Individual Rationality and Nash's Solution to the Bargaining Problem

The classical two person bargaining problem, as set forth by Nash in 1950, consists of a compact convex subset S of the plane, and a point s E S. A point (x^, X2) in S represents the von Neumann-Morgenstern utility available to each player as a result of some feasible agreement, and the set S represents the set of all feasible utility payoffs. The point s = (s^, Sj) represents the utility of the "status quo"—that is, (^i, .v̂ ) is the payoff to the players in the absence of any agreement. We will only consider bargaining problems in which there is some possibiHty of mutual benefit; i.e., problems for which there is some x E S such that' (x,, ^2) > (5|, '̂2)Since the origin of each player's utility scale is arbitrary, we may without loss of generality normalize each player's utility function so that (5,, ^2) = (0, 0). Of course the units of each player's utility function may still be varied arbitrarily (see Condition 1 below). Let B denote the class of bargaining problems with status quo at the origin. We will denote elements of B by S, rather than by (S, 0). Nash defined a solution of the bargaining problem to be a function / defined on B which associates with each bargaining problem a single feasible outcome (i.e., f(S) G sy and which obeys the following four conditions.^ (We state the first condition somewhat more explicitly than Nash did.) I. Independence of Linear Transformations. For any bargaining problem S and positive real numbers a and b, if T = {{aXf, bxj) \ (Xy, X2) E S), then f(T)