All-pay auctions with certain and uncertain prizes

We study all-pay auctions with multiple prizes. The players have the same value for all the certain prizes except for one uncertain prize for which each player has a private value. We characterize the equilibrium strategy and show that, independent of the ranking of the uncertain prize, if the uncertain prize is not the lowest one, a player's effort as well as his expected payoff increase in his value for the uncertain prize. Otherwise, if this prize is the lowest one, we obtain that a player's effort may decrease in his value for the uncertain prize but his expected utility increases.