Infinite-Duration All-Pay Bidding Games
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A graph game is a two-player zero-sum game in which the players move a token throughout a graph to produce an infinite path, which determines the winner or payoff of the game. In "bidding games", in each turn, we hold an 'auction' (bidding) to determine which player moves the token. The players simultaneously submit bids and the higher bidder moves the token. Several different payment schemes have been considered. In "first-price" bidding, only the higher bidder pays his bid, while in "all-pay" bidding, both players pay their bids. Bidding games were largely studied with variants of first-price bidding. In this work, we study, for the first time, infinite-duration all-pay bidding games, and show that they exhibit the elegant mathematical properties of their first-price counterparts. This is in stark contrast with reachability games, which are known to be much more complicated under all-pay bidding than first-price bidding. Another orthogonal distinction between the bidding rules is in the recipient of the payments: in "Richman" bidding, the bids are paid to the other player, and in "poorman" bidding, the bids are paid to the 'bank'. We focus on strongly-connected games with "mean-payoff" and "parity" objectives. We completely solve all-pay Richman games: a simple argument shows that deterministic strategies cannot guarantee anything in this model, and it is technically much more challenging to find optimal probabilistic strategies that achieve the same expected guarantees in a game as can be obtained with deterministic strategies under first-price bidding. Under poorman all-pay bidding, in contrast to Richman bidding, deterministic strategies are useful and guarantee a payoff that is only slightly lower than the optimal payoff under first-price poorman bidding. Our proofs are constructive and based on new and significantly simpler constructions for first-price bidding.
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