Some remarks on equilibria in semi-Markov games

• Autorzy: Andrzej S. Nowak
• Języki publikacji: EN

Abstrakty

EN

This paper is a first study of correlated equilibria in nonzero-sum semi-Markov stochastic games. We consider the expected average payoff criterion under a strong ergodicity assumption on the transition structure of the games. The main result is an extension of the correlated equilibrium theorem proven for discounted (discrete-time) Markov games in our joint paper with Raghavan. We also provide an existence result for stationary Nash equilibria in the limiting average payoff semi-Markov games with state independent and nonatomic transition probabilities. A similar result was proven for discounted Markov games by Parthasarathy and Sinha.

Słowa kluczowe

EN

correlated equilibrium; Nash equilibrium; general state space; nonzero-sum semi-Markov game; long run expected average payoff criterion

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 2000
• Tom: 27
• Numer: 4
• Strony: 385-394

Daty

wydano

2000

otrzymano

1999-03-26

poprawiono

2000-03-30

Twórcy

autor

Andrzej S. Nowak

• Institute of Mathematics, Zielona Góra University of Technology, Podgórna 50, 65-246 Zielona Góra, Poland

Bibliografia

• [1] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12.
• [2] T. R. Bielecki, Approximations of dynamic Nash games with general state and action spaces and ergodic costs for the players, Appl. Math. (Warsaw) 24 (1996), 195-202.
• [3] P. Billingsley, Probability and Measure, Wiley, New York, 1979.
• [4] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, New York, 1977.
• [5] N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, Wiley-Interscience, New York, 1958.
• [6] E. B. Dynkin and A. A. Yushkevich, Controlled Markov Processes, Springer, New York, 1979.
• [7] F. Forges, An approach to communication equilibria, Econometrica 54 (1986), 1375-1385.
• [8] I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), 170-174.
• [9] H.-U. Küenle, Stochastic games with complete information and average cost criterion, in: Advances in Dynam. Games and Applications ( Kanagawa, 1996), Ann. Internat. Soc. Dynam. Games 5, Birkhäuser, Boston, 2000, 325-338.
• [10] M. Kurano, Semi-Markov decision processes and their applications in replacement models, J. Oper. Res. Soc. Japan 28 (1985), 18-30.
• [11] K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. 13 (1965), 397-403.
• [12] H.-C. Lai and K. Tanaka, A noncooperative n-person semi-Markov game with a separable metric state space, Appl. Math. Optim. 11 (1984), 23-42.
• [13] H.-C. Lai and K. Tanaka, On an n-person noncooperative Markov game with a metric state space, J. Math. Anal. Appl. 101 (1984), 78-96.
• [14] A. K. Lal and S. Sinha, Zero-sum two-person semi-Markov games, J. Appl. Probab. 29 (1992), 56-72.
• [15] J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, 1965.
• [16] A. S. Nowak, Stationary equilibria for nonzero-sum average payoff ergodic stochastic games with general state space, in: Advances in Dynamic Games and Applications, T. Basar and A. Haurie (eds.), Birkhäuser, New York, 1994, 231-246.
• [17] A. S. Nowak, On approximations of nonzero-sum uniformly continuous ergodic stochastic games, Appl. Math. (Warsaw) 26 (1999), 221-228.
• [18] A. S. Nowak and E. Altman, ε-Nash equilibria for stochastic games with uncountable state space and unbounded cost, technical report, Inst. Math., Wrocław Univ. of Technology, 1998.
• [19] A. S. Nowak and T. E. S. Raghavan, Existence of stationary correlated equilibria with symmetric information for discounted stochastic games, Math. Oper. Res. 17 (1992), 519-526.
• [20] A. S. Nowak and K. Szajowski, Nonzero-sum stochastic games, in: Stochastic and Differential Games, Ann. Internat. Soc. Dynam. Games 4, Birkhäuser, Boston, 1999, 297-342.
• [21] T. Parthasarathy and S. Sinha, Existence of stationary equilibrium strategies in non-zero-sum discounted stochastic games with uncountable state space and state independent transitions, Internat. J. Game Theory 18 (1989), 189-194.
• [22] W. Połowczuk, Nonzero-sum semi-Markov games with countable state spaces, this issue, 395-402.
• [23] S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970.