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 ( N, v) be a cooperative game in characteristic function form, where N is the set of players and where the value function is defined on N's power set (the set of all subsets of N). The core of ( N, v ) is non-empty if and only if for every function where the following condition holds: