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",{\em   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 schemeshave been considered. In "{\em first-price " bidding , only the higher bidder pays his bid , while in "{\em 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{\em   mathematical properties of their first-price counterparts . This is in stark contrast with reachability games, which are known to be much more complicated under{\em   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.
In a two-player zero-sum graph game the players move a token throughout a graph to produce an infinite path, which determines the winner or payoff of the game. Traditionally, the players alternate turns in moving the token. In{\em bidding games , however, the players have budgets, and in each turn, we hold an "auction" (bidding) to determine which player moves the token : both players simultaneously submit bids and the higher bidder moves the token. The bidding mechanisms differ in their payment schemes. Bidding games were largely studied with variants of{\em first-price bidding in which only the higher bidder pays his bid . We focus on{\em all-pay bidding, where both players pay their bids. Finite-duration all-pay bidding games were studied and shown to be technically more challenging than their first-price counterparts. We study for the first time, infinite-duration all-pay bidding games . Our most interesting results are for{\em mean-payoff objectives: we portray a complete picture for games played on strongly-connected graphs. We study both pure (deterministic) and mixed (probabilistic) strategies and completely characterize the optimal sure and almost-sure (with probability 1) payoffs that the players can respectively guarantee. We show that mean-payoff games under all-pay bidding exhibit the intriguing mathematical properties of their first-price counterparts ; namely, an equivalence with{\em random-turn games in which in each turn, the player who moves is selected according to a (biased) coin toss. The equivalences for all-pay bidding are more intricate and unexpected than for first-price bidding.
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