Communication Leading to Coalition Nash Equilibrium I

This paper is to introduce the new concept of coalition Nash equilibrium of a strategic game, and to show 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 partition of 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.