To fight or to give up? Dynamic contests with a deadline

We study dynamic contests between two players whose performance is determined jointly by effort and luck. The players observe each other's positions in real time. There is a fixed deadline, and the player with a higher performance at the deadline wins the contest. We fully characterize the Markov perfect equilibrium for heterogeneous players. Effort is high when the players are tied but collapses quickly when one of them assumes a lead, due to a dynamic momentum effect. Therefore, total expected effort does not necessarily increase in the prize or in the players' abilities. We discuss implications for contest design and propose splitting the contest to cool off competition, and introducing heterogeneous players with optimal head-starts, as possible solutions.