Nash Bargaining with Coalitional Threats

We axiomatically characterize bargaining outcomes in the presence of coalitional threats. As in Nash’s solution, these involve the product of payoffs net of disagreement points, but coalitional threats appear as conventional constraints, and are not netted out from payoffs as disagreement points are. This asymmetry is implied by a new “expansion axiom†(along with standard axioms), one that is automatically satisfied in the standard bargaining problem. We then endogenize coalitional threats using internal consistency, requiring coalitions to be constrained by their subcoalitions just as the grand coalition is. For games with convex payoff sets, this consistent solution coincides with one in which the only threat from each coalition is their “standard†Nash solution, unconstrained by subcoalitions. For transferable-utility games, this observation uncovers a connection between the coalitional solution and the egalitarian solution of Dutta and Ray (1989, 1991).