Stochastic Games with a Single Controller and Incomplete Information

Abstract

Abstract. We study stochastic games with incomplete information on one side, in which the transition is controlled by one of the players.We prove that if the informed player also controls the transitions, the game has a value, whereas if the uninformed player controls the transitions, the max-min value as well as the min-max value exist, but they may differ.We discuss the structure of the optimal strategies, and provide extensions to the case of incomplete information on both sides.

The Value of Repeated Games with an Informed Controller

“…Studying this model has lead to the present paper, and some ideas developed here already come from Renault 2006. It also contains stochastic games with a single controller and incomplete information on the side of his opponent, as studied in Rosenberg et al, 2004. So the present paper generalizes both theorem 2.3 in Renault 2006, and theorem 6 in Rosenberg et al 2004, and as a consequence it also generalizes the original existence result of Aumann and Maschler (1995) for the value of (non stochastic) repeated games with incomplete information on one side and perfect monitoring.…”

LP formulation of asymmetric zero-sum stochastic games

“…If the transition law is Markovian [4], or Markovian controlled by the informed player [5], then stochastic games have values. Recently, [6] generalized the results to the case when players observe signals rather than actions, and showed that the value of the stochastic game existed if one of the players was fully informed and controlled the transition of the state.…”

Stochastic Games: Existence of the Minmax

“…The statement of the lemma is essentially a reformulation of an implicit result in [2] and its proof is essentially identical to the proof there. Without needing to repeat and replicate the proof, the reformulation enables us to use an implicit result of [2] in various other applications, like (1) the present existence of the minmax in an n-player stochastic game, (2) the existence of the minmax of two-player stochastic games with imperfect monitoring [1], [4], [5], and (3) the existence of an extensive-form correlated equilibrium in n-player stochastic games [6].…”

“…, z t ). In stochastic games with imperfect monitoring the σ-algebra H t may stand for (describe) the information available to a given player prior to his choosing an action at stage t; see [1], [4] and [5].…”