Bondareva–Shapley theorem - Alchetron, the free social encyclopedia

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

Bondareva–Shapley theorem Wikipedia

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