The computation of the Nash bargaining solution
type: Item
instance of: scholarly article
title: The computation of the Nash bargaining solution
zbMATH Open document ID: 0688.90063
DOI: 10.1016/0898-1221(89)90136-3
author: K. Appert
publication date: 1989-01-01T00:00:00Z
review text: An n-player decision problem is described by the decision set \(\Omega_ i\) and the objective functional \(J_ i:\) \(\Omega\) \(\to {\mathbb{R}}\) for any decision maker i, \(1\leq i\leq n\). Here \(\Omega =\Omega_ 1\times \Omega_ 2\times...\times \Omega_ n\) and further, let \((D_ 1,D_ 2,...,D_ n)\in {\mathbb{R}}^ n\) be the noncooperative Nash outcome where \(D_ i\in {\mathbb{R}}\) represents a status quo outcome for decision maker i. The element \(u^*\in \Omega\) is called the cooperative Nash bargaining solution for the n-player decision problem if it is a solution to the problem \[ \max imize\quad \prod^{n}_{i=1}(J_ i(u)-D_ i) \] subject to \(u\in \Omega\) and \(J_ i(u)\geq D_ i\) for all \(1\leq i\leq n\). The authors present two methods to compute the Nash bargaining solution. Firstly, the Nash bargaining problem is considered as a product-form optimization problem in a Hilbert space with functional inequality constraints and the feasible direction method is applied to find its solution. Secondly, the Nash bargaining problem is considered as a hierarchical decision problem and it is solved by an indirect hierarchical method of finding bargaining solutions. Under certain conditions, the Nash bargaining problem is equivalent to a two-level hierarchical decision problem. At the lower level of the hierarchy a weighted sum of the decision makers' objectives is maximized. At the top level a set of nonlinear algebraic equations is iteratively solved to obtain the Nash bargaining solution. The computation of the Nash bargaining solution in both methods is carried out by means of an algorithm.
Mathematics Subject Classification ID: 91A12
Mathematics Subject Classification ID: 90C90
zbMATH DE Number: 4128679
zbMATH Keywords: n-player decision problem
zbMATH Keywords: noncooperative Nash outcome
zbMATH Keywords: cooperative Nash bargaining solution
zbMATH Keywords: Hilbert space
zbMATH Keywords: feasible direction method
zbMATH Keywords: two-level hierarchical decision
reviewed by: Theo S. H. Driessen
MaRDI profile type: Publication
full work available at URL: 0898-1221(89)90136-3
OpenAlex ID: W2068844701
cites work: Axiomatic models of bargaining
cites work: Solution for a dynamic bargaining problem with an application to resource management
cites work: Rate of Convergence of a Class of Methods of Feasible Directions
cites work: Constrained optimization methods in Hilbert spaces and their applications to optimal control problems with functional inequality constraints
cites work: Nondominated decisions and cone convexity in dynamic multicriteria decision problems
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 4128679