Abstract
We study infinite stochastic games played by two-players on a finite graph with goals specified by sets of infinite traces. The games are concurrent (each player simultaneously and independently chooses an action at each round), stochastic (the next state is determined by a probability distribution depending on the current state and the chosen actions), infinite (the game continues for an infinite number of rounds), nonzero-sum (the players’ goals are not necessarily conflicting), and undiscounted. We show that if each player has an ω-regular objective expressed as a parity objective, then there exists an ε-Nash equilibrium, for every ε> 0. However, exact Nash equilibria need not exist. We study the complexity of finding values (payoff profile) of an ε-Nash equilibrium. We show that the values of an ε-Nash equilibrium in nonzero-sum concurrent parity games can be computed by solving the following two simpler problems: computing the values of zero-sum (the goals of the players are strictly conflicting) concurrent parity games and computing ε-Nash equilibrium values of nonzero-sum concurrent games with reachability objectives. As a consequence we establish that values of an ε-Nash equilibrium can be computed in TFNP (total functional NP), and hence in EXPTIME.
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Chatterjee, K.: Two-player nonzero-sum ω-regular games. Technical Report: UCB/CSD-04-1364 (2004)
Chatterjee, K., de Alfaro, L., Henzinger, T.A.: The complexity of quantitative concurrent parity games. Technical Report: UCB/CSD-04-1354 (2004)
Chatterjee, K., de Alfaro, L., Henzinger, T.A.: The complexity of stochastic Rabin and Streett games. Technical Report: UCB/CSD-04-1355 (2004)
Chatterjee, K., de Alfaro, L., Henzinger, T.A.: Trading memory for randomness. In: QEST 2004. IEEE Computer Society Press, Los Alamitos (2004)
Chatterjee, K., Majumdar, R., Jurdziński, M.: On Nash equilibria in stochastic games. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 26–40. Springer, Heidelberg (2004)
de Alfaro, L., Henzinger, T.A.: Concurrent omega-regular games. In: LICS 2000, pp. 141–154. IEEE Computer Society Press, Los Alamitos (2000)
de Alfaro, L., Majumdar, R.: Quantitative solution of omega-regular games. In: STOC 2001, pp. 675–683. ACM Press, New York (2001)
Filarand, J., Vrieze, K.: Competitive Markov Decision Processes. Springer, Heidelberg (1997)
Fink, A.M.: Equilibrium in a stochastic n-person game. Journal of Science of Hiroshima University 28, 89–93 (1964)
Nash Jr., J.F.: Equilibrium points in n-person games. Proceedings of the National Academny of Sciences USA 36, 48–49 (1950)
Martin, D.A.: The determinacy of Blackwell games. The Journal of Symbolic Logic 63(4), 1565–1581 (1998)
Owen, G.: Game Theory. Academic Press, London (1995)
Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. JCSS 48(3), 498–532 (1994)
Papadimitriou, C.H.: Algorithms, games, and the internet. In: STOC 2001, pp. 749–753. ACM Press, New York (2001)
Secchi, P., Sudderth, W.D.: Stay-in-a-set games. International Journal of Game Theory 30, 479–490 (2001)
Shapley, L.S.: Stochastic games. Proc. Nat. Acad. Sci. USA 39, 1095–1100 (1953)
Thomas, W.: Languages, automata, and logic. In: Handbook of Formal Languages. Beyond Words, ch. 7, vol. 3, pp. 389–455. Springer, Heidelberg (1997)
Vardi, M.Y.: Automatic verification of probabilistic concurrent finite-state systems. In: STOC 1985, pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)
Vieille, N.: Two player stochastic games I: a reduction. Israel Journal of Mathematics 119, 55–91 (2000)
Vieille, N.: Two player stochastic games II: the case of recursive games. Israel Journal of Mathematics 119, 93–126 (2000)
von Neumann, J., Morgenstern, O.: Theory of games and economic behavior. Princeton University Press, Princeton (1947)
von Stengel, B.: Computing equilibria for two-person games. In: Handbook of Game Theory, ch. 45, vol. 3, pp. 1723–1759 (2002)
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Chatterjee, K. (2005). Two-Player Nonzero-Sum ω-Regular Games. In: Abadi, M., de Alfaro, L. (eds) CONCUR 2005 – Concurrency Theory. CONCUR 2005. Lecture Notes in Computer Science, vol 3653. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11539452_32
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DOI: https://doi.org/10.1007/11539452_32
Publisher Name: Springer, Berlin, Heidelberg
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