Optimal Prize Allocations in Contests

Abstract

Given a fixed prize budget for a contest, what is the optimal prize allocation among contestants? The answer depends on the objective of the contest designer, which typically is either to maximize the total performance of all contestants or simply the champion's performance. We try to shed light on this question for both objectives in a standard model in which contestants are heterogeneous in skill and exert effort to win a prize. We show that weak concavity of the reduced-form cost function leads to optimality of single prize for both objectives, which generalizes the previous results in the literature. We find a dual relationship between the cost function and the principal's utility function (in particular, risk attitude), which not only helps to provide intuition for the optimality but also directly provides results for a principal with a different risk attitude. Surprisingly, with the traditional Cobb-Douglas functional form, optimality of single prize, when the number of contestants is three, continues to hold for arbitrary degree of convexity under maximal performance objective. On the contrary, if the reduced-form cost function is piecewise linear, then it may be optimal to reward the runners-up if the function is convex enough. When the number of prizes under consideration is two, there is an interesting relationship between the two objectives. In the derivation of the results, a series of simple facts about the winning probability functions are presented, which may be useful for future works in contest theory and multi-object auction theory.