Prizes in Contests with Many Entrants

We characterize the optimal prize distribution in an all-pay contest under organizer objectives of expected outcome, expected maximum outcome, and expected mth highest outcome. Multiple prizes can be optimal for small contests, but as the number of entrants grows large a single prize becomes optimal. A large number of entrants is not optimal in all cases. Every feasible integer is optimal for some cost function. We characterize the limiting value of contest outcomes; exactly in the case of linear production and up to bounds in the concave/convex case. In all cases the limiting value is bounded away from zero.