Extending the Nash solution to choice problems with reference points

- We treat bargaining problems and rationing problems with non-convex feasible sets.
- We generalize the Nash solution to choice problems with reference points.
- Aumannʼs result on the Shapley NTU value may be modified for the Nash solution.
- The characterization of the Nash solution remains valid for interesting subclasses.
- We provide an axiomatization of the Nash solution on classical bargaining problems.

In 1985 Aumann axiomatized the Shapley NTU value by non-emptiness, efficiency, unanimity, scale covariance, conditional additivity, and independence of irrelevant alternatives. We show that, when replacing unanimity by “unanimity for the grand coalition” and translation covariance, these axioms characterize the Nash solution on the class of n-person choice problems with reference points. A classical bargaining problem consists of a convex feasible set that contains the disagreement point here called reference point. The feasible set of a choice problem does not necessarily contain the reference point and may not be convex. However, we assume that it satisfies some standard properties. Our result is robust so that the characterization is still valid for many subclasses of choice problems, among those is the class of classical bargaining problems. Moreover, we show that each of the employed axioms - including independence of irrelevant alternatives - may be logically independent of the remaining axioms.