Subsidy and taxation in all-pay auctions under incomplete information

We study all-pay auctions under incomplete information with n contestants who have non-linear cost functions. The designer may award two kinds of subsidy (taxation): one that decreases (increases) each contestant's marginal cost of effort and another that increases (decreases) each contestant's value of winning. The designer's expected payoff is the contestants' expected total effort minus the cost of subsidy or, alternatively, plus the tax payment. We show that when the resource of subsidy (the marginal taxation rate) is relatively small and the cost function is concave (convex), the designer's expected payoff in all-pay auctions with both kinds of subsidy (taxation) is higher than in the same contest without any subsidy (taxation).