  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.
