Abstract
Zero-sum two-person finite undiscounted (limiting ratio average) semi-Markov games are considered where the transition probabilities and the transition times are controlled by a fixed player in all states. We prove that if such a game is unichain, the value and optimal stationary strategies can be obtained from an optimal solution of a linear programming algorithm for the associated undiscounted unichain single controller stochastic game (obtained by a data transformation method). The single controller undiscounted unichain semi-Markov games have been formulated as a linear complementarity problem and solved using a stepwise principal pivoting algorithm. We provide necessary and sufficient conditions for such games to be completely mixed (i.e., every optimal stationary strategy for each player assigns a positive probability to every action in every state). Some properties analogous to completely mixed matrix games are also established in this paper.
Similar content being viewed by others
References
Blackwell D, Girshick MA (1954) Theory of games and statistical decisions. Wiley, New York
Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Academic, New York
Federgruen A, Hordijk A, Tijms HC (1978) A note on simultaneous recurrence conditions on a set of denumerable stochastic matrices. J Appl Prob 15(4):842–847
Filar JA (1985) The completely mixed single-controller stochastic game. Proc Am Math Soc 95(4):585–594
Filar JA, Raghavan TES (1984) A matrix game solution of the single-controller stochastic game. Math Oper Res 9:356–362
Howard RA (1963) Semi-Markovian decision processes. Proceedings International Statistics Institute, Ottawa
Jaskiewicz A (2002) Zero-sum semi-Markov games. SIAM J Control Optim 41(3):723–739
Jianyong L, Xiaobo Z (2004) On average reward semi-Markov decision processes with a general multichain structure. Math Oper Res 29(2):339–352
Jurdzinski M, Savani R (2008) A simple P-matrix linear complementarity problem for discounted games. In: Beckmann A, Dimitracopoulos C, Lwe B (eds) Lecture Notes In Computer Science, CiE 2008, LNCS 5028. Springer, Berlin, Heidelberg, pp 283–293
Jurg AP, Jansen MJM, Parthasarathy T, Tijs SH (1990) On weakly completely mixed bimatrix games. Linear Algebra Appl 141:61–74
Kaplansky I (1945) A contribution to von Neumann’s theory of games. Ann Math 46(2):474–479
Lal AK, Sinha S (1992) Zero-sum two-person semi-Markov games. J Appl Prob 29(1):56–72
Lemke CE (1965) Bimatrix equilibrium points and mathematical programming. Manag Sci 11:681–689
Mertens JF, Neyman A (1981) Stochastic games. Int J Game Theory 10(2):53–66
Mohan SR, Neogy SK, Parthasarathy T (2001) Pivoting algorithms for some classes of stochastic games: a survey. Int Game Theory Rev 3(2 & 3):253–281
Mondal P (2015) Linear programming and zero-sum two-person undiscounted semi-Markov games. Asia Pac J Oper Res 32(6):1550043-1–1550043-20
Mondal P (2016) On undiscounted semi-Markov decision processes with absorbing states. Math Methods Oper Res 83(2):161–177
Mondal P, Neogy SK, Sinha S, Ghorui D (2016) Completely mixed strategies for two structured classes of semi-Markov games, principal pivot transform and its generalizations. Appl Math Optim. doi:10.1007/s00245-016-9362-4
Mondal P, Sinha S (2015) Ordered field property for semi-Markov games when one player controls transition probabilities and transition times. Int Game Theory Rev 17(2):1540022-1–1540022-26
Neogy SK, Dubey D, Ghorui D (2015) Completely mixed generalized bimatrix game, vertical linear complementarity problem and completely mixed switching controller stochastic game. Preprints, isid/SQCORD/2015/03
Neogy SK, Das AK, Gupta A (2012) Generalized principal pivot transforms, complementarity theory and their applications in stochastic games. Optim Lett 6:339–356
Parsons TD (1970) Applications of principal pivoting. In: Kuhn HW (ed) Proceedings of the Princeton symposium on mathematical programming. Princeton University Press, Princeton, pp 561–581
Parthasarathy T, Raghavan TES (1981) An orderfield property for stochastic games when one player controls transition probabilities. J Optim Theory Appl 33(3):375–392
Raghavan TES (1970) Completely mixed strategies in bimatrix games. J Lond Math Soc 2(2):709–712
Raghavan TES, Filar JA (1991) Algorithms for stochastic games—a survey. Math Methods Oper Res 35(6):437–472
Ross SM (1970) Applied probability models with optimization applications. Holden-Day, San Francisco
Schultz TA (1992) Linear complementarity and discounted switching controller stochastic games. J Optim Theory Appl 73(1):89–99
Schweitzer PJ (1971) Iterative solution of the functional equations of undiscounted Markov renewal programming. J Math Anal App 34(3):495–501
Shapley L (1953) Stochastic games. Proc Natl Acad Sci 39(10):1095–1100
Shapley LS, Snow RN (1950) Basic solutions of discrete games. Annals of mathematics studies, vol 24. Princeton University Press, Princeton
Tsatsomeros M (2000) Principal pivot transforms: properties and applications. Linear Algebra Appl 307:151–165
Tucker AW (1963) Principal pivotal transforms of square matrices. SIAM Rev 5:305
Vega-Amaya O (2003) Zero-sum average semi-Markov games: fixed-point solutions of the Shapley equation. SIAM J Control Optim 42(5):1876–1894
Vrieze OJ (1981) Linear programming and undiscounted stochastic games in which one player controls transitions. OR Spektrum 3(1):29–35
Acknowledgments
The author wishes to thank the unknown referees who have patiently gone through this paper and whose constructive suggestions have improved its presentation and readability considerably.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mondal, P. Completely mixed strategies for single controller unichain semi-Markov games with undiscounted payoffs. Oper Res Int J 18, 451–468 (2018). https://doi.org/10.1007/s12351-016-0272-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12351-016-0272-7
Keywords
- Semi-Markov games
- Undiscounted payoffs
- Single controller semi-Markov games
- Completely mixed strategies
- Linear complementarity problem
- Principal pivot transform