Bondareva–Shapley theorem について

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Bondareva–Shapley theorem ：ウィキペディア英語版

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. 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 \;

\; \; N \;

is the set of players and where the ''value function''

\; v: 2^N \to \mathbb \;

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 \ \to ()

where

\forall i \in N : \sum_ \alpha (S) = 1

the following condition holds:
:

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

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