Contest with Attack and Defence: Does Negative Campaigning Increase or Decrease Voters' Turnout?

We present a general model of two players contest with two types of efforts. Contrary to the classical models of contest, where each player chooses a unique effort, and where the outcome depends on the efforts of all the players, contestants are allowed to reduce the effort of the opponent. Defence increases one's chance of winning while attack annihilates the defence of the opponent. This model has many applications like political campaigning, wars, competition among lobbies, job promotion competitions, or sport contests. We study the general model of contest with attacks and defence and propose an application to negative political campaigns, where two candidates arbitrate between disparaging their opponent or enhancing their own image. We propose sufficient conditions for the existence and uniqueness of a symmetric Nash equilibrium of the contest game. In the application, we contribute to the empirically debated question dealing with the effect of attack on voters turnout, and show that the conclusion depends on the distribution of voters sensitivity to defence and attack. Furthermore, contrary to the literature, we show that an underdog candidate may be less aggressive than his opponent.