# Linear Complementarity and the Irreducible Polystochastic Game with the Average Cost Criterion When One Player Controls Transitions

## Abstract

We consider the polystochastic game in which the transition probabilities depend on the actions of a single player and the criterion is the limiting average of the expected costs for each player. Using linear complementarity theory, we present a computational scheme for computing a set of stationary equilibrium strategies and the corresponding costs for this game with the additional assumption that under any choice of stationary strategies for the players the resulting one step transition probability matrix is irreducible. This work extends our previous work on the computation of a set of stationary equilibrium strategies and the corresponding costs for a polystochastic game in which the transition probabilities depend on the actions of a single player and the criterion is the total discounted expected cost for each player.

## Keywords

Nash Equilibrium Linear Complementarity Problem Stationary Strategy Stochastic Game Basic Feasible Solution## Preview

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## References

- 1.R. W. Cottle and G.B. Dantzig (1970) “A generalization of the linear complementarity problem” , Journal of Combinatorial Theory 8 pp. 79–90.CrossRefGoogle Scholar
- 2.R. W. Cottle, J. S. Pang, and R. E. Stone (1992)
*The Linear Complemeniarity Problem*Academic Press, New York.Google Scholar - 3.B. C.Eaves (1971) “The linear complementarity problem” , Management Science 17 pp. 612–634.CrossRefGoogle Scholar
- 4.A. A. Ebiefung and M. M. Kostreva (1991) “Z-matrices and the generalized linear com plementarity problem” , Technical Report #608, Department of Mathematical Sciences, Clemson University, South Carolina.Google Scholar
- 5.A. A. Ebiefung and M. M. Kostreva (1993) “Generalized
*P*_{0}and Z-matrices”, Linear Algebra and Its Applications 195 pp.165–179.CrossRefGoogle Scholar - 6.A. M. Fink (1964) “Equilibrium in a stochastic n-person game”, Journal of Science of Hiroshima University, Series A-I 28 pp.89–93.Google Scholar
- 7.M. Fiedler and V. Pták (1962) “On matrices with non-positive off-diagonal elements and positive principal minors” , Czechoslovak Mathematical Journal 12 pp.382–400.CrossRefGoogle Scholar
- 8.D. Gilette (1957) “Stochastic games with zero stop probabilities”, In: A. W. T. M. Dresher and P. Wolfe (eds), Contributions to the theory of games, Princeton University Press, Annals of Mathematical Studies 39 (1957) .Google Scholar
- 9.C. E. Lemke (1965) “Bimatrix equilibrium points and mathematical programming”, Management Science 11 pp.681–689.CrossRefGoogle Scholar
- 10.C. E. Lemke and J. T. Howson (1964) “Equilibrium points of bimatrix games”, SIAM Journal on Applied Mathematics 12 pp.413–423.CrossRefGoogle Scholar
- 11.S. R. Mohan, S. K. Neogy and T. Parthasarathy (1997) “Linear complementarity and discounted polystochastic game when one player controls transitions” , in
*Corrmplementarity and Variational Problems*eds. M. C. Ferris and Jong -Shi Pang, SIAM, Philedelphia, pp 284–294.Google Scholar - 12.S. R. Mohan and S. K. Neogy (1996) “Algorithms for the generalized linear complementarity problem with a vertical block Z-matrix”, SIAM Journal On Optimization 6 pp. 994–1006.CrossRefGoogle Scholar
- 13.A. S. Nowak and T. E. S. Raghavan (1993) “A finite step algorithm via a bimatrix game to a single controller non-zerosum stochastic game” , Mathematical Programming 59 pp. 249–259.CrossRefGoogle Scholar
- 14.T. Parthasarathy and T. E. S. Raghavan (1971).
*Some topics in two-person games*, American Elsevier Publishing Company, Inc., New York.Google Scholar - 15.T. Parthasarathy and T. E. S. Ragavan (1981) “An orderfield property for stochastic games when one player controls transition probabilities” , Journal of Optimization theory and Applications 33 pp.375–392.CrossRefGoogle Scholar
- 16.T. E. S. Raghavan and J. A. Filar (1991) “Algorithms for stochastic games, a survey”, Zeitschrift fi’ür Operations Research 35 pp. 437–472.Google Scholar
- 17.L. S. Shapley (1953) “Stochastic games”, Proceedings of the National Academy of Sciences 39 pp.1095–1100.CrossRefGoogle Scholar
- 18.M. A. Stern (1975) “On stochastic games with limiting average payoff” , Ph. D. thesis in Mathematics submitted to the Graduate College of the University of Illinois, Chicago, Illinois.Google Scholar
- 19.M. Takahasi (1964) “Equilibrium points of stochastic noncooperative n-person game”, Journal of Science of Hiroshima University, Series A-I 28 pp.95–99.Google Scholar