Egalitarianism in ordinal bargaining: the Shapley–Shubik rule

Abstract

A bargaining rule is ordinally invariant if its solutions are independent of which utility functions are chosen to represent the agents' preferences. For two agents, only dictatorial bargaining rules satisfy this property (Shapley, L., La Décision: Agrégation et Dynamique des Ordres de Préférence, Editions du CNRS (1969) 251). For three agents, we construct a “normalized subclass” of problems through which an infinite variety of such rules can be defined. We then analyze the implications of various properties on these rules. We show that a class of monotone path rules uniquely satisfy ordinal invariance, Pareto optimality, and “monotonicity” and that the Shapley–Shubik rule is the only symmetric member of this class. We also show that the only ordinal rules to satisfy a stronger monotonicity property are the dictatorial ones.