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Discrete All-Pay Bidding Games

In an all-pay auction, only one bidder wins but all bidders must pay the auctioneer. All-pay bidding games arise from attaching a similar bidding structure to traditional combinatorial games to determine which player moves next. In contrast to the established theory of single-pay bidding games, optimal play involves choosing bids from some probability distribution that will guarantee a minimum probability of winning. In this manner, all-pay bidding games wed the underlying concepts of economic and combinatorial games. We present several results on the structures of optimal strategies in these games. We then give a fast algorithm for computing such strategies for a large class of all-pay bidding games. The methods presented provide a framework for further development of the theory of all-pay bidding games.