Asymptotic Collusion-Proofness of Voting Rules: The Case of Large Number of Candidates

TL;DR

This paper investigates how the likelihood of manipulation in voting rules diminishes as the number of candidates grows, introducing asymptotic notions of strategy-proofness and collusion-proofness under different cultural assumptions.

Contribution

It introduces asymptotic strategy-proofness and collusion-proofness concepts and analyzes their behavior for various voting rules under IC and ISC assumptions with many candidates.

Findings

Manipulability bounds decrease as candidates increase under certain conditions.

Tie-breaking rules significantly influence the robustness of voting rules.

Coalition size impacts the likelihood of successful manipulation.

Abstract

Classical results in voting theory show that strategic manipulation by voters is inevitable if a voting rule simultaneously satisfy certain desirable properties. Motivated by this, we study the relevant question of how often a voting rule is manipulable. It is well known that elections with a large number of voters are rarely manipulable under impartial culture (IC) assumption. However, the manipulability of voting rules when the number of candidates is large has hardly been addressed in the literature and our paper focuses on this problem. First, we propose two properties (1) asymptotic strategy-proofness and (2) asymptotic collusion-proofness, with respect to new voters, which makes the two notions more relevant from the perspective of computational problem of manipulation. In addition to IC, we explore a new culture of society where all score vectors of the candidates are equally likely. This new notion has its motivation in computational social choice and we call it impartial scores culture (ISC) assumption. We study asymptotic strategy-proofness and asymptotic collusion-proofness for plurality, veto, $k$-approval, and Borda voting rules under IC as well as ISC assumptions. Specifically, we prove bounds for the fraction of manipulable profiles when the number of candidates is large. Our results show that the size of the coalition and the tie-breaking rule play a crucial role in determining whether or not a voting rule satisfies the above two properties.