Nash Bargaining in Ordinal Environments

We analyze the implications of Nash’s (1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (1977), we introduce a weaker independence of irrelevant alternatives axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley-Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker independence of irrelevant alternatives axiom. We also analyze the implications of other independence axioms.