Proportional bargaining solutions, strictly comprehensive sets and the axiom of continuity

Continuity is an axiom of Kalai (Econometrica 45(7):1623–1630, 1977)’s and Roth (Econometrica 47(3):775–777, 1979b)’s characterizations of the proportional bargaining. In this paper we establish that, in the class of strictly comprehensive sets, continuity is a corollary of the other axioms. Consequently the proportional solution can be axiomatically defined in the class of strictly comprehensive sets without the axiom of continuity. We also show it is possible to tighten Kalai (Econometrica 45(7):1623–1630, 1977)’s and Roth (Econometrica 47(3):775–777, 1979b)’s axiomatization of the proportional bargaining solution, assuming a weaker version of the axiom of continuity.