The computation of the Nash bargaining solution (Q1264100)

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.