Ton Storcken

Strategy-Proof Probabilistic Decision Schemes for One-Dimensional Single-Peaked Preferences

Abstract Collective decision problems are considered with a finite number of agents who have single-peaked preferences on the real line. A probabilistic decision scheme assigns a probability distribution over the real line to every profile of reported preferences. The main result of the paper is a characterization of the class of unanimous and strategy-proof probabilistic schemes with the aid of fixed probability distributions that play a role similar to that of the phantom voters in H. Moulin ( Public Choice 35 (1980), 437–455). Thereby, the work of Moulin (1980) is extended to the probabilistic framework. Journal of Economic Literature Classification Numbers: D71, D81.

Strategy-proof division of a private good when preferences are single-dipped

Abstract Pareto-optimal and strategy-proof distributions of a perfectly divisible good among agents with single-dipped preferences are studied. In order to satisfy these two properties and, in addition, either a so-called replacement property or a property of consistency, the whole amount should be assigned to one of the agents. Characterizations of the two classes of division rules satisfying the above conditions are provided.

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.

Pareto optimality, anonymity, and strategy-proofness in location problems

Generalized location problems withn agents are considered, who each report a point inm-dimensional Euclidean space. A solution assigns a compromise point to thesen points, and the individual utilities for this compromise point are equal to the negatives of the Euclidean distances to the individual positions. Form = 2 andn odd, it is shown that a solution is Pareto optimal, anonymous, and strategy-proof if, and only if, it is obtained by taking the coordinatewise median with respect to a pair of orthogonal axes. Further, for all other situations withm≥2, such a solution does not exist. A few results concerning other solution properties, as well as different utility functions, are discussed.

Minimal manipulability: anonymity and unanimity

This paper is concerned with the minimal number of profiles at which a unanimous and anonymous social choice function for three alternatives is manipulable. The lower bound is derived and examples of social choice functions attaining the lower bound are given. It is conjectured that these social choice functions are in fact all minimally manipulable social choice functions. Since some of these social choice functions are Pareto optimal, it follows that the lower bound also holds for Pareto optimal and anonymous social choice functions. Some of the minimally manipulable Pareto optimal and anonymous social choice functions can be interpreted as status quo voting.

Nash Consistent Representation of Constitutions: A Reaction to the Gibbard Paradox

The concept of an effectivity function is adopted as a formal model of a constitution. A game form models the actions available and permissible to individuals in a society. As a representation of the constitution such a game form should endow each group in society with the same power as it has under the constitution. Another desirable property is Nash consistency of the game form: whatever the individual preferences, the resulting game should be minimally stable in the sense of possessing a Nash equilibrium. A first main result of the paper is a characterization of all effectivity functions that have a Nash consistent representation for the case without special structure on the set of alternatives (social states). Next, a similar result is derived for the case where the set of alternatives is a topological space and the effectivity function is topological. As a special case, veto functions are considered. Further results concern Pareto optimality of Nash equilibrium outcomes.

Centers in Connected Undirected Graphs: An Axiomatic Approach

A center is a function that associates with every finite connected and undirected graph a nonempty subset of its vertices. These functions play an important role in networks such as social or interorganizational networks. Centers capture notions like: being a focal point of communication, being strategically located, ability and willingness to participate in strategic alliances, and the like. We focus on the conceptual issue of what makes a position in a graph a central one and investigate some possible concepts of centrality in relation to various properties. Characterizations of the uncovered center, the median, and degree center are presented, where each of these centers is defined for arbitrary connected undirected simple, and possibly cyclic, graphs.

Threshold strategy-proofness: on manipulability in large voting problems

In voting problems where agents have well behaved (Lipschitz continuous) utility functions on a multidimensional space of alternatives, a voting rule is threshold strategy-proof if any agent can only obtain a limited utility gain by not voting for a most preferred alternative,given that the number of agents is large enough. For anonymous voting rules it is shown that this condition is not only implied by but in fact equivalent to the influence of any single agent reducing to zero as the number of agents grows. If there are at least five agents, the mean rule (taking the average vote) is shown to be the unique anonymous and unanimous voting rule that meets a lower bound with respect to the number of agents needed to obtain threshold strategy-proofness.

Arrow's Possibility Theorem for one-dimensional single-peaked preferences

In one-dimensional environments with single-peaked preferences we consider social welfare functions satisfying Arrows requirements, i.e. weak Pareto and independence of irrelevant alternatives. When the policy space is a one-dimensional continuum such a welfare function is determined by a collection of 2N strictly quasi-concave preferences and a tie-breaking rule. As a corollary we obtain that when the number of voters is odd, simple majority voting is transitive if and only if each voters preference is strictly quasi-concave.

Maximal Domains for Strategy-Proof or Maskin Monotonic Choice Rules

Domains of individual preferences for which the well-known impossibility Theorems of Gibbard-Satterthwaite and Muller-Satterthwaite do not hold are studied. First, we introduce necessary and sufficient conditions for a domain to admit non-dictatorial, Pareto efficient and either strategy-proof or Maskin monotonic social choice rules. Next, to comprehend the limitations the two Theorems imply for social choice rules, we search for the largest domains that are possible. Put differently, we look for the minimal restrictions that have to be imposed on the unrestricted domain to recover possibility results. It turns out that, for such domains, the conditions of inseparable pair and of inseparable set yield the only maximal domains on which there exist non-dictatorial, Pareto efficient and strategy-proof social choice rules. Next, we characterize the maximal domains which allow for Maskin monotonic, non-dictatorial and Pareto-optimal social choice rules.