Existence of a coalitionally strategyproof social choice function: A constructive proof

This paper gives a concrete example of a nondictatorial, coalitionally strategyproof social choice function for countably infinite societies. The function is defined for those profiles such that for each alternative, the coalition that prefers it the most is “describable.” The “describable” coalitions are assumed to form a countable Boolean algebra. The paper discusses oligarchical characteristics of the function, employing a specific interpretation of an infinite society. The discussion clarifies within a single framework a connection between the negative result (the Gibbard-Satterthwaite theorem) for finite societies and the positive result for infinite ones. Received: 10 August 1998/Accepted: 29 February 2000     

On strategy-proof social choice under categorization

I consider social choice problems such that (i) the set of alternatives can be partitioned into categories based on a prominent and objective feature and (ii) agents have strict preferences over the alternatives. Main results are characterizations of the structure of the strategy-proof social choice functions. I prove that each social choice function is strategy-proof if and only if it is decomposable into “small” strategy-proof social choice functions; one of them chooses one category and each of the others chooses one alternative from a category.     

Strategy-proof Cardinal Decision Schemes

This paper analyses strategy-proof mechanisms or decision schemes which map profiles of cardinal utility functions to lotteries over a finite set of outcomes. We provide a new proof of Hylland’s theorem which shows that the only strategy-proof cardinal decision scheme satisfying a weak unanimity property is the random dictatorship. Our proof technique assumes a framework where individuals can discern utility differences only if the difference is at least some fixed number which we call the grid size. We also prove a limit random dictatorship result which shows that any sequence of strategy-proof and unanimous decision schemes defined on a sequence of decreasing grid sizes approaching zero must converge to a random dictatorship. We are most grateful to an Associate Editor and two referees for very helpful comments on an earlier version of the paper. An erratum to this article can be found at     

On the manipulation of social choice correspondences

Duggan and Schwartz (Soc Choice and Welfare 17: 85–93, 2000) have proposed a generalization of the Gibbard–Satterthwaite Theorem to multivalued social choice rules. They show that only dictatorial rules are strategy-proof and satisfy citizens sovereignty and residual resoluteness. Citizens sovereignty requires that each alternative is chosen at some preference profile. Residual resoluteness compels the election to be single-valued when the preferences of the voters are “similar”. We propose an alternative proof to the Duggan and Schwartz’s Theorem. Our proof highlights the crucial role of residual resoluteness. In addition, we prove that every strategy-proof and onto social choice correspondence concentrates the social decision power in the hands of an arbitrary group of voters. Finally, we show that this result still holds in a more general framework in which voters report their preferences over sets of alternatives.     

Generic difference of expected vote share and probability of victory maximization in simple plurality elections with probabilistic voters

In this paper I examine single member, simple plurality elections with n ≥ 3 probabilistic voters and show that the maximization of expected vote share and maximization of probability of victory are “generically different” in a specific sense. More specifically, I first describe finite shyness (Anderson and Zame in Adv Theor Econ 1:1–62, 2000), a notion of genericity for infinite dimensional spaces. Using this notion, I show that, for any policy in the interior of the policy space and any candidate j, the set of n-dimensional profiles of twice continuously differentiable probabilistic voting functions for which simultaneously satisfies the first and second order conditions for maximization of j’s probability of victory and j’s expected vote share at is finitely shy with respect to the set of n-dimensional profiles of twice continuously differentiable probabilistic voting functions for which satisfies the first and second order conditions for maximization of j’s expected vote share.     

Opportunity sets and individual well-being

 An opportunity set ranking rule assigns an ordering of opportunity sets to each individual utility function (defined on the universal set of alternatives) within the domain of this rule. Using an axiomatic approach, this paper characterizes a general class of opportunity set ranking rules which are based on the utilities associated with the elements of an opportunity set. It is argued that the addition of an alternative to a given opportunity set is not necessarily desirable in terms of overall well-being, and this position is reflected in replacing a commonly used monotonicity axiom with an alternative condition. Received: 15 May 1995/Accepted: 14 December 1995     

The replacement principle for the provision of multiple public goods on tree networks

This article considers the provision of two public goods on tree networks where each agent has a single-peaked preference. We show that if there are at least four agents, then no social choice rule exists that satisfies efficiency and replacement-domination. In fact, these properties are incompatible, even if agents’ preferences are restricted to a smaller domain of symmetric single-peaked preferences. However, for rules on an interval, we prove that Miyagawa’s (Soc Choice Welf 18:527–541, 2001) characterization that only the left-peaks rule and the right-peaks rule satisfy both of these properties also holds on the domain of symmetric single-peaked preferences. Moreover, if agents’ peak locations are restricted to either the nodes or the endpoints of trees, rules exist on a subclass of trees. We provide a characterization of a family of such rules for this tree subclass.     

Single-Crossing, Strategic Voting and the Median Choice Rule

This paper studies the strategic foundation of the Representative Voter Theorem (Rothstein in: Pub Choice 72:193–212, 1991), also called the “second version” of the Median Voter Theorem. As a by-product, it also considers the existence of strategy-proof social choice functions over the domain of single-crossing preferences. The main result shows that single-crossing constitutes a domain restriction over the real line that allows not only majority voting equilibria, but also non-manipulable choice rules. In particular, this is true for the median rule, which is found to be group strategic-proof over the full set of alternatives and over every nonempty subset. In addition, the paper also examines the relation between single-crossing and order-restriction. And it uses this relation together with the strategy-proofness of the median rule to prove that the outcome predicted by the Representative Voter Theorem can be implemented in dominant strategies through a simple mechanism. This mechanism is a two-stage voting procedure in which, first, individuals select a representative among themselves, and then the winner chooses a policy to be implemented by the planner.     

Contraction consistent stochastic choice correspondence

We model a general choice environment via probabilistic choice correspondences, with (possibly) incomplete domain and infinite universal set of alternatives. We offer a consistency restriction regarding choice when the feasible set contracts. This condition, ‘contraction consistency’, subsumes earlier notions such as Chernoff’s Condition, Sen’s α and β, and regularity. We identify a restriction on the domain of the stochastic choice correspondence (SCC), under which contraction consistency is equivalent to the weak axiom of revealed preference in its most general form. When the universal set of alternatives is finite, this restriction is also necessary for such equivalence. Analogous domain restrictions are also identified for the special case where choice is deterministic but possibly multi-valued. Results due to Sen (Rev Econ Stud 38:307–317, 1971) and Dasgupta and Pattanaik (Econ Theory 31:35–50, 2007) fall out as corollaries. Thus, conditions are established, under which our notion of consistency, articulated only in reference to contractions of the feasible set, suffices as the axiomatic foundation for a general revealed preference theory of choice behaviour.     

Anonymity and neutrality in Arrow’s Theorem with restricted coalition algebras

 In the very general setting of Armstrong (1980) 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. Received: 11 August 1994/Accepted: 1 April 1996     

The majoritarian compromise is majoritarian-optimal and subgame-perfect implementable

It is shown that the Majoritarian Compromise of Sertel (1986) is subgame-perfect implementable on the domain of strict preference profiles, although it fails to be Maskin-monotonic and is hence not implementable in Nash equilibrium. The Majoritarian Compromise is Pareto-optimal and obeys SNIP (strong no imposition power), i.e. never chooses a strict majority's worst candidate. In fact, it is “majoritarian approving” i.e. it always picks “what's good for a majority” (alternatives which some majority regards as among the better “effective” half of the available alternatives). Thus, being Pareto-optimal and majoritarian approving, it is majoritarian-optimal. Finally, the Majoritarian Compromise is measured against various criteria, such as consistency and Condorcet-consistency. Received: 31 January 1995/Accepted: 22 July 1998     

The geometry of implementation: a necessary and sufficient condition for straightforward games

 We characterize games which induce truthful revelation of the players’ preferences, either as dominant strategies (straightforward games) or in Nash equilibria. Strategies are statements of individual preferences on R ⁿ . Outcomes are social preferences. Preferences over outcomes are defined by a distance from a bliss point. We prove that g is straightforward if and only if g is locally constant or dictatorial (LCD), i.e., coordinate-wise either a constant or a projection map locally for almost all strategy profiles. We also establish that: (i) If a game is straightforward and respects unanimity, then the map g must be continuous, (ii) Straightforwardness is a nowhere dense property, (iii) There exist differentiable straightforward games which are non-dictatorial. (iv) If a social choice rule is Nash implementable, then it is straightforward and locally constant or dictatorial. Received: 30 December 1994/Accepted: 22 April 1996     

Topological aggregation, the case of an infinite population

 The literature on infinite Chichilnisky rules considers two forms of anonymity: a weak and a strong. This note introduces a third form: bounded anonymity. It allows us to prove an infinite analogue of the “Chichilnisky– Heal-resolution” close to the original theorem: a compact parafinite CW-complex X admits a bounded anonymous infinite rule if and only if X is contractible. Furthermore, bounded anonymity is shown to be compatible with the finite and the [0, 1]-continuum version of anonymity and allows the construction of convex means in infinite populations. With X=[0, 1], the set of linear bounded anonymous rules coincides with the set of medial limits. Received: 30 October 1993/Accepted: 22 April 1996     

Social aggregators

This paper proposes a general framework for analyzing a class of functions called social aggregators, which map profiles of linear orders to a set of binary relations. This class of aggregators includes aggregators that yield a preference relation (social welfare functions) and those which yield a choice of an alternative (social choice functions). Equipped with this framework, I identify a property called Preference Reversal (PR) such that any Pareto efficient aggregator having this property must be dictatorial. This allows me to state a general impossibility theorem, which includes Arrows Theorem and the Gibbard Satterthwaite Theorem as two special examples. Furthermore, I show that monotonicity and IIA are closely linked, by demonstrating that both are actually special cases of PR in specific environments.Received: 15 October 2001     

Aggregation of smooth preferences

Suppose p is a smooth preference profile (for a society, N) belonging to a domain P ^N . Let σ be a voting rule, and σ(p)(x) be the set of alternatives in the space, W, which is preferred to x. The equilibrium E(σ(p)) is the set {x∈W:σ(p)(x) is empty}. A sufficient condition for existence of E(σ(p)) when p is convex is that a “dual”, or generalized gradient, dσ(p)(x), is non-empty at all x. Under certain conditions the dual “field”, dσ(p), admits a “social gradient field”Γ(p). Γ is called an “aggregator” on the domain P ^N if Γ is continuous for all p in P ^N . It is shown here that the “minmax” voting rule, σ, admits an aggregator when P ^N is the set of smooth, convex preference profiles (on a compact, convex topological vector space, W) and P ^N is endowed with a C ¹-topology. An aggregator can also be constructed on a domain of smooth, non-convex preferences when W is the compact interval. The construction of an aggregator for a general political economy is also discussed. Some remarks are addressed to the relationship between these results and the Chichilnisky-Heal theorem on the non-existence of a preference aggregator when P ^N is not contractible. Received: 4 July 1995 / Accepted: 26 August 1996     

Theory-bashing and answer-improving in the study of social movements

Over the 1970s and 1980s, a “theory-bashing” mindset gained popularity among sociologists of social movements and, for a period, overshadowed the alternative mindset of seeking to improve answers to questions about social movements. Now on the wane, theory-bashing nonetheless retains a significant presence. This mindset has a number of attractions and virtues and it is, broadly speaking, legitimate. But, it also has negative features and consequences that I want to point out. I begin by showing how the theory-bashing differs from the answer-improving mindset and I then explain ways in which the former hinders the analysis of social movements even though it can also be helpful. Finally, I offer a sociological account of why theory-bashing has been so popular in movement studies. His most recent book isPolite Protesters: The American Peace Movement of the 1980s (Syracuse University Press, 1993).     

Reference functions and possibility theorems for cardinal social choice problems

 In this paper, we provide axiomatic foundations for social choice rules on a domain of convex and comprehensive social choice problems when agents have cardinal utility functions. We translate the axioms of three well known approaches in bargaining theory (Nash 1950; Kalai and Smorodinsky 1975; Kalai 1977) to the domain of social choice problems and provide an impossibility result for each. We then introduce the concept of a reference function which, for each social choice set, selects a point from which relative gains are measured. By restricting the invariance and comparison axioms so that they only apply to sets with the same reference point, we obtain characterizations of social choice rules that are natural analogues of the bargaining theory solutions. Received: 8 August 1994/Accepted: 12 February 1996     

An informationally parsimonious impartial observer theorem

In Harsanyi's impartial observer theorem, an impartial observer determines a social ordering of the lotteries on the set of social alternatives based on a sympathetic but impartial concern for all individuals in society. This ordering is derived from a more primitive ordering on the set of all extended lotteries. An extended lottery is a lottery which determines both the observer's personal identity and the social alternative. We establish a version of Harsanyi's theorem in which the observer is only required to have preferences on the extended lotteries in which there is an equal chance of being any person in society. Received: 19 June 1996 / Accepted: 30 December 1996     

The structure of continuous-valued neutral monotonic social functions

This article consists of several results characterizing neutral monotonic social functions over alternative sets that are compact, convex subsets of Euclidean space. One major result is that a neutral monotonic social function is continuous-valued (i.e., does not make abrupt reversals) for all profiles of continuous-valued weak orderings if and only if it is simple (i.e., completely determined by a single set of decisive coalitions). A second major result is that a continuous-valued neutral monotonic social function will guarantee the existence of a socially undominated alternative for all profiles of continuousvalued, almost convex weak orderings if and only if the smallest empty intersection within the set of decisive coalitions is at least m+2 in size where m is the dimension of the alternative set.Associate Professor of Law, University of Southern California and Assistant Professor of Law and Economics, California Institute of Technology.This article is taken almost directly from a chapter of my Ph.D. dissertation (Strnad 1982). That chapter itself is substantially the same as an earlier paper, Strnad (1981). I have profited greatly from the suggestions, guidance and encouragement of Donald Brown in this work. Strnad (1981) was presented at the California Institute of Technology Theory Workshop in January 1982 and at the annual Public Choice Society Meetings in San Antonio, Texas on March 5–7, 1982. I am grateful to participants at both the Workshop and the Meetings for their comments. Comments by the referee and by Jerry Kramer, the editor, also were very helpful. Any remaining errors are solely my responsibility.Substantial financial support for this work has been provided by Summer Research Grants for 1981 and 1982 from the University of Southern California Law Center     

Election of vice-president and adding new members to representative bodies

 In Tanguiane (1991, 1993, 1994) we have introduced quantitative indicators of representativeness, with which we have estimated the capacity of individuals and limited groups to represent a collective preference. We have studied three forms of representation: (a) single representative (president); (b) cabinet which consists of representatives personally responsible for certain domains of competence (government); and (c) council which makes collective decisions by means by voting (parliament). In this paper we examine the appointment of president and vice-president. In our model it corresponds to the appointment of a cabinet with two members. We show that it may be impossible to make an optimal appointment successively, finding first the most representative president, and matching the vice-president afterwards. The only way which guarantees their optimal appointment, is choosing them together as a team. We prove, however, that successively chosen president and vice-president, as a cabinet with two members, have the indicators of representativeness greater than or equal to 75% of their maximal value. Besides we investigate a recursive construction of cabinets and councils by optimally adding new members one by one. We prove that the indicators of representativeness of such a recursively constructed cabinet with k members are greater or equal to (1–2^-k) ⋅ 100% of their maximal value. This estimate has the same exponent as that for the optimal cabinets, meaning that such a recursive construction provides, if not optimal, still rather good results. The recursive construction of representative councils is restricted to particular cases, so that an optimal council should be chosen simultaneously. In conclusion we discuss the applicability of the results obtained to real politics. Received: 27 December 1994/Accepted: 15 November 1995