We consider concurrent games played by two players on a finite-state graph, where in every round the players simultaneously choose a move, and the current state along with the joint moves determine the successor state. We study the most fundamental objective for concurrent games, namely, mean-payoff or limit-average objective, where a reward is associated to each transition, and the goal of player 1 is to maximize the long-run average of the rewards, and the objective of player 2 is strictly the opposite (i.e., the games are zero-sum). The path constraint for player 1 could be qualitative, i.e., the mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative, i.e., a given threshold between the minimal and maximal reward. We consider the computation of the almost-sure (resp. positive) winning sets, where player 1 can ensure that the path constraint is satisfied with probability 1 (resp. positive probability). Almost-sure winning with qualitative constraint exactly corresponds to the question of whether there exists a strategy to ensure that the payoff is the maximal reward of the game. Our main results for qualitative path constraints are as follows: (1) we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost-sure (resp. positive) winning against all player-2 strategies, or player 2 has a spoiling strategy to falsify almost-sure (resp. positive) winning against all player-1 strategies; (2) we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost-sure and positive winning for both players; and (3) we present quadratic time algorithms to compute the almost-sure and the positive winning sets, matching the best known bound of the algorithms for much simpler problems (such as reachability objectives). For quantitative constraints we show that a polynomial time solution for the almost-sure or the positive winning set would imply a solution to a long-standing open problem (of solving the value problem of turn-based deterministic mean-payoff games) that is not known to be solvable in polynomial time.
Information and Computation
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Chatterjee K, Ibsen-Jensen R. Qualitative analysis of concurrent mean payoff games. Information and Computation. 2015;242(6):2-24. doi:10.1016/j.ic.2015.03.009
Chatterjee, K., & Ibsen-Jensen, R. (2015). Qualitative analysis of concurrent mean payoff games. Information and Computation. Elsevier. https://doi.org/10.1016/j.ic.2015.03.009
Chatterjee, Krishnendu, and Rasmus Ibsen-Jensen. “Qualitative Analysis of Concurrent Mean Payoff Games.” Information and Computation. Elsevier, 2015. https://doi.org/10.1016/j.ic.2015.03.009.
K. Chatterjee and R. Ibsen-Jensen, “Qualitative analysis of concurrent mean payoff games,” Information and Computation, vol. 242, no. 6. Elsevier, pp. 2–24, 2015.
Chatterjee K, Ibsen-Jensen R. 2015. Qualitative analysis of concurrent mean payoff games. Information and Computation. 242(6), 2–24.
Chatterjee, Krishnendu, and Rasmus Ibsen-Jensen. “Qualitative Analysis of Concurrent Mean Payoff Games.” Information and Computation, vol. 242, no. 6, Elsevier, 2015, pp. 2–24, doi:10.1016/j.ic.2015.03.009.
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