Infinite-Duration Poorman-Bidding Games
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In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner or payoff of the game. Such games are central in formal verification since they model the interaction between a non-terminating system and its environment. We study bidding games in which the players bid for the right to move the token. Two bidding rules have been defined. In Richman bidding, in each round, the players simultaneously submit bids, and the higher bidder moves the token and pays the other player. Poorman bidding is similar except that the winner of the bidding pays the "bank" rather than the other player. While poorman reachability games have been studied before, we present, for the first time, results on infinite-duration poorman games. A central quantity in these games is the ratio between the two players' initial budgets. The questions we study concern a necessary and sufficient ratio with which a player can achieve a goal. For qualitative objectives such as parity, we show that the properties of poorman games are largely similar to the properties of their Richman counterparts. Our most interesting results concern poorman mean-payoff games, where we construct optimal strategies depending on the ratio. Unlike in the Richman case, where the optimal value is determined by the structure of the game, in poorman bidding, with a higher ratio, a player can achieve a better payoff. This is shown via a connection with probabilistic games, which in itself is surprising, given that such a connection is not known for poorman reachability games. We also solve the complexity problems that arise from these games.
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