 November 1994/Accepted: 22 April 1996     

Abstention as an escape from Arrow's theorem

There are non-dictatorial social welfare functions satisfying the Pareto principle and Arrow's independence of irrelevant alternatives when voters can abstain. In particular, with just seven voters, the number of dictatorial social welfare functions satisfying Arrow's conditions could be deemed, relative to the total number of social welfare functions satisfying Arrow's conditions, negligible.     

Strategy-proofness of social welfare functions: The use of the Kemeny distance between preference orderings

The Kemeny distance for preference orderings is used to determine individual rankings of social preferences. Based on this distance function, the strategy-proofness of social welfare functions is examined. Our main result is an impossibility theorem stating that no social welfare function can be strategy-proof, if some additional properties are required.The work on this paper initiated while Walter Bossert was a visitor at the CentER for Economic Research at Tilburg University. The hospitality and the support of CentER and its members are gratefully acknowledged. Ton Storcken's research was supported by a grant from the Cooperation Centre Tilburg and Eindhoven University. An earlier version of the paper was presented at the 1991 Canadian Economic Theory Meeting in Toronto. We thank the participants and especially Ingrid Peters-Fransen for comments. Substantial improvements are due to the suggestions of Bernard Monjardet and two referees.     

Continuity of social choice functions with restricted coalition algebras

Social choice functions are studied with an infinite population framework that includes algebras over the set of individuals. It is shown that continuous social choice functions are either dictatorial or manipulable. Weak anonymity is shown to be incompatible with continuity and neutrality. An earlier version of this paper was presented at the Logic, Game Theory and Social Choice 4, Caen 2005, and at the First Spain Italy Netherlands Meeting on Game Theory, Maastricht 2005.     

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.     

Multi-profile intergenerational social choice

In an infinite-horizon setting, Ferejohn and Page showed that any social welfare function satisfying Arrow’s axioms and stationarity must be a dictatorship of the first generation. Packel strengthened this result by proving that no collective choice rule generating complete social preferences can satisfy unlimited domain, weak Pareto and stationarity. We prove that this impossibility survives under a domain restriction and without completeness. We propose an alternative stationarity axiom and show that a social welfare function on a specific domain satisfies this modified version and some standard social choice axioms if and only if it is a chronological dictatorship.     

Utilitarianism or welfarism: does it make a difference?

We show that it is possible to reconcile the utilitarian and welfarist principles under the requirement of unanimity provided that the set of profiles over which the consensus is attained is rich enough. More precisely, we identify a closedness condition which, if satisfied by a class of n-tuples of utility functions, guarantees that the rankings of social states induced by utilitarian and welfarist unanimities over that class are identical. We illustrate the importance of the result for the measurement of unidimensional as well as multidimensional inequalities from a dominance point of view.     

Topological aggregation of inequality preorders

An inequality preorder is defined as a complete preorder on a simplex which satisfies the properties of continuity and strict Schur-convexity (the mathematical equivalent of Dalton's principle of transfers). The paper shows that it is possible to aggregate individual inequality preorders into a collective one if we are interested in continuous anonymous aggregation rules that respect unanimity. The aggregation problem is studied within a topological framework introduced by Chichilnisky.We are grateful to J. Laine for helpful discussions and to an anonymous referee of this Journal for valuable comments     

Characterizations of majoritarianism: a unified approach

Defining regularity as the conjunction of unanimity, anonymity, neutrality and monotonicity, we show that every regular social choice rule can be expressed as a function of vote differences and the maximal number of canceling couples. This equivalence not only identifies the class of regular social choice rules but also allows a unified conception of various majority characterizations. I acknowledge the support of the Turkish Academy of Sciences Distinguished Young Scientist Award Program (TUBA-GEBIP).     

Empirical evidence on preference formation

To explain the empirical fact that different individuals have different individual welfare functions, a theory of preference formation has been developed by the first author. The theory essentially states that an individual's welfare function is identical to the distribution of consumption patterns the individual has observed over time. This includes both his own consumption and the consumption by others in his social reference group. The paper reviews the evidence collected with respect to directly measured individual welfare functions in Europe. Evidence obtained in related research by economists, psychologists, and sociologists is also discussed.