# Some remarks on equilibria in semi-Markov games

Applicationes Mathematicae (2000)

- Volume: 27, Issue: 4, page 385-394
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topNowak, Andrzej. "Some remarks on equilibria in semi-Markov games." Applicationes Mathematicae 27.4 (2000): 385-394. <http://eudml.org/doc/219281>.

@article{Nowak2000,

abstract = {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.},

author = {Nowak, Andrzej},

journal = {Applicationes Mathematicae},

keywords = {correlated equilibrium; Nash equilibrium; general state space; nonzero-sum semi-Markov game; long run expected average payoff criterion},

language = {eng},

number = {4},

pages = {385-394},

title = {Some remarks on equilibria in semi-Markov games},

url = {http://eudml.org/doc/219281},

volume = {27},

year = {2000},

}

TY - JOUR

AU - Nowak, Andrzej

TI - Some remarks on equilibria in semi-Markov games

JO - Applicationes Mathematicae

PY - 2000

VL - 27

IS - 4

SP - 385

EP - 394

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/219281

ER -

## References

top- [1] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1-12. Zbl0163.06301
- [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. Zbl0865.90146
- [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. Zbl0084.10402
- [6] E. B. Dynkin and A. A. Yushkevich, Controlled Markov Processes, Springer, New York, 1979. Zbl0073.34801
- [7] F. Forges, An approach to communication equilibria, Econometrica 54 (1986), 1375-1385. Zbl0605.90146
- [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. Zbl0046.12103
- [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. Zbl0564.90090
- [11] K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. 13 (1965), 397-403. Zbl0152.21403
- [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. Zbl0532.90105
- [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. Zbl0615.90101
- [14] A. K. Lal and S. Sinha, Zero-sum two-person semi-Markov games, J. Appl. Probab. 29 (1992), 56-72. Zbl0761.90111
- [15] J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, 1965. Zbl0137.11301
- [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. Zbl0820.90145
- [17] A. S. Nowak, On approximations of nonzero-sum uniformly continuous ergodic stochastic games, Appl. Math. (Warsaw) 26 (1999), 221-228. Zbl1050.91009
- [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. Zbl0761.90112
- [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. Zbl0940.91014
- [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. Zbl0674.90108
- [22] W. Połowczuk, Nonzero-sum semi-Markov games with countable state spaces, this issue, 395-402. Zbl1050.91012
- [23] S. M. Ross, Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970. Zbl0213.19101

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.