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 core of is non-empty if and only if for every function where
the following condition holds:
- 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.