Optimal Accuracy of Unbiased Tullock Contests with Two Heterogeneous Players

I characterize the optimal accuracy level r of an unbiased Tullock contest between two players with heterogeneous prize valuations. The designer maximizes the winning probability of the strong player or the winner's expected valuation by choosing a contest with an all-pay auction equilibrium (r >= 2). By contrast, if she aims at maximizing the expected aggregate effort or the winner's expected effort, she will choose a contest with a pure-strategy equilibrium, and the optimal accuracy level r < 2 decreases in the players' heterogeneity. Finally, a contest designer who faces a tradeoff between selection quality and minimum (maximum) effort will never choose a contest with a semi-mixed equilibrium.