The Group All-Pay Auction with Heterogeneous Impact Functions

We analyze a group all-pay auction with a group specific public good prize in which one group follows a weakest-link and the other group follows a best-shot impact function. This type of game depicts situations in which the best-shot group is an attacker and the other group is a defender. We show that when the per-capita valuations are equal across groups, there exists a continuum of mixed strategy equilibria in which both groups randomize continuously without a gap over the same interval whose lower bound is zero. There are two further types of equilibria with discontinuous strategies. For the first type, each player in the best-shot group puts mass at the upper bound of the support whereas each player in the other group puts mass at the lower bound of the support. For the second type, players in the best-shot group put masses at both the lower and the upper bounds, while the other group randomizes without an atom. If players in the best-shot group could coordinate on the mass they put at the upper bound of the support, they would want to make it as large as possible (within the relevant range).