Existence and Uniqueness of Ordinal Nash Outcomes

Abstract

In tiffs paper we present necessary and sufficient conditions Ibr existence and uniqueness of ordinal Nash outcomes. These outcomes are derived from the ordinal Nash solution a reinterpretation and an extension of the Nash bargaining solution that allows bargainers to have preference relations that are more general than expected utility. Out task is undertaken by the construction of a new notion called "induced utilities'.Journaltt/Ecommfic LiteratmeClassification Number: C78. ~!, 2000Academic Press

Bargaining and the Theory of Cooperative Games: John Nash and Beyond

“…An early contribution to the issue is by Rubinstein, Safra, and Thomson (1992) who propose a definition and a characterization of an ordinal version of the Nash solution in this context. Follow-up papers are by Burgos (1993Burgos ( , 1995, Hanany and Safra (1998), Grant and Kajii (1995), Safra and Zilcha (1993), Valenciano andZarzuelo (1994, 1997), Houba, Tieman, and Brinksman (1998), Denicolò (2000), Burgos, Grant, and Kajii (2002), Shalev (2002), Zhou (2007), Hanany (2007), and de Clippel (2008.…”

The ordinal Nash social welfare function

“…Other profiles in the domain are not restricted up-front in any way. The analysis utilizes the notion of induced utilities, as introduced in Hanany and Safra (2000).…”

Appeals immune bargaining solution with variable alternative sets

“…Since such a maximizing outcome exists for all problems X, D ∈ B, then for the EU case, the appeals-immune solution is well-defined on B. However, in Hanany and Safra (2000), examples were given of bargaining problems with non-expected utility preferences, for which no appeals-immune outcome exists. In these cases, the bargaining solution cannot be represented by the weak appeals relation, since there exist sets of alternatives for which the outcome set is empty (see Example 1 below).…”

“…In order to use this result to analyze extensions to non-expected utility, we define the notion of induced utilities similarly to Hanany and Safra (2000). This definition allows a useful connection between the weak appeals relation and the players' induced utilities, as shown in Lemma 1 below.…”

“…Given non-EU preferences, there exist bargaining problems for which no ordinal Nash outcome exists. Hanany and Safra (2000) presented necessary and sufficient conditions on players' preferences for existence and uniqueness of ordinal Nash outcomes. The ordinal Nash solution methodology analyzed bargaining problems with varying individual preferences on a fixed set of alternatives.…”