Background and summary
One of the main axioms that characterizes the Shapley value is the axiom of symmetry. However, in many applications the assumption that, except for the parameters of the games the players are completely symmetric, seems unrealistic. Thus, the use of nonsymmetric generalizations of the Shapley value was proposed in such cases.
Weighted Shapley values were discussed in the original Shapley (1953a) Ph.D. dissertation. Owen (1968, 1972) studied weighted Shapley values through probabilistic approaches. Axiomatizations of nonsymmetric values were done by Weber (Chapter 7 this volume), Shapley (1981), Kalai and Samet (1987), and Hart and Mas-Colell (1987).
Consider, for example, a situation involving two players. If the two players cooperate in a joint project, they can generate a unit profit that is to be divided between them. On their own they can generate no profit. The Shapley value views this situation as being symmetric and would allocate the profit from cooperation equally between the two players. However, in some applications lack of symmetry may be present. It may be, for example, that for the project to succeed, a greater effort is needed on the part of player 1 than on the part of player 2. Another example arises in situations where player 1 represents a large constituency with many individuals and player 2's constituency is small (see, for example, Kalai 1977 and Thomson 1986).