Generalized Convex Games

Economists have long argued that the existence of increasing returns to scale or unique complementary inputs may lead to indeterminacy in how the gains to team production are shared among team members. Cooperative games provide a framework in which to formalize and explore this intuition. In games with side payments, the notion of a convex game introduced by Shapley (1971) provides a natural way to formalize these ideas. A coalition form game (N, v) with side payments is a convex game if for all coalitions S and T , v(S) + v(T ) ≤ v(S ∩ T ) + v(S ∪ T ). This condition arises when each player provides some number of units of a homogeneous input and production displays increasing returns to scale. It also arises when each player possesses a unique input and the inputs are complementary (Topkis (1981)). Shapley showed that the core of a convex game is nonempty, and that its extreme points can be computed by the greedy algorithm, that is, by listing the players in some order and giving each player in turn his or her marginal contribution v(S ∪ {i})− v(S) to the coalition S of preceding players. The fact that any ordering of the players in the greedy algorithm yields a payoff vector in the core suggests that the core places weak restrictions on the way the fruits of cooperation are shared in convex games. Sharkey (1982) further explores this idea. He introduces the notion of a “large” core, which is characterized by the property that for every unblocked (but not necessarily feasible) payoff vector y there exists a payoff vector x in the core such that x ≤ y. He shows that the core in a convex game is large in this sense. ∗We thank Tatsuro Ichiishi for pointing us to the literature on generalizations of convex games, John Roberts for