Second price all-pay auctions, how much money do players get or lose?

The paper studies second price all-pay auctions - wars of attrition - in a new way, based on class room experiments and Kosfeld, Droste and Voorneveld’s (2002) best reply matching equilibrium. Two players fight over a prize of value V, have a budget M, submit bids lower or equal to M; both pay the lowest bid and the prize goes to the highest bidder. The behaviour probability distributions in the class room experiments are strikingly different from the mixed Nash equilibrium. They fit with best reply matching or generalized best reply matching, an ordinal logic according to which, if bid A is the best response to bid B, and if B is played with probability p, then A is also played with probability p. In the mixed Nash equilibrium, the expected payoff is never negative and close to 0. In the best reply and generalized best reply matching equilibria, players may lose money, up to 1/12th of the budget when M is large in comparison to V, but they can also get a lot of money, especially if V is large. The study leads to examine possible bifurcations in the bidding behaviour and gives some insights into how to regulate games to avoid pathological gambling with a huge waste of money.