borkar - J OURNAL OF OPTIMIZATION THEORY AND APPLICATIONS...

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 76, No. 3, MARCH 1993 Denmmerable State Stochastic Games with Limiting Average Payoff 1 V. S. BORKAR x AND M. K. GHOSH 3 Communicated by P. Varaiya Abstract. We study stochastic games with countable state space, com- pact action spaces, and limiting average payoff. For N-person games, the existence of an equilibrium in stationary strategies is established under a certain Liapunov stability condition. For two-person zero-sum games, the existence of a value and optimal strategies for both players are established under the same stability condition. Key Words. Ergodic occupation measure, stationary strategies, Shapley equation, equilibrium. 1. Introduction We study noncooperative stochastic games with countable state space, compact action spaces, and with ergodic or limiting average payoff. The existing literature in stochastic games with limiting average payoff seems to be very limited. To the best of our knowledge, notable contributions are due to Gillette (Ref. t), Sobel (Ref. 2), Bewley and Kohlberg (Ref. 3), Federgruen (Ref. 4), and Mertens and Neyrnan (Ref. 5). As in Markov decision processes (MDP), the standard approach to stochastic games with limiting average payoff is to treat it as a limiting case of B-discount payoff as the discount factor fl~ 1. For B-discount payoff criterion, much more is known. For two-person zero-sum games with B-discount payoff, the 1The authors wish to thank Prof. T. Parthasarathy for pointing out an error in an earlier version of this paper. M, K. Ghosh wishes to thank Prof. A. Arapos~athis and Prof. S. I. Marcus for their hospitality and support. 2Associate Professor, Department of Electrical Engineering, Indian Institute of Science, Banga- lore, India. 3Assistant Professor, Department of Mathematics, Indian lr~stitute of Science, Bangalore, India. 539 0022-3239/93/0300-0539507.00/0 © t993 Plenum Publlsl~ing Corporation
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540 JOTA: VOL. 76, NO, 3, MARCH 1993 existence of stationary fl-discount optimal strategies for both players is established in Ref. 6 for Borel state space. For N-person game, the existence of fl-discount Nash equilibrium in stationary strategies is established in Ref. 4 for countable state space. For more general state space the problem is much more involved. Parthasarathy and Sinha (Ref. 7) have established the existence of fl-discount equilibrium in stationary strategies for Borel state space and finite action spaces, but with a transition law which is independent of the state. Mertens and Parthasarathy (Ref. 8) have proved the existence of subgame perfect equilibrium for fl-discount criterion for general state and action spaces. The limiting average payoffcase is drastically different from other cases, because here the finite-time evolution of the processes is irrelevant in some sense; it is only the asymptotic behavior of the time-averaged processes that matters. For the big match, Blackwell and Ferguson (Ref. 9) have estab- lished the nonexistence of an optimal strategy for the maximizer. Therefore,
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borkar - J OURNAL OF OPTIMIZATION THEORY AND APPLICATIONS...

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