New Results on Equilibria in Strategic Candidacy

TL;DR

This paper investigates the existence of pure Nash equilibria in candidacy games across various voting rules, revealing that Condorcet-consistent rules and Copeland rule often guarantee equilibria, unlike most scoring rules.

Contribution

It provides new theoretical results on the existence of pure Nash equilibria in candidacy games for different voting rules, including conditions under which they are guaranteed or not.

Findings

Most scoring rules (except Borda) lack pure Nash equilibrium guarantees for four candidates.

Condorcet-consistent rules with an odd number of voters often guarantee pure Nash equilibria.

The Copeland rule guarantees the existence of pure Nash equilibria for any number of candidates.

Abstract

We consider a voting setting where candidates have preferences about the outcome of the election and are free to join or leave the election. The corresponding candidacy game, where candidates choose strategically to participate or not, has been studied %initially by Dutta et al., who showed that no non-dictatorial voting procedure satisfying unanimity is candidacy-strategyproof, that is, is such that the joint action where all candidates enter the election is always a pure strategy Nash equilibrium. Dutta et al. also showed that for some voting tree procedures, there are candidacy games with no pure Nash equilibria, and that for the rule that outputs the sophisticated winner of voting by successive elimination, all games have a pure Nash equilibrium. No results were known about other voting rules. Here we prove several such results. For four candidates, the message is, roughly, that most scoring rules (with the exception of Borda) do not guarantee the existence of a pure Nash equilibrium but that Condorcet-consistent rules, for an odd number of voters, do. For five candidates, most rules we study no longer have this guarantee. Finally, we identify one prominent rule that guarantees the existence of a pure Nash equilibrium for any number of candidates (and for an odd number of voters): the Copeland rule. We also show that under mild assumptions on the voting rule, the existence of strong equilibria cannot be guaranteed.