A split is the unique fair settlement if it is efficient, symmetric, and unchanged by irrelevant alternatives or rescaling.
Nash's 1950 solution, and its axioms are the contested part — independence of irrelevant alternatives in particular. Kalai and Smorodinsky's alternative drops it and yields a different split.
In practice
- Two departments splitting a shared saving find a settlement that is efficient, symmetric, and independent of the third proposal nobody wanted.
- A divorce settlement that would change if an irrelevant third option were introduced has failed one of the axioms, which is usually where mediation stalls.