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.
be a cooperative game in characteristic function form, where
is the set of players and where the value function
is non-empty if and only if for every function
the following condition holds: