New Results on Equilibria in Strategic Candidacy
This addresses a theoretical gap in voting theory for researchers, providing new equilibrium results for strategic candidacy, but it is incremental as it builds on prior work by Dutta et al.
The paper tackles the problem of whether voting rules guarantee pure Nash equilibria in strategic candidacy games, showing that for four candidates most scoring rules lack this guarantee except Borda, while Condorcet-consistent rules do, but for five candidates most rules lose this property, and identifies the Copeland rule as guaranteeing equilibrium for any number of candidates.
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.