Many-Candidate Nash Equilibria for Elections Involving Random Selection

We consider various voter game theory models which involve some form of random selection, including random tie-breaking, and single elimination, and runoff of the top two candidates. Under certain rules for resolving ties, we prove that with any number of candidates, each such model has a Nash equilibrium in which all candidates attempt to contest the election at the median policy. For models which do not permit ties, we prove that each such model has a Nash equilibrium in which the number of candidates contesting the election is essentially equal to the ratio of the positive payoff for winning divided by the negative of the payoff for losing. All of these model variations thus predict lots of candidates. This result contrasts with Duverger’s Law, which asserts that only two (major) candidates will contest the election at all, and which has been confirmed in some other voter game theory models. However, it is consistent with recent primary and leadership and runoff elections where the number of major candidates reached two figures. We close with a simulation study showing that, through repeated elections and averaging and tweaking, candidates’ actions will sometimes converge to their predicted equilibrium behaviour.