Nash-reinforcement learning (N-RL) for developing coordination strategies in non-transferable utility games

Social (central) planning is normally used in the literature to optimize the system-wide efficiency and utility of multi-operator systems. Central planning tries to maximize system's benefits by coordinating the operators' strategies and reduce the externalities, assuming that all parties are willing to cooperate. This assumption implies that operators are willing to base their decisions based on group rationality rather than individual rationality, even if increased group benefits results in reduced benefits for some agents. This assumption limits the applicability of social planner's solutions, as perfect cooperation among agents is often infeasible in real world. Recognizing the fact that decisions are normally based on individual rationality in human systems, cooperative game theory methods are normally employed to address the major limitation of social planner's methods. Game theory methods revise the social planner's solution such that not only group benefits are increased, but also there exists no agent whose cooperative gain is less than his non-cooperative gain. However, in most cases, utility is assumed to be transferrable and the literature has not sufficiently focused on non-transferrable utility games. In such games parties are willing to cooperate and coordinate their strategies to increase their benefits, but have no ability to compensate each other to promote cooperation. To a good extent, the transferrable utility assumption is due to the complexity of calculations to find the best response strategies of agents in non-cooperative and cooperative modes, especially in multi-period games. By combining Reinforcement Learning and Nash bargaining solution, this paper develops a new method for applying cooperative game theory to complex multi-period non-transferrable utility games. For illustration, the suggested method is applied to two numerical examples in which two hydropower operators seek developing a fair and efficient cooperation mechanism to increase their gains.