Coincidence of two solutions to Nash's bargaining problem

- A summary is given for the Nash and Kalai-Smorodinsky solutions to the bargaining problem.
- An account is given for the precise conditions under which the two solutions coincide.
- In normalized situations, these solutions coincide only when they satisfy a number of notions of fairness.

In 1950, John Nash gave an elegant solution to the bargaining problem using his somewhat controversial IIA axiom. Twenty-five years later, Ehud Kalai and Meir Smorodinsky gave a different solution replacing the IIA condition with their own Monotonicity condition. While the two solutions obviously coincide under certain conditions (e.g. when the problem is symmetric), they do not in general agree. This paper presents a complete account of the precise conditions under which the two solutions coincide.