Weak independence of irrelevant alternatives and generalized Nash bargaining solutions

TL;DR

This paper explores the implications of a weak form of independence of irrelevant alternatives in non-convex bargaining problems, revealing solutions that unify Nash and Kalai-Smorodinsky concepts through endogenous weighting.

Contribution

It introduces a representation of bargaining solutions satisfying weak IIA, linking Nash and Kalai-Smorodinsky solutions via a two-stage optimization process.

Findings

Solutions can be represented using weighted products of normalized utilities.

Weights are determined endogenously through a two-stage optimization.

The approach applies to bargaining over linear production technologies.

Abstract

In Nash's (1950) seminal result, independence of irrelevant alternatives (IIA) plays a central role, but it has long been a subject of criticism in axiomatic bargaining theory. This paper examines the implication of a weak version of IIA in multi-valued bargaining solutions defined on non-convex bargaining problems. We show that if a solution satisfies weak IIA together with standard axioms, it can be represented, like the Nash solution, using weighted products of normalized utility levels. In this representation, the weight assigned to players for evaluating each agreement is determined endogenously through a two-stage optimization process. These solutions bridge the two dominant solution concepts, the Nash solution and the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975). Furthermore, we consider special cases of these solutions in the context of bargaining over linear production technologies.