Institute for Economics and Mathematics at St. Petersburg Russian Academy of Sciences Vladimir Matveenko Bargaining Powers, Weights of Individual Utilities, and Implementation of the Nash Bargaining Solution 10 th International Meeting of the Society for Social Choice and Welfare Moscow, July 21-24, 2010

n-person bargaining problem with bargaining powers - the feasible set of utilities d = 0 - the disagreement point Asymmetric Nash bargaining solution (N.b.s.): Axiomatized by Roth, 1979, Kalai, 1977 Nash Program / Implementation (Support for the solution)

Plan: 1.Introduction: Relations between weights of individual utilities and the bargaining powers. 2.A 2-stage game: I. Formation of a surface of weights Λ II. An arbitrator finds The solution is the asymmetric N.b.s.

Utilitarian, Egalitarian, and Nash Solutions THEOREM. Let the set S be restricted by coordinate planes and by a surface where is a smooth strictly convex function, and let be positive bargaining powers. Then the following two statements are equivalent: 1. For a point such weights exist that is simultaneously: (a) a “utilitarian” solution with weights (b) an “egalitarian” solution (what is the same, ) 2. is an asymmetric N.b.s. with bargaining powers

Shapley, 1969: “(III) An outcome is acceptable as a “value of the game” only if there exist scaling factors for the individual (cardinal) utilities under which the outcome is both equitable and efficient”. Binmore, 2009: “A small school of psychologists who work on “modern equity theory” [Deutsch, Kayser et al., Lerner, Reis, Sampson, Schwartz, Wagstaff, Walster et al.]... They find experimental support for Aristotle’s ancient contention that “what is fair is what is proportional”. More precisely, they argue that an outcome is regarded as fair when each person’s gain over the status quo is proportional to that person’s “social index”. The “social index” of the player i is the number inverse to the weight

Geometric characterization of the asymmetric N.b.s

- transfer coefficients - bargaining powers A case of the (asymmetric) Shapley value for TU and NTU games (Kalai and Samet, 1987, Levy and McLean, 1991)

Does the iteration sequence converge? THEOREM. A stability condition is the inequality where E is the elasticity of substitution of function in the solution point

A new approach to the N.b.s. A 2-stage game: On the 1 st stage the players form a surface of weights Λ. On the 2 nd stage an arbitrator in a concrete situation chooses weights and an outcome x following a Rawlsian principle.

Formation of the surface of weights (which can be used in many bargains) Under this mechanism, the utility is negatively connected with the “own” weight and positively - with the another’s. That is why each participant is interested in diminishing the weight of his “own” utility and in increasing the weight of another’s. However participant i agrees in a part of the surface on a decrease in another’s weight at the expense of an increase in his own weight, as soon as a partner similarly temporizes in another part of surface Λ. So far as the system of weights is essential only to within a multiplier, the participants may start bargaining from an arbitrary vector of weights and then construct the surface on different sides of the initial point.

Formation of the surface of weights (which can be used in many bargains) On what increase of his own weight (under a decrease in the another’s weight) will the participant agree? Bargaining powers become apparent here. We suppose that a constancy of bargaining powers of participants mean a constancy of elasticities of in respect to. The more is the relative bargaining power of the participant the less increase in his utility can he achieve. Differential equation: Its solution is the curve of weights: The arbitrator’s problem: THEOREM: it is the asymmetric N.b.s.

Curves of weights with different relative bargaining powers

INTERPRETATION: A role of a community (a society) in decision making Community (society) serves as an arbitrator takes into account moral-ethical valuations confesses a maximin (Rawlsian) principle of fairness is manipulated Participants form moral-ethical valuations using all accessible means (propaganda through media, Internet, meetings, rumours) Bargaining powers of the participants depend on their military power, access to media and to government, on their propagandist and imagemaking talent, and earlier reputation Moral-ethical valuations are not one-valued. Although in any bargain concrete valuations act, in general the society is conformist and there is a whole spectrum of valuations which can be used in case of need. There is a correspondence between acceptable outcomes and vectors of valuations (weights).

THEOREM A mathematical comment is provided in: V.Matveenko Ekonomika i Matematicheskie Metody

Mechanism (simultaneous change in weights and in a supposed outcome) Another version of the model: in a concrete situation of bargaining a process of changing valuations and, correspondingly, assumed outcomes. The society encourages those valuations (and those outcomes) which correspond to the maximin principle of fairness.

Mechanism (simultaneous change in weights and in a supposed outcome)

The egalitarian solution (Kalai) and the Kalai-Smorodinsky solution No 1 st stage of the game: For these solutions the question of the choice of weights is solved already in a definite way