Bondareva–Shapley theorem
# 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 $v:2^{N}\to \mathbb {R}$ 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 $\alpha :2^{N}\setminus \{\emptyset \}\to [0,1]$ where

$\forall i\in N:\sum _{S\in 2^{N}:\;i\in S}\alpha (S)=1$ the following condition holds:

    $\sum _{S\in 2^{N}\setminus \{\emptyset \}}\alpha (S)v(S)\leq v(N).$
