We consider a voting setting where candidates have preferences about theoutcome of the election and are free to join or leave the election. Thecorresponding candidacy game, where candidates choose strategically toparticipate or not, has been studied %initially by Dutta et al., who showedthat no non-dictatorial voting procedure satisfying unanimity iscandidacy-strategyproof, that is, is such that the joint action where allcandidates enter the election is always a pure strategy Nash equilibrium. Duttaet al. also showed that for some voting tree procedures, there are candidacygames with no pure Nash equilibria, and that for the rule that outputs thesophisticated winner of voting by successive elimination, all games have a pureNash equilibrium. No results were known about other voting rules. Here we proveseveral such results. For four candidates, the message is, roughly, that mostscoring rules (with the exception of Borda) do not guarantee the existence of apure Nash equilibrium but that Condorcet-consistent rules, for an odd number ofvoters, do. For five candidates, most rules we study no longer have thisguarantee. Finally, we identify one prominent rule that guarantees theexistence of a pure Nash equilibrium for any number of candidates (and for anodd number of voters): the Copeland rule. We also show that under mildassumptions on the voting rule, the existence of strong equilibria cannot beguaranteed.
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