The role of the second prize in all-pay auctions with two heterogeneous prizes

We study complete information all-pay contests with n players and two heterogeneous prizes with distinct values. Among the players, n−1 are symmetric (i.e., they evaluate the prizes in a similar manner), whereas the remaining player has different valuations for each of the prizes. Our analysis focuses on the equilibrium profiles and expected payoffs for the case of three players. We also partially extend our analysis for cases with additional players. Our results show that in all-pay auctions with heterogeneous prizes, the ordering of the players according to their expected payoffs in equilibrium might vary significantly, depending on both prizes. Furthermore, although the values for the first (larger) prize have the greatest effect on the identity of the players with positive expected payoffs, the value of the second prize might have a non-negligible effect.