Instance-optimal information-based voting

Abstract

The classical Condorcet Jury Theorem considers a voting scenario in which there exists a candidate whose election would be ideal for each voter; each voter, though, has only a limited understanding of the world and is thus unable to determine exactly who this candidate is. The main question in this scenario is whether the voters, acting individually, can cast their ballots so that the unknown optimal candidate wins the election, and the welfare of the group of voters is maximized.

In this setting, each candidate is represented by a known probability distribution over signals about the world that the voters can perceive, that is, over bits. One of these candidates is chosen (secretively, by an adversary) to be the ideal candidate. Afterwards, each voter samples this unknown candidate's distribution once and casts a ballot with the hope that the unknown ideal candidate wins the election.

In this paper, we consider the famous Condorcet voting system, as well as some of its variants. First, we give a positive answer to an open question of Chierichetti and Kleinberg [8], and show that, with Condorcet voting, there exists a uniform voting strategy that makes the group of voters succeed with probability $1-\delta$ provided that $\Theta ({ϵ}_{\mathrm{tv}}^{-2}\cdot \ln {\delta }^{-1})$ voters take part in the election — here, ${ϵ}_{\mathrm{tv}}$ is the minimum total variation distance between the distributions of two candidates.

We also give a uniform voting strategy for the Copeland voting system (a variant of Condorcet) that makes the group succeed with probability $1-\delta$ with $\Theta ({ϵ}_H^{-2}\cdot \ln {\delta }^{-1})$ voters, where ${ϵ}_H$ is the minimum Hellinger distance between the distributions. Our uniform Copeland strategy, then, is an instance-optimal hypothesis testing algorithm: constants aside, the strategy is as efficient as the optimal omniscient algorithm which determines the unknown candidate after having directly observed each of the signals perceived by the voters. Then, we “derandomize” our uniform Copeland strategy, and obtain a Condorcet strategy that achieves instance-optimality at the cost of losing uniformity; finally, we prove that this loss of uniformity is necessary: no uniform Condorcet strategy can achieve instance-optimality, in general.

Thus, the right voting strategies let these classical combinatorial voting systems attain the same efficiency of centralized, optimal, hypothesis testers.