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'' $v: 2^N \to \mathbb$ 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 \ \to ,1/math> where


$\forall i \in N : \sum_ \alpha (S) = 1$

the following condition holds:
:$\sum_ \alpha (S) v (S) \leq v (N).$

References

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{{DEFAULTSORT:Bondareva-Shapley theorem
Game theory
Economics theorems
Cooperative games
Lloyd Shapley