We study simple majority rule for a fixed profile of individuals' preferences and a variable society. Four properties each linking decisions by a group to decisions by its various subgroups are introduced, and are used for characterizing simple majority rule.

A new axiom, symmetry-based compromise, is introduced to study solutions to convex bargaining problems. It is shown that any efficient solution that is also scale invariant and satisfies symmetry-based compromise must be Nash.

Three alternative characterizations of the proportional solution defined on compact and comprehensive bargaining problems with claims are provided. Two new contraction-type and expansion-type axioms are used. Moreover, the single-valuedness axiom is dispensable if the classical symmetry axiom is...