Intermediate prizes in multi-dimensional contests

We study n symmetric agents engaged in simultaneous k-dimensional contests. We demonstrate that there is no symmetric pure-strategy equilibrium when there are a sufficient number of agents and a prize is awarded only if the agent wins all k sub-contests. Therefore, intermediate prizes (a prize for a win in a single sub-contest) are required for the existence of symmetric pure-strategy equilibrium. We characterize the symmetric equilibrium for either simultaneous or sequential two-dimensional contests and show that the agents’ expected effort increases with the value of the intermediate prizes, and that when there are more than two agents, the optimal total effort in both types of contests is the same.