The Bargaining problem: Nash and other solutions

Abstract

[en] The main objective of this project is to analyze the bargaining problem from a mathematical perspective, as Nash (1950) did. A bargaining problem is a situation where a (finite) set of agents may cooperate to their mutual benefit. They must reach an unanimous agreement. If they do not reach such an agreement, they get the status quo, the disagreement outcome. This problem was first analyzed in an axiomatic form by Nash (1950). Our object of study in bargaining theory is to find a bargaining solution and characterize it. We consider a set of axioms, motivated by a particular application, and we identify the bargaining solution that satisfy them. To this end, we provide some mathematical tools and concepts, mainly separation theorems and some concepts of preferences and utility functions. This is needed for the following chapters. Next, we present the bargaining model of Nash (1950). We prove that the unique solution he proposes of the bargaining problem, the Nash solution, is the one that is characterized by four axioms, which have a nice interpretation. Finally, other bargaining solutions are presented such as the Kalai-Smorodinsky (1975) solution, or other solutions proposed in the literature, which use different sets of axioms.