Social choice and telling the truth

This paper presents a straightforward proof of Gibbard's theorem that any non-manipulable choice rule with at least three outcomes is dictatorial. The proof is similar to proofs of Arrow's impossibility theorem, but does not rely on that theorem. The last part of the paper discusses the relation between Gibbard's and Arrow's theorems. The paper contains no new result, and is intended to make this important area of social choice theory accessible to a wider audience.     

Arrovian aggregation in economic environments: how much should we know about indifference surfaces?

Arrow's celebrated theorem of social choice shows that the aggregation of individual preferences into a social ordering cannot make the ranking of any pair of alternatives depend only on individual preferences over that pair, unless the fundamental weak Pareto and non-dictatorship principles are violated. In the standard model of division of commodities, we investigate how much information about indifference surfaces is needed to construct social ordering functions satisfying the weak Pareto principle and anonymity. We show that local information such as marginal rates of substitution or the shapes “within the Edgeworth box” is not enough, and knowledge of substantially non-local information is necessary.     

Geometry and impossibility

Summary. This note presents a simple proof of Arrow's impossibility theorem using Saari's [3, 4] “geometry of voting”. Received: March 5, 2001; revised version: August 16, 2001     

A note on introducing a "zero‐line" of welfare as an escape route from Arrow's theorem

Since Sen's insightful analysis of Arrow's Impossibility Theorem, Arrow's theorem is often interpreted as a consequence of the exclusion of interpersonal information from Arrow's framework. Interpersonal comparability of either welfare levels or welfare units is known to be sufficient for circumventing Arrow's impossibility result. But it is less well known whether one of these types of comparability is also necessary or whether Arrow's conditions can already be satisfied in much narrower informational frameworks. This note explores such a framework: the assumption of (ONC + 0), ordinal measurability of welfare with the additional measurability of a "zero‐line", is shown to point towards new, albeit limited, escape routes from Arrow's theorem. Some existence and classification results are established, using the condition that social orderings be transitive as well as the condition that social orderings be quasi‐transitive.     

A unifying impossibility theorem

This paper identifies and illuminates a common impossibility principle underlying a number of impossibility theorems in social choice. We consider social choice correspondences assigning a choice set to each non-empty subset of social alternatives. Three simple axioms are imposed as follows: unanimity, independence of preferences over infeasible alternatives, and choice consistency with respect to choices out of all possible alternatives. With more than three social alternatives and the universal preference domain, any social choice correspondence that satisfies our axioms is serially dictatorial. A number of known impossibility theorems—including Arrow’s Impossibility Theorem, the Muller–Satterthwaite Theorem, and the impossibility theorem under strategic candidacy—follow as corollaries.     

Aggregation of binary evaluations with abstentions

A general model of aggregation of binary evaluations over interrelated issues, introduced by Wilson and further studied by Rubinstein and Fishburn and by the authors, is extended here to allow for abstentions on some of the issues. It is shown that the same structural conditions on the set of feasible evaluations that lead to dictatorship in the model without abstentions, lead to oligarchy in the presence of abstentions. Arrow's impossibility theorem for social welfare functions, Gibbard's oligarchy theorem for quasi-transitive social decision functions, as well as some apparently new theorems on preference aggregation, are obtained as corollaries.     

Three brief proofs of Arrow’s Impossibility Theorem

Summary. Arrows original proof of his impossibility theorem proceeded in two steps: showing the existence of a decisive voter, and then showing that a decisive voter is a dictator. Barbera replaced the decisive voter with the weaker notion of a pivotal voter, thereby shortening the first step, but complicating the second step. I give three brief proofs, all of which turn on replacing the decisive/pivotal voter with an extremely pivotal voter (a voter who by unilaterally changing his vote can move some alternative from the bottom of the social ranking to the top), thereby simplifying both steps in Arrows proof. My first proof is the most straightforward, and the second uses Condorcet preferences (which are transformed into each other by moving the bottom alternative to the top). The third proof proceeds by reinterpreting Step 1 of the first proof as saying that all social decisions are made the same way (neutrality).Received: 9 July 2001, Revised: 2 September 2004, JEL Classification Numbers: D7, D70, D71.John Geanakoplos: I wish to thank Ken Arrow, Chris Avery, Don Brown, Ben Polak, Herb Scarf, Chris Shannon, Lin Zhou, and especially Eric Maskin for very helpful comments and advice. I was motivated to think of reproving Arrows theorem when I undertook to teach it to George Zettler, a mathematician friend. After I presented this paper at MIT, a graduate student there named Luis Ubeda-Rives told me he had worked out the same neutrality argument as I give in my third proof while he was in Spain nine years ago. He said he was anxious to publish on his own and not jointly, so I encourage the reader to consult his forthcoming working paper. The proofs appearing here appeared in my 1996 CFDP working paper. Proofs 2 and 3 originally used Mays notation, which I have dropped on the advice of Chris Avery.     

Candidate stability and nonbinary social choice

Summary. The requirement that a voting procedure be immune to the strategic withdrawal of a candidate for election can be formalized in different ways. Dutta, Jackson, and Le Breton (Econometrica, 2001) have recently shown that two formalizations of this candidate stability property are incompatible with some other desirable properties of voting procedures. This article shows that Grether and Plott's nonbinary generalization of Arrow's Theorem can be used to provide simple proofs of two of their impossibility theorems. Received: August 15, 2001; revised version: March 11, 2002 RID="*" ID="*" Parts of this article were previously circulated in somewhat different form in a working paper with the same title by the second author. We are grateful to Michel Le Breton and an anonymous referee for their comments. Correspondence to:J.A. Weymark     

Factoring out the impossibility of logical aggregation

According to a theorem recently proved in the theory of logical aggregation, any nonconstant social judgment function that satisfies independence of irrelevant alternatives (IIA) is dictatorial. We show that the strong and not very plausible IIA condition can be replaced with a minimal independence assumption plus a Pareto-like condition. This new version of the impossibility theorem likens it to Arrow's and arguably enhances its paradoxical value.     

Impossibility Theorems with Interpersonally Comparable Welfare Levels: The "Extended Sympathy Approach" Reconsidered

We re-examine a type of interpersonal welfare comparison, called the "extended sympathy" approach, which Arrow (1977), Hammond (1976) and Roberts (1980a) introduced in order to escape from Arrovian impossibility theorems. In particular, we extend the positional dictatorship theorem due to Roberts to the case where the domain of social choice rules satisfies the axiom of identity. We show that there is a positional dictator if the rule with the domain satisfies independence, Suppes unanimity and monotonicity
JEL Classification Numbers: 022, 025, 026     

Preference aggregation under uncertainty: Savage vs. Pareto

Following Mongin [J. Econ. Theory 66 (1995) 313; J. Math. Econ. 29 (1998) 331], we study social aggregation of subjective expected utility preferences in a Savage framework. We argue that each of Savage's P3 and P4 are incompatible with the strong Pareto property. A representation theorem for social preferences satisfying Pareto indifference and conforming to the state-dependent expected utility model is provided.     

Social welfare orderings for ratio-scale measurable utilities

Summary. This article characterizes all of the continuous social welfare orderings which satisfy the Weak (resp. Strong) Pareto principle when utilities are ratio-scale measurable. With Weak Pareto, on both the nonnegative and positive orthants the social welfare ordering must be representable by a weakly increasing Cobb-Douglas social welfare function while on the whole Euclidean space the social welfare ordering must be strongly dictatorial. With Strong Pareto, on the positive orthant the social welfare ordering must be representable by a strictly increasing Cobb-Douglas social welfare function but on the other two domains an impossibility theorem is obtained. Received: July 31, 1995; revised version August 7, 1996     

Social Welfare Functionals on Restricted Domains and in Economic Environments

Arrow's “impossibility” and similar classical theorems are usually proved for an unrestricted domain of preference profiles. Recent work extends Arrow's 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 Wilson's theorem and of the strong and weak “welfarism” results due to d'Aspremont 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.     

SOCIAL CHOICE FROM AN n‐DIMENSIONAL EUCLIDEAN SPACE: A SHORT IMPOSSIBILITY PROOF

Most presentations of social choice theory are difficult for a non‐specialist to follow. This paper attempts to provide some feel for what is going on by providing a short proof of an impossibility theorem in terms of the techniques that are familiar to political‐economists. We illustrate the advantages of this approach with a short proof of an impossibility theorem.     

Introduction to judgment aggregation

This introduces the symposium on judgment aggregation. The theory of judgment aggregation asks how several individuals' judgments on some logically connected propositions can be aggregated into consistent collective judgments. The aim of this introduction is to show how ideas from the familiar theory of preference aggregation can be extended to this more general case. We first translate a proof of Arrow's impossibility theorem into the new setting, so as to motivate some of the central concepts and conditions leading to analogous impossibilities, as discussed in the symposium. We then consider each of four possible escape-routes explored in the symposium.     

Implementation in undominated strategies by bounded mechanisms: Some results on compromise alternatives

We consider an impossibility result in Börgers (1991), which says in a restricted environment with two players or three alternatives it is impossible to implement a social choice correspondence that is efficient, enforces compromises at a profile and is implementable in undominated strategies by a bounded mechanism. We extend and refine this result in many ways- we generalize the impossibility result for any number of players and alternatives when the compromises are enforced at a near-unanimous preference profile. We further show that the impossibility result in Börgers (1991) holds good if we replace efficiency of SCCs by neutrality. Also the impossibility result holds good for two agents and any number of alternatives when the SCC is unanimous and minimal. Interestingly, we get a possibility result when we relax the assumption of minimality.     

Individual decisions and group decisions: The fundamental differences

The paper contains a unified development of Arrow's impossibility theorem for rational group decisions, Gibbard and Satterthwaite's impossibility theorem for strategy-proof group decisions, and the close reciprocal relationship that exists between these two theorems.     

Weak Independence and Social Semi‐Orders

This paper provides variants of Arrow's impossibility theorem, which states that there exists no non‐dictatorial aggregation rule satisfying weak Pareto, independence of irrelevant alternatives and collective rationality. In this paper, independence of irrelevant alternatives and collective rationality are simultaneously relaxed. Weak independence is imposed instead of independence of irrelevant alternatives. Social preferences are assumed to satisfy the semi‐order properties of semi‐transitivity and the interval‐order property. We prove that there exists a vetoer when the number of alternatives is greater than or equal to six.     

Policy Discrimination with and without Interpersonal Comparisons of Utility

Can the Pareto criterion guide policymakers who do not know the true model of the economy? If policymakers specify ex ante preferences for agents, then Pareto improvements from a distorted status quo are usually possible, and with more commodities than states, one can implement almost every Pareto optimum. Unlike the standard second welfare theorem, planners cannot dictate allocations: agents must trade. Unfortunately ex ante preferences impose interpersonal comparisons. If policymakers merely aim to maximize some social welfare function then optimal policies form an open set; hence small changes in the environment do not necessitate any policy response. Planners with symmetric information about agents can sometimes intervene without making interpersonal comparisons.     

The veil of public ignorance

A theory of cooperative choice under incomplete information is developed in which agents possess private information at the time of contracting and have agreed on a utilitarian “standard of evaluation” governing choices under complete information. The task is to extend this standard to situations of incomplete information. Our first main result generalizes Harsanyi's (J. Polit. Econ. 63 (1955) 309) classical result to situations of incomplete information, assuming that group preferences satisfy Bayesian Coherence and Interim Pareto Dominance. These axioms are mutually compatible if and only if a common prior exists. We argue that this result partly resolves the impossibility of Bayesian preference aggregation under complete information.