A coalitional game of N-person dilemma

As it is well known, non-cooperative N-person dilemma game has a unique dominant strategy solution where all players choose defecting strategy. In this paper, we analyze an N-person dilemma game by taking the point of view of coalition and try to find the condition that all players choose cooperating strategy in a grand coalition. First, we show that if we transform N-person dilemma game into a coalitional form, then the core is non empty when a characteristic function is defined by the maximin value. We also show the condition that players do not choose cooperating strategy in a grand coalition. Next we specify the condition of existence of core by generalizing a characteristic function. We define a characteristic function by the subjective probability of choosing cooperative strategy. As a result we find that the core is empty when a characteristic function is defined by the maximax value. In general, if players in a coalition believe that other players, who are not members of the coalition, will choose cooperative strategy with a high probability, then the core will be empty by a deviation of a single coalition.