Existence of Equilibria in All-Pay Auctions

For an all-pay sealed-bid auction of an item for which each bidder’s realized value can depend on every bidder’s privately observed signal, existence of equilibria in behavioral strategies is established using only the assumption that bidders’ value functions and the density function of signals are positive and continuous on a product of intervals. Such equilibria have atomless distributions of bids and thus are not affected by how tied bids are resolved.