On the relation between the Nash bargaining solution and the weighting method

Abstract

A set of n-person convex bargaining problems is considered where the feasible payoff set is the same, but the disagreement payoff vector is −αr with a given positive vector r and positive real parameter α. It is shown that as α→∞, the limit of the Nash solution is an optimal solution obtained by the weighting method where the weights are reciprocals of the components of vector r. The limit of the Nash solution is axiomatized in the multicriteria decision context and is determined as the unique minimum of a convex programming problem.

Introduction

An n-person conflict is defined by a pair (F,d) where $\text{F⊂}\text{R}{}^{\mathrm{n}}$ is a compact, convex set with at least one positive element and d is a nonpositive vector. F and d are called the feasible (payoff) set and the disagreement vector, respectively. In the case when the participants in the conflict (players) agree to select a payoff from F we will call this unique payoff vector solution, whereas when they are unable to reach an agreement, the payoffs are determined by the corresponding components of the disagreement vector.

There are many solution concepts and methods to find solutions for conflicts. The axiomatic approach requires the solution to satisfy a certain set of axioms and the existence and uniqueness of the solution is usually proved. The most common axiomatic solutions are the Nash solution (Nash, 1950), the non-symmetric Nash solution (Harsanyi and Selten, 1972), the Kalai–Smorodinsky solution (Kalai and Smorodinsky, 1975), the reference function solution (Anbarci, 1995), the egalitarian solution (Kalai, 1977), the superadditive solution (Perles and Maschler, 1981), the equal sacrifices solution (Chun, 1988), and the equal area solution (Anbarci, 1993, Calvo and Peters, 2000). A comprehensive review of cooperative models of bargaining can be found in Thomson (1994).

One can also consider an n-person conflict as a multiobjective optimization problem, where the payoff functions of the players are the objective functions. An early example of this approach can be found in Forgó (1983). There are many different solution concepts and algorithms for solving multiobjective optimization problems. The most frequently used methods are the sequential optimization method, the ε-constraint method, the weighting method, the distance-based and direction-based methods. A comprehensive summary of these techniques can be found, for example, in Szidarovszky et al. (1986).

Several researchers have pointed out the analogy between certain methods of conflict resolution and multiobjective optimization. For example, the Nash bargaining solution is a distance-based method where the disagreement vector is the nadir, and the geometric distance from the nadir is maximized. The non-symmetric Nash solution can also be considered as a distance-based method, where a weighted geometric distance is used. The Kalai–Smorodinsky solution is equivalent to a direction-based method where the selected direction of improvement points away from the disagreement vector to the ideal point (the vector with the maxima of the payoff functions in its components).

Dependence of a solution to a conflict on the disagreement point has been extensively studied as it is reviewed in Thomson (1994). The main focus of the investigations has been on disagreement point monotonicity and disagreement point concavity (linearity). Less attention has been paid to limiting properties of solutions.

In Forgó (1983) and later in Forgó et al. (1999), it is shown for special feasible sets that the limit of the Nash bargaining solution (L-Nash solution) is uniquely determined as the disagreement vector tends to infinity in a given negative direction and the limit can be obtained by solving at most n mathematical programming problems. In this paper, we will first show that it is enough to solve at most two optimization problems to get the L-Nash solution. It will then be pointed out that if we consider the multicriteria decision problem with the same feasible set and use the reciprocals of the components of the disagreement vector as weights, then the L-Nash solution can be obtained by the simple linear weighting method (LWM). If LWM produces multiple solutions, then the L-Nash solution is the minimum of a strictly convex quadratic function over the optimum set obtained by LWM. This two-step procedure will be called the extended weighting method (EWM).

EWM produces for every convex, compact set $\text{F⊂}\text{R}{}^{\mathrm{n}}$ and positive weight vector w a unique point. We will show that a set of axioms slightly different from Nash’s uniquely determines EWM if n=2. In the general case (n>2), a new consistency axiom is required which together with utilitarianism guarantees the uniqueness of EWM.