Communication leading to coalition Nash equilibrium II - S4n-knowledge case

In this paper the new concept of coalition Nash equilibrium of a strategic game is introduced, and it is shown that a communication among the players in a coalition leads to the equilibrium through messages. A coalition Nash equilibrium for a strategic game consists of (1) a subset S of players, (2) independent mixed strategies for each member of S, (3) the conjecture of the actions for the other players not in S with the condition that each member of S maximises his/her expected payoff according to the product of all mixed strategies for S and the other players´ conjecture. However, this paper stands on the Bayesian point of view as follows: The players start with the same prior distribution on a state-space. In addition they have private information which is given by a reflexive and transitive binary relation on the state space. Each player in a coalition S predicts the other players´ actions as the posterior of the others´ actions given his/her information. He/she communicates privately their beliefs about the other players´ actions through messages among all members in S according to the communication network in S, which message is information about his/her individual conjecture about the others´ actions. The recipients update their belief by the messages. Precisely, at every stage each player communicates privately not only his/her belief about the others´ actions but also his/her rationality as messages according to a protocol and then the recipient updates their private information and revises her/his prediction. In this circumstance, we show that the conjectures of the players in a coalition S regarding the future beliefs converge in the long run communication, which lead to a coalition Nash equilibrium for the strategic game.