This paper by Lloyd S. Shapley discusses the core of convex games. The core of an n-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one based on a convex set function, meaning that the incentives for joining a coalition increase as the coalition grows. Shapley shows that the core of a convex game is not empty and has a regular structure. He also demonstrates that certain cooperative solution concepts are related to the core. Specifically, the value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core. In a subsequent paper, similar results will be presented for the kernel and the bargaining set. The paper uses notation where n, s, t, etc., denote the number of elements in finite sets N, S, T, etc. The letter "O" denotes the empty set, and "⊂", "⊃" denote strict inclusion. Payoff vectors are elements of the n-dimensional linear space E^N with coordinates indexed by the elements of N. If a ∈ E^N and S ⊆ N, then a(S) is the sum of a_i over S. The hyperplane in E^N defined by the equation a(S) = v(S), 0 ⊂ S ⊆ N, is denoted by H_S. A game is a function v from a lower-case ring N to the reals. It is superadditive if v(S) + v(T) ≤ v(S ∪ T) for all S, T ∈ N with S ∩ T = O. It is convex if v(S) + v(T) ≤ v(S ∪ T) + v(S ∩ T) for all S, T ∈ N. It is strictly convex if the inequality holds whenever neither S ⊆ T nor T ⊆ S. Convex games form a convex cone in a suitable linear space, and this cone contains the subspace of additive games.This paper by Lloyd S. Shapley discusses the core of convex games. The core of an n-person game is the set of feasible outcomes that cannot be improved upon by any coalition of players. A convex game is defined as one based on a convex set function, meaning that the incentives for joining a coalition increase as the coalition grows. Shapley shows that the core of a convex game is not empty and has a regular structure. He also demonstrates that certain cooperative solution concepts are related to the core. Specifically, the value of a convex game is the center of gravity of the extreme points of the core, and the von Neumann-Morgenstern stable set solution of a convex game is unique and coincides with the core. In a subsequent paper, similar results will be presented for the kernel and the bargaining set. The paper uses notation where n, s, t, etc., denote the number of elements in finite sets N, S, T, etc. The letter "O" denotes the empty set, and "⊂", "⊃" denote strict inclusion. Payoff vectors are elements of the n-dimensional linear space E^N with coordinates indexed by the elements of N. If a ∈ E^N and S ⊆ N, then a(S) is the sum of a_i over S. The hyperplane in E^N defined by the equation a(S) = v(S), 0 ⊂ S ⊆ N, is denoted by H_S. A game is a function v from a lower-case ring N to the reals. It is superadditive if v(S) + v(T) ≤ v(S ∪ T) for all S, T ∈ N with S ∩ T = O. It is convex if v(S) + v(T) ≤ v(S ∪ T) + v(S ∩ T) for all S, T ∈ N. It is strictly convex if the inequality holds whenever neither S ⊆ T nor T ⊆ S. Convex games form a convex cone in a suitable linear space, and this cone contains the subspace of additive games.