Optimal allocations of prizes and punishments in Tullock contests

We study Tullock contests with n symmetric players. We show that in a contest without an exit option, if prizes and punishments (negative prizes) have the same cost, it is optimal for the designer who wants to maximize the players’ total effort to allocate the entire prize sum to a single punishment without any prize. On the other hand, in a contest with an exit option, it is optimal to allocate the entire prize sum to a single prize and a single punishment, where independent of the costs of the prize and the punishment, the optimal value of the prize is larger than the optimal value of the punishment. We also show that allocating a prize and a punishment in a two-stage contest yields a higher expected total effort than in a one-stage contest.