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 ${\displaystyle \;\langle N,v\rangle \;}$ be a cooperative game in characteristic function form, where ${\displaystyle \;\;N\;}$ is the set of players and where the value function ${\displaystyle \;v:2^{N}\to \mathbb {R} \;}$ is defined on $N$'s power set (the set of all subsets of $N$).
The core of ${\displaystyle \;\langle N,v\rangle \;}$ is non-empty if and only if for every function ${\displaystyle \alpha :2^{N}\setminus \{\emptyset \}\to [0,1]}$ where
${\displaystyle \forall i\in N:\sum _{S\in 2^{N}:\;i\in S}\alpha (S)=1}$
the following condition holds:

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

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.