Stationary, completely mixed and symmetric optimal and equilibrium strategies in stochastic games

Sujatha Babu (), Nagarajan Krishnamurthy () and T. Parthasarathy ()
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Sujatha Babu: Indian Institute of Technology Madras
Nagarajan Krishnamurthy: Indian Institute of Management Indore
T. Parthasarathy: Indian Statistical Institute, Chennai Centre

International Journal of Game Theory, 2017, vol. 46, issue 3, No 8, 782 pages

Abstract: Abstract In this paper, we address various types of two-person stochastic games—both zero-sum and nonzero-sum, discounted and undiscounted. In particular, we address different aspects of stochastic games, namely: (1) When is a two-person stochastic game completely mixed? (2) Can we identify classes of undiscounted zero-sum stochastic games that have stationary optimal strategies? (3) When does a two-person stochastic game possess symmetric optimal/equilibrium strategies? Firstly, we provide some necessary and some sufficient conditions under which certain classes of discounted and undiscounted stochastic games are completely mixed. In particular, we show that, if a discounted zero-sum switching control stochastic game with symmetric payoff matrices has a completely mixed stationary optimal strategy, then the stochastic game is completely mixed if and only if the matrix games restricted to states are all completely mixed. Secondly, we identify certain classes of undiscounted zero-sum stochastic games that have stationary optima under specific conditions for individual payoff matrices and transition probabilities. Thirdly, we provide sufficient conditions for discounted as well as certain classes of undiscounted stochastic games to have symmetric optimal/equilibrium strategies—namely, transitions are symmetric and the payoff matrices of one player are the transpose of those of the other. We also provide a sufficient condition for the stochastic game to have a symmetric pure strategy equilibrium. We also provide examples to show the sharpness of our results.

Keywords: Completely mixed stochastic game; Symmetric optimal and equilibrium strategy; Stationary strategy; Undiscounted and discounted stochastic game; Single-player controlled stochastic game; Switching control stochastic game; Separable reward-state independent transition (SER-SIT) game (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s00182-016-0555-5

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