New Characterizations of Strategy-Proofness under Single-Peakedness

TL;DR

This paper introduces new simple representations of strategy-proof voting rules under single-peaked preferences, unifying and extending existing characterizations while analyzing their computational properties and implications for various desirable voting properties.

Contribution

It provides novel simple representations of strategy-proof rules, compares their computational complexity, and derives new characterizations when combined with other voting properties.

Findings

Grading curve representation is computationally superior.

New characterizations for strategy-proofness with properties like anonymity and participation.

Ordinal median and linear median are key methods in the anonymous case.

Abstract

We provide novel simple representations of strategy-proof voting rules when voters have uni-dimensional single-peaked preferences (as well as multi-dimensional separable preferences). The analysis recovers, links and unifies existing results in the literature such as Moulin's classic characterization in terms of phantom voters and Barber\`a, Gul and Stacchetti's in terms of winning coalitions ("generalized median voter schemes"). First, we compare the computational properties of the various representations and show that the grading curve representation is superior in terms of computational complexity. Moreover, the new approach allows us to obtain new characterizations when strategy-proofness is combined with other desirable properties such as anonymity, responsiveness, ordinality, participation, consistency, or proportionality. In the anonymous case, two methods are single out: the -- well know -- ordinal median and the -- most recent -- linear median.