On Completely Mixed Stochastic Games

• Published in: Operations Research Forum 2022-12, Vol.3 (4), p.57, Article 57
• Main Authors: Das, Purba, Parthasarathy, T., Ravindran, G.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Business and Management; Equilibrium; Game theory; Games; Math Applications in Computer Science; Mathematical analysis; Mathematical and Computational Engineering; Matrices (mathematics); Operations Research/Decision Theory; Optimization; Original Research; Probability; Transition probabilities
• ISSN: 2662-2556; 2662-2556
• DOI: 10.1007/s43069-022-00150-y

In this paper, we consider a two-person finite state stochastic games with finite number of pure actions for both players in all the states. In particular, for a large number of results we also consider one-player controlled transition probability and show that if all the optimal strategies of the undiscounted stochastic game are completely mixed then for β sufficiently close to 1; all the optimal strategies of β -discounted stochastic games are also completely mixed. A counterexample is provided to show that the converse is not true. Further, for single-player controlled completely mixed stochastic games if the individual payoff matrices are symmetric in each state, then we show that the individual matrix games are also completely mixed. For the non-zerosum single-player controlled stochastic game under some non-singularity conditions, we show that if the undiscounted game is completely mixed, then the Nash equilibrium is unique. For non-zerosum β -discounted stochastic games when Nash equilibrium exists, we provide equalizer rules for corresponding value of the game.