We prove that the maximal bid in asymmetric first-price and all-pay auctions is the same for all bidders. Our proof is elementary, and does not require that bidders are risk neutral, or that the distribution functions of their valuations are independent or smooth.

We consider a battle between two groups D (defenders) and A (attackers). Each group has the same number of agents, but they differ by their abilities. Every agent confronts only one opponent from the other group once in either an all-pay contest or a Tullock contest such that the number of the...