Equilibrium Behavior in an All-Pay Auction with Complete Information

A widely used sealed-bid auction is the first-price auction. In this auction, the highest bidder wins the item and pays the price submitted; the other bidders get and pay nothing. The all-pay auction is similar to the first-price auction, except that losers must also pay their submitted bids. The Nash equilibria of the all-pay auction involves the use of randomized strategies, which protect bidders from being overbid by a small amount. This paper generalizes the standard Nash equilibrium analysis of the all-pay auction to allow for endogenously determined decision ''errors''. Such errors may either be due to mistakes or to unobserved random variations in payoff functions. The error distributions depend on equilibrium expected payoffs, which in turn determine the error distributions as a fixed point. A striking result derived in this paper is that for any structure of the error terms the generalized Nash equilibrium and the nash equilibrium of the all-pay auction are equivalent if the error terms are identically and independently distributed. In addition, this paper shows how the generalized Nash equilibrium can be computed for two particular parametrizations, which we call the power function and the logit equilibrium. It is also shown how Nash-mixed equilibria for the all-pay auction with discrete bid choices can be computed. In the process, we derive theoretical results for the symmetric and asymmetric models with discrete choices. The results derived in this paper are appealing since can be used to establish differences in the qualitative properties of Nash and the generalized Nash equilibrium in games.