Optimal Prize Allocations in Contests with Maximal Performance Objective

We show that weak concavity of the cost function leads to optimality of single prize in contests with maximal performance objective, which generalizes the previous result in Chawla et al. (2015). Moldovanu and Sela (2001) show that, with the constant elasticity functional form, enough convexity can provide a rationale for multiple prizes under total performance objective. Surprisingly, we find optimality of single prize continues to hold for arbitrary degree of convexity under maximal performance objective when the number of contestants is three. On the contrary, if the cost function is piecewise linear, then the convexity argument for multi-prize can be restored. Furthermore, We find that in terms of optimal prize allocations 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 is presented, which may be useful for future works in contest theory and multi-object auction theory