Ex ante heterogeneity in all-pay many-player auctions with Pareto distribution of costs

We analyze an all-pay auction as a game of incomplete information, in which every player knows the cost of his effort and believes that the cost to each of his rivals is an independent Pareto random variable. The lower bound of its support, the cost parameter, can be different for each player and all are commonly known. Such an auction has a unique Bayesian-Nash equilibrium. The players with a relatively high cost parameter are not active in equilibrium, i.e., do not exert any effort, no matter what its cost. We study the effects of heterogeneity measured as a mean-preserving spread of cost parameters of the active players. There are two main findings. (1) Heterogeneity has no effect on the maximal effort, the distribution of which is fully determined by the average of the cost parameters. (2) Increase in heterogeneity lowers the total expected effort and the distribution of the minimal effort in terms of first-order stochastic dominance.