Construction of a strong Nash equilibrium in a class of infinite non-zero-sum games

In our previous papers (2002, 2017), we derived conditions for the existence of a strong Nash equilibrium in multistage non-zero-sum games under additional constraints on the possible deviations of coalitions from their agreed-upon strategies. These constraints allowed only one-time simultaneous deviations of all the players in a coalition. However, it is clear that in real-world problems the deviations of different members of a coalition may occur at different times (at different stages of the game), which makes the punishment strategy approach proposed by the authors earlier inapplicable in the general case. The fundamental difficulty is that in the general case the players who must punish the deviating coalition know neither the members of this coalition nor the times when each player performs the deviation. In this paper we propose a new punishment strategy, which does not require the full information about the deviating coalition but uses only the fact of deviation of at least one player of the coalition. Of course, this punishment strategy can be realized only under some additional constraints on simultaneous components of the game in an infinite-stage game. Under these additional constraints it was proved that the punishment of the deviating coalition can be effectively realized. As a result, the existence of a strong Nash equilibrium was established.