Contests with discontinuous payoffs

We consider a contest model with non-monotonic payoff and cost functions as introduced by Siegel (2014b). In this paper we generalize this set-up to also allow for cases in which payoffs and costs of the players can have some kind of discontinuities. Relevant real-life examples include open markets where firm decisions lead to discontinuous costs and competitions where, for instance, an extra reward is allocated to the contestant when a particular performance threshold is reached. Players compete by investing some effort in order to win one of several identical prizes. We show that one can relax the assumptions in Siegel (2009, 2014b) so that the Payoff Equivalence Theorem of Siegel still holds with the aforementioned discontinuities. We present relevant examples which show that the relaxed assumptions are indispensable, and therefore one cannot expect to refine the assumptions of Siegel more than the ones presented in this paper.