Prize formation and sharing in multi-stage contests

This paper examines an infinite-horizon multi-stage contest between two players. In each period, their effort may contribute to a joint prize pool, but also generates private costs. The player who works harder is awarded a larger share of the current prize pool, while his opponent receives the remainder. The stages of the supergame are linked through delayed effects of effort on subsequent prizes. The state equation and the prize pool are assumed to have a piecewise linear structure, allowing for both static and intertemporal rank-order effects. Under specific conditions, our game has an equilibrium in which the players use state-contingent mixed strategies with continuous support. We characterize the equilibrium mixing distributions, expected effort levels and payoffs. Then we illustrate our analysis with several examples.