Strategy-proof voting rules on a multidimensional policy space for a continuum of voters with elliptic preferences

Hans Peters,Ton Storcken,Souvik Roy

doi:10.1007/s13209-011-0048-5

Hans Peters, Ton Storcken + Show 1 more

Open Access

https://doi.org/10.1007/s13209-011-0048-5

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Journal: SERIEs	Publication Date: Feb 24, 2011
Citations: 8	License type: CC BY 2.0

Affiliation: Maastricht University, Université de Caen Normandie

Abstract

We consider voting rules on a multidimensional policy space for a continuum of voters with elliptic preferences. Assuming continuity, γ-strategy-proofness—meaning that coalitions of size smaller or equal to a small number γ cannot manipulate—and unanimity, we show that such rules are decomposable into one-dimensional rules. Requiring, additionally, anonymity leads to an impossibility result. The paper can be seen as an extension of the model of Border and Jordan (1983) to a continuum of voters. Contrary, however, to their finite case where single voters are atoms, in our model with nonatomic voters even a small amount of strategy-proofness leads to an impossibility.

Highlights

We consider voting rules for situations with a large number of voters, who have singlepeaked preferences on a multidimensional policy space
In the context of a national election, such a point may represent a specific political party; in this context, voters normally vote for a finite number of parties, but allowing them to vote for any position in the hypercube is again an approximation of the finite party case
Our model extends the model of Border and Jordan (1983), where the number of voters is finite, to a continuum of voters

Summary

Introduction

We consider voting rules for situations with a large number of voters, who have singlepeaked preferences on a multidimensional policy space. Border and Jordan impose strategy-proofness and unanimity on a voting rule and obtain decomposability: such a voting rule is completely determined by one-dimensional voting rules applied to each coordinate separately, and these one-dimensional voting rules are of the type as characterized earlier in Moulin (1980) It is well-known that, these one-dimensional voting rules are group-strategy proof (cannot be manipulated by coalitions of voters) this property is lost as soon as the dimension is higher than one. We obtain an impossibility result: there is no unanimous, anonymous and continuous rule for more than one dimension which is non-manipulable, even if we require this only for coalitions of arbitrarily small size.

Preliminaries

The one-dimensional case

The general case: decomposability and impossibility

Without unanimity

Without anonymity

Without continuity

Different domains