We study all-pay auctions with discrete strategy sets and analyze the equilibrium strategies when players have asymmetric values of winning as well as asymmetric effort constraints. We prove that for any number of players if one of them has the highest effort constraint then, independent of the players’ values of winning, he is the only player with a positive expected payoff. However, when two players have the same highest effort constraint then they do not necessarily have the highest expected payoffs. By several examples we show a significant distinction between the equilibrium strategies of two players and a larger number of players, particularly when the player with the highest effort constraint is not unique.
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