Voting under Constraints (original) (raw)

An extension of May's Theorem to three alternatives: axiomatizing Minimax voting

2023

May’s Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May’s axioms, we can uniquely determine how to vote on three alternatives (setting aside tiebreaking). In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, must choose from among the Minimax winners in all three- alternative elections. When applied to more than three alternatives, our axioms also distinguish Minimax from other known voting methods that coincide with or refine Minimax for three alternatives.

A unified approach to strategy-proofness for single-peaked preferences

Series, 2011

This article establishes versions of Moulin’s (Public Choice 35:437–455, 1980) characterizations of various classes of strategy-proof social choice functions when the domain consists of all profiles of single-peaked preferences on an arbitrary subset of the real line. Two results are established that show that the median of 2n + 1 numbers can be expressed using a combination of minimization and maximization operations applied to subsets of these numbers when either these subsets or the numbers themselves are restricted in a particular way. These results are used to show how Moulin’s characterizations of generalized median social choice functions can be obtained as corollaries of his characterization of min–max social choice functions.

Single-peakedness and strategy-proofness of generalized median voter schemes

Social Choice and Welfare, 2002

We identify, in a continuous multidimensional framework, a maximal domain of preferences compatible with strategy-proofness for a given generalized median voter scheme. It turns out that these domains are a variation of single-peakedness. A similar but stronger result for the discrete case and singlepeakedness has been already obtained by Barbera Á et al. (1999). However, both results are independent and their proofs involve di¨erent arguments. This work is part of my Thesis dissertation and it would not have come out without the help of Jordi Masso Â. He gave me very helpful comments and suggestions. I am also grateful to three anonymous referees and an associate editor for very helpful comments. This work has been partially ®nanced by a grant from the European Community through the Universite  de Caen, MRSH and Aura Phenix.

Some Comments on Strategic Voting Extended Abstract

2006

Whether made explicit or implicit, knowledge theoretic properties such as common knowledge of rationality are important in understanding and modeling game-theoretic, or strategic, situations. There is a large literature devoted to exploring these and other issues related to the epistemic foundations of game theory. Much of the literature focuses on what the agents need to know about the other agents' strategies, rationality or knowledge in order to guarantee that a particular solution concept, such as the Nash equilibrium, is realized. This paper, which is based on two recent papers 1 [7] and [16], develops a framework that looks at similar issues relevant to the field of voting theory. Our analysis suggests that an agent must possess information about the other agents' preferences in order for the agent to decide to vote strategically. In a sense, our claim is that the agents need a certain amount of information in order for the Gibbard-Satterthwaite theorem to be "effective".

New characterizations of strategy-proofness under single-peakedness

Mathematical Programming

We provide novel representations of strategy-proof voting rules applicable when voters have uni-dimensional single-peaked preferences. In particular, we introduce a ‘grading curve’ representation which is particularly useful when introducing variable electorates. Our analysis recovers, links and unifies existing results in the literature, and provides new characterizations when strategy-proofness is combined with other desirable properties such as ordinality, participation, consistency, and proportionality. Finally, the new representations are used to compute the strategy-proof methods that maximize the ex-ante social welfare for the L_2$$ L 2 -norm and a uniform prior. The resulting strategy-proof welfare maximizer is the linear median (or ‘uniform median’), that we also characterize as the unique proportional strategy-proof voting rule.

Manipulation in elections with uncertain preferences

J Math Econ, 2008

A decision scheme ) is a function mapping profiles of strict preferences over a set of social alternatives to lotteries over the social alternatives. Motivated by conditions typically prevailing in elections with many voters, we say that a decision scheme is weakly strategy-proof if it is never possible for a voter to increase expected utility (for some vNM utility function consistent with her true preferences) by misrepresenting her preferences when her belief about the preferences of other voters is generated by a model in which the other voters are i.i.d. draws from a distribution over possible preferences. We show that if there are at least three alternatives, a decision scheme is necessarily a random dictatorship if it is weakly strategy-proof, never assigns positive probability to Pareto dominated alternatives, and is anonymous in the sense of being unaffected by permutations of the components of the profile. This result is established in two settings: a) a model with a fixed set of voters; b) the Poisson voting model of Meyerson (Abstract A decision scheme (Gibbard ) is a function mapping profiles of strict preferences over a set of social alternatives to lotteries over the social alternatives. Motivated by conditions typically prevailing in elections with many voters, we say that a decision scheme is weakly strategy-proof if it is never possible for a voter to increase expected utility (for some vNM utility function consistent with her true preferences) by misrepresenting her preferences when her belief about the preferences of other voters is generated by a model in which the other voters are i.i.d. draws from a distribution over possible preferences. We show that if there are at least three alternatives, a decision scheme is necessarily a random dictatorship if it is weakly strategy-proof, never assigns positive probability to Pareto dominated alternatives, and is anonymous in the sense of being unaffected by permutations of the components of the profile. This result is established in two settings: a) a model with a fixed set of voters; b) the Poisson voting model of