All-pay auctions with budget constraints and fair insurance.

We study all-pay auctions with budget-constrained bidders who have access to fair insurance before bidding simultaneously over a prize. We characterize a unique equilibrium for the special cases of two bidders and one prize, show existence and a heuristic for finding an equilibrium in the case of multiple bidders and multiple prizes. We end with an example of non-uniqueness of equilibria for the general case of multiple prizes and multiple players.all-pay auctions; fair lotteries; political campaigning; oligopoly; regional competition; patent races