Luca de Alfaro and Thomas A. Henzinger
We consider two-player games which are played on a finite state space for an infinite number of rounds. The games are concurrent, that is, in each round, the two players choose their moves independently and simultaneously; the current state and the two moves determine a successor state. We consider omega-regular winning conditions on the resulting infinite state sequence. To model the independent choice of moves, both players are allowed to use randomization for selecting their moves. This gives rise to the following qualitative modes of winning, which can be studied without numerical considerations concerning probabilities: sure-win (player 1 can ensure winning with certainty), almost-sure-win (player 1 can ensure winning with probability 1), limit-win (player 1 can ensure winning with probability arbitrarily close to 1), bounded-win (player 1 can ensure winning with probability bounded away from 0), positive-win (player 1 can ensure winning with positive probability), and exist-win (player 1 can ensure that at least one possible outcome of the game satisfies the winning condition).
We provide algorithms for computing the sets of winning states for each of these winning modes. In particular, we solve concurrent Rabin-chain games in n^O(m) time, where n is the size of the game structure and m is the number of pairs in the Rabin-chain condition. While this complexity is in line with traditional turn-based games, where in each state only one of the two players has a choice of moves, our algorithms are considerably more involved than those for turn-based games. This is because concurrent games violate two of the most fundamental properties of turn-based games. First, concurrent games are not determined, but rather exhibit a more general duality property which involves multiple modes of winning. Second, winning strategies for concurrent games may require infinite memory.
Proceedings of the 15th Annual Symposium on Logic in Computer Science (LICS), IEEE Computer Society Press, 2000, pp. 141-154.