 # Bondareva–Shapley theorem

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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 v : 2 N R 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 α : 2 N { } [ 0 , 1 ] where
i N : S 2 N : i S α ( S ) = 1
the following condition holds:

S 2 N { } α ( S ) v ( S ) v ( N ) .

## References

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