Other Solutions to Nash's Bargaining Problem

A two-person bargaining problem is considered. It is shown that under four axioms that describe the behavior of players there is a unique solution to such a problem. The axioms and the solution presented are different from those suggested by Nash. Also, families of solutions which satisfy a more limited set of axioms and which are continuous are discussed. WE CONSIDER a two-person bargaining problem mathematically formulated as follows. To every two-person game we associate a pair (a, S), where a is a point in the plane and S is a subset of the plane. The pair (a, S) has the following intuitive interpretation: a = (a1, a2) where ai is the level of utility that player i receives if the two players do not cooperate with each other. Every point x = (x1, x2) e S represents levels of utility for players 1 and 2 that can be reached by an outcome of the game which is feasible for the two players when they do cooperate. We are interested in finding an outcome in S which will be agreeable to both players. This problem was considered by Nash 3 and his classical result was that under certain axioms there is a unique solution. However, one of his axioms of independence of irrelevant alternatives came under criticism (see 2, p. 128). In this paper we suggest an alternative axiom which leads to another unique solution. Also, it was called to our attention by the referee that experiments conducted by H. W. Crott 1 led to the solution implied by our axioms rather than to Nash's solution. We also consider the class of continuous solutions which are required to satisfy only the axioms of Nash which are usually accepted. We give examples of families of such solutions.