Bondareva–Shapley theorem

The Bondareva–Shapley theorem, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form. Specifically, the game's core is non-empty if and only if the game is balanced. The Bondareva–Shapley theorem implies that market games and convex games have non-empty cores. The theorem was formulated independently by Olga Bondareva and Lloyd Shapley in the 1960s.

Theorem
Let the pair $$\langle N, v\rangle$$ be a cooperative game in characteristic function form, where $$ N$$ is the set of players and where the value function $$ v: 2^N \to \mathbb{R} $$ is defined on $$N$$'s power set (the set of all subsets of $$N$$).

The core of $$\langle N, v \rangle $$ is non-empty if and only if for every function $$\alpha : 2^N \setminus \{\emptyset\} \to [0,1]$$ where $$\forall i \in N : \sum_{S \in 2^N : \; i \in S} \alpha (S) = 1$$ the following condition holds:
 * $$\sum_{S \in 2^N\setminus\{\emptyset\}} \alpha (S) v (S) \leq v (N).$$