Bargaining Powers, a Surface of Weights, and Implementation of the Nash Bargaining Solution

In the present paper a new approach to the Nash bargaining solution (N.b.s.) is proposed. (Shapley, 1969) introduced weights of individual utilities and linked the N.b.s. with utilitarian and egalitarian solutions. This equivalence leaves open a positive question of a possible mechanism of weights formation. Can the weights be constructed in result of a recurrent procedure of reconciliation of utilitarian and egalitarian interests? Can a set of feasible bundles of weights be a result of a procedure or a game independent on a concrete bargaining situation? We answer these questions in the paper. A two-stage $n$-person game is considered, where on the first stage the players on base of their bargaining powers elaborate a set of all possible bundles of weights $\Lambda = \{ \left(\lambda_1,\ldots,\lambda_n \right) \}.$ This surface of weights can be used by an arbitrator for evaluation outcomes in different concrete bargains. On the second stage, for a concrete bargain, the arbitrator chooses a vector of weights and an outcome by use of a maximin criterion. We prove that this two-stage game leads to the well-known asymmetric N.b.s.