In What Sense is the Nash Solution Fair?

Abstract

The interpretation is that two players have to agree on a joint payoff vector in T , the ith coordinate for the ith player, i = 1, 2 in order to receive these payoffs or, else, to fall back to the status quo point d. Assuming, that this scenario results as the image under the two players’ concave von Neumann-Morgenstern utility functions on an underlying economic or social scenario, this model is determined only up to affine transformations of both players’ payoffs. Accordingly, d is sometimes assumed to be 0 ∈ R (0-normalization), sometimes in addition it is assumed that maxx∈T xi = 1, i = 1, 2 (0−1−1-normalization). Moreover, every part of T not in R+ is skipped representing the fact that the interest focusses only on individually rational payoff vectors. The resulting S ⊂ R+ is then a 0− 1− 1-normalized bargaining situation, whose boundary is often assumed to be smooth. The Nash bargaining solution has been introduced by John F. Nash (1953) as a solution for two person bargaining games. Nash already presented three approaches to the solution that are methodologically and in spirit quite different. One is the definition of the Nash solution as the maximizer of the Nash product, i.e. the product of the two players’ payoffs. This might be seen as maximizing some social planners’ preference relation on the set of players’ utility allocations. So whatever fairness is represented by the Nash solution it should be hidden in this planners’ preferences.