On the strong $$\beta$$ β -hybrid solution of an N-person game

We propose a new notion of coalitional equilibria, the strong $$\beta$$ β -hybrid solution, which is a refinement of the hybrid solution introduced by Zhao. Zhao’s solution is well suited to study situations where people cooperate within coalitions but where coalitions compete with one another. This paper’s solution, as opposed to the hybrid solution, assigns to each coalition a strategy profile that is strongly Pareto optimal. Moreover, like the $$\beta$$ β -core, deviations by subcoalitions of any existing coalition are deterred by the threat of a unique counter-strategy available to the non-deviating players. Zhao proved the existence of existence of strong $$\beta$$ β -hybrid solution for transferable utility games with compact and convex strategy spaces and concave continuous payoff functions. Here, we extend his result to non-transferable utility games.