A Little Bit Nasty, Some of the Time: Mixed Strategy Equilibria in Political Campaigns with Continuous Negativity

Reputation interacts with electoral outcomes in both directions: a candidate with a higher reputation is more likely to win, and winning directly raises reputation. In this context, we assume that negative campaigning is a continuous variable that increases a candidate’s chances of winning by lowering the opponent’s reputation. However, it also lowers the candidate’s own reputation, either due to voters’ dislike of candidates who go negative, or because of the opportunity cost of forgoing positive ads. The candidate who ends the campaign with the higher reputation wins the election. For candidates who seek to maximize their post campaign reputations, we derive necessary conditions for mixed strategy equilibria. These conditions are not always compatible, so equilibria do not always exist. For a linear specification, we specify when an equilibrium exists and derive properties when it does. When no continuous equilibrium exists, we consider equilibria for discrete approximations to the game.