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 by Dutta et al. [6], 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. In [7] 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.

We study equilibrium dynamics in candidacy games, in which candidates may strategically decide to enter the election or withdraw their candidacy, following their own preferences over possible outcomes. Focusing on games under Plurality, we extend the standard model to allow for situations where voters may refuse to return their votes to those candidates who had previously left the election, should they decide to run again. We show that if at the time when a candidate withdraws his candidacy, with some positive probability each voter takes this candidate out of his future consideration, the process converges with probability 1. This is in sharp contrast with the original model where the very existence of a Nash equilibrium is not guaranteed. We then consider the two extreme cases of this setting, where voters may block a withdrawn candidate with probabilities 0 or 1. In these scenarios, we study the complexity of reaching equilibria from a given initial point, converging to an equilibrium with a predermined winner or to an equilibrium with a given set of running candidates. Except for one easy case, we show that these problems are NP-complete, even when the initial point is fixed to a natural---truthful---state where all potential candidates stand for election.