Allocation games with caps: from Captain Lotto to all-pay auctions

A Lotto game is a two-person zero-sum game where each player chooses a distribution on nonnegative real numbers with given expectation, so as to maximize the probability that his realized choice is higher than his opponent’s. These games arise in various competitive allocation setups (e.g., contests, research and development races, political campaigns, Colonel Blotto games). A Captain Lotto game is a Lotto game with caps, which are upper bounds on the numbers that may be chosen. First, we solve the Captain Lotto games. Second, we show how to reduce all-pay auctions to simpler games—expenditure games—using the solution of the corresponding Lotto games. As a particular application we solve all-pay auctions with unequal caps, which may yield a significant increase in the seller’s revenue (or, the players’ efforts).