Bondareva-Shapley theorem

The Bondareva-Shapley theorem describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game. 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 1960's.

Theorem

Let the pair $;\; langle\; N,\; v\; angle\; ;$ be a cooperative game ($;\; ;\; N\; ;$ is the set of players, and $;\; v:\; 2^N\; o\; mathbb\{R\}\; ;$ is the value function).
The core of $;\; langle\; N,\; v\; angle\; ;$ is non-empty if and only if for every function $alpha\; :\; 2^N\; o\; mathbb\{R\}\_+$ where
$forall\; i\; in\; N\; :\; sum\_\{S\; subseteq\; N\; :\; ;\; i\; in\; S\}\; alpha\; (S)\; =\; 1$
the following condition holds: $sum\_\{S\; subseteq\; N\}\; alpha\; (S)\; v\; (S)\; leq\; v\; (N)$

References

*cite journal
last =Bondareva
first =Olga N.
authorlink = Olga N. Bondareva
coauthors =
title =Some applications of linear programming methods to the theory of cooperative games (In Russian)
journal =Problemy Kybernetiki
volume =10
issue =
pages =119–139
date =1963
url =
doi =
id =
accessdate =
*citation
last =Kannai
first =Y
authorlink =
editor-last =Aumann |editor-first=Robert J. |editor-link=Robert J. Aumann
editor2-last=Hart |editor2-first=Sergiu|editor2-link=Sergiu Hart
contribution=The core and balancedness
title =Handbook of Game Theory with Economic Applications, Volume I.
publisher =Elsevier
date =1992
location =Amsterdam
pages =355-395
url =
doi =
id =
isbn = 978-0-444-88098-7
*cite journal
last = Shapley
first = Lloyd S.
authorlink =Lloyd S. Shapley
coauthors =
title =On balanced sets and cores
journal =Naval Research Logistics Quarterly
volume =14
issue =
pages =453–460
date =1967
url =
doi =
id =
accessdate =