An Axiomatization of the Shapley Value using a Fairness Property
by
Rene van den Brink
Department of Econometrics and CentER, Tilburg University

A famous solution for cooperative games with transferable utility -or simply TU-games- is the Shapley value (Shapley, 1953). In this paper we provide an axiomatization of the Shapley value using a fairness property. This fairness property states that if to a game we add another game in which two players are symmetric then the payoffs of these two players change by the same amount. We show that efficiency, the null player property and fairness characterize the Shapley value on the class of all TU-games on a given set of players on N.

This fairness property is related to fairness as introduced by Myerson (1977) for games in which the possibilities of coalition formation in a TU-game are limited because the players are part of a limited communication structure. In that model fairness means that deleting a communication relation between two players has the same effect on both their payoffs. A similar fairness axiom is used in van den Brink (1997) for games in which the cooperation possibilities in a TU-game are limited because the players are part of a hierarchical permission structure in which there are players who need permission from certain other players before they are allowed to cooperate. In that model fairness means that deleting a permission relation between two players has the same effect on both their payoffs. In van den Brink (1995) a fairness axiom for relational power measures for directed graphs is introduced. In that context fairness means that deleting a dominance relation between two nodes in a digraph changes their relational power by the same amount.

As already noted by Dubey (1975), axioms that characterize the Shapley value on the class of all games (on given set of players N) not necessarily characterize the Shapley value on important subclasses of games such as the class of simple games. It turns out that efficiency, the null player property and fairness also characterize the Shapley value on the class of simple games (on N).

Besides the literature on fairness as introduced in Myerson (1977), this paper also is related to the axiomatization of the Shapley value by efficiency, symmetry and strong monotonicity given by Young (1985). Strong monotonicity states that the payoff of a player does not decrease if we transform the game in a way such that the marginal contribution of this player to every coalition does not decrease. As argued by Chun (1991), it is sufficient to require that the payoff of a player does not change if we transform the game in a way such that its marginal contributions do not change. So, strong monotonicity essentially compares the payoff of a player if we add a game in which this player is a null player, while fairness compares the change in payoff of two players if we add a game in which these players are symmetric.

Date received: May 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafc-26.