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:

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

References

• Bondareva, Olga N. (1963). "Some applications of linear programming methods to the theory of cooperative games (In Russian)" (PDF). Problemy Kybernetiki 10: 119–139.
• Kannai, Y (1992), "The core and balancedness", in Aumann, Robert J.; Hart, Sergiu, Handbook of Game Theory with Economic Applications, Volume I., Amsterdam: Elsevier, pp. 355–395, ISBN 978-0-444-88098-7
• Shapley, Lloyd S. (1967). "On balanced sets and cores". Naval Research Logistics Quarterly 14: 453–460. doi:10.1002/nav.3800140404.