Simple vs Optimal Contests with Convex Costs

We study an optimal contest design problem where contributors abilities are private, their costs are convex as a function of their effort, and the designer seeks to maximize their total productivity. We address the design of approximately-optimal mechanisms that are robust, in that they are independent of the ability distribution and the precise form of the cost function. We show that a very simple all-pay contest where the prize is distributed equally among the top quartile of contributors is a constant-factor approximation to the optimal, for a large class of convex cost functions, when the number of contributors is larger than some constant. This result stands in contrast to contests with linear costs, where awarding a prize to a single top contributor ("winner-takes-all»») is approximately-optimal; when costs are convex, winner-takes-all is far from optimal. We validate the performance of our approximately-optimal contest designs via simulation experiments, which uncover much better empirical performance than the worst-case guarantees. Our results are enabled by novel results in the space of optimal mechanism design with convex costs, which could be of independent interest.