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Solution for a Class of Stochastic Coalitional Games

Xeniya Grigorieva

St. Petersburg State University, Faculty of Applied Mathematics and Control Processes, University pr. 35, St. Petersburg, 198504, Russia

Abstract: The stochastic game $\Gamma$ under consideration is repetition of the same stage game $G$ which is played on each stage with different coalitional partitions. The probability distribution over the coalitional structures of each stage game depends on the initial stage game $G$ and the $n$-tuple of strategies realized in this game. The payoffs in stage games (which is a simultaneous game with a given coalitional structure) are computed as components of the generalized PMS-vector (see (Grigorieva and Mamkina, 2009), (Petrosjan and Mamkina, 2006)). The total payoff of each player in game $\Gamma$ is equal to the mathematical expectation of payoffs in different stage games $G$ (mathematical expectation of the components of PMS-vector). The concept of solution for such class of stochastic game is proposed and the existence of this solution is proved. The theory is illustrated by 3-person 3-stage stochastic game with changing coalitional structure.

Keywords: stochastic games, coalitional partition, Nash equilibrium, Shapley value, PMS-vector.

Language: English