The value of Markov chain games with lack of information on both sides

Abstract

We consider zero-sum repeated games with incomplete information on both sides, where the states privately observed by each player follow independent Markov chains. It generalizes the model, introduced by Aumann and Maschler in the sixties and solved by Mertens and Zamir in the seventies, where the private states of the players were fixed. It also includes the model introduced in Renault [19] [Renault J (2006) The value of Markov chain games with lack of information on one side. Math. Oper. Res. 31(3):490–512.], of Markov chain repeated games with lack of information on one side, where only one player privately observes the sequence of states. We prove here that the limit value exists, and we obtain a characterization via the Mertens-Zamir system, where the “nonrevealing value function” plugged in the system is now defined as the limit value of an auxiliary “nonrevealing” dynamic game. This nonrevealing game is defined by restricting the players not to reveal any information on the limit behavior of their own Markov chain, as in Renault [19]. There are two key technical difficulties in the proof: (1) proving regularity, in the sense of equicontinuity, of the T-stage nonrevealing value functions and (2) constructing strategies by blocks in order to link the values of the nonrevealing games with the original values.

Keywords

Reference

Fabien Gensbittel, and Jérôme Renault, “The value of Markov chain games with lack of information on both sides”, Mathematics of Operations Research, vol. 40, n. 4, November 2015, pp. 820–841.