A cardinal characterization of the Rubinstein-Safra-Thomson axiomatic bargaining theory

In a recent paper Rubinstein, Safra, and Thomson (RST) have provided an interesting re-examination of the widely applied Nash solution for a two-person bargaining problem. They recast the usual Nash bargaining problem into a more 'natural' setting of feasible alternatives with a disagreement outcome. The two players are then described by their risk preferences defined on the set of lotteries over the alternatives and the disagreement outcome. This enables them to define an ordinal Nash solution in terms of the agents' risk preferences. Essentially, their ordinal solution is an outcome that is 'immune' against possible objections. Freeing the definition of the Nash solution from 'utility' naturally led RST to extending its scope to Non-Expected Utility (NEU) preferences. We contend, however, that the family of NEU preferences considered by RST is unduly restrictive. The assumptions imposed on the risk preferences by RST essentially exclude any members of the Rank Dependent Expected Utility (RDEU) and betweenness families that can accommodate the very choice paradoxes that stimulated the development of NEU theory. As these are two of the most extensively analyzed and widely applied NEU models in the literature, this seems to cast doubt on how broad an extension to NEU preferences the RST approach affords. We demonstrate, however, that RST's analysis can be modified so that their conclusion is valid in a wider class of preferences that can include examples of RDEU preferences. This class consists of preferences that admit what we term a disagreement linear representation.