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. 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. Let the pair ⟨ N , v ⟩ {displaystyle langle N,v angle } be a cooperative game in characteristic function form, where N ; {displaystyle N;} is the set of players and where the value function v : 2 N → R {displaystyle v:2^{N} o mathbb {R} } is defined on N {displaystyle N} 's power set (the set of all subsets of N {displaystyle N} ). The core of ⟨ N , v ⟩ {displaystyle langle N,v angle } is non-empty if and only if for every function α : 2 N ∖ { ∅ } → [ 0 , 1 ] {displaystyle alpha :2^{N}setminus {emptyset } o } where ∀ i ∈ N : ∑ S ∈ 2 N : i ∈ S α ( S ) = 1 {displaystyle forall iin N:sum _{Sin 2^{N}:;iin S}alpha (S)=1} the following condition holds: