Bondareva–Shapley theorem

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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 be a cooperative game in characteristic function form, where is the set of players and where the value function is defined on 's power set (the set of all subsets of ).

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 (eds.), 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 (4): 453–460. doi:10.1002/nav.3800140404. hdl:10338.dmlcz/135729.