Теорема Бондаревой — Шепли

В теории игр теорема Бондаревой — Шепли описывает необходимые и достаточные условия для непустоты ядра в кооперативной игре. В частности, ядро игры непусто тогда и только тогда, когда игра сбалансирована. Теорема была независимо сформулирована Ольгой Бондаревой и Ллойдом Шепли в 1960-х.

Теорема

Пусть дана кооперативная игра $\;\langle N,v\rangle \;$ , в которой $\;\;N\;$ — множество игроков, а функция полезности $\;v:2^{N}\to \mathbb {R} \;$ определена на множестве всех подмножеств $N$ .
Ядро игры $\;\langle N,v\rangle \;$ непусто тогда и только тогда, когда для любой функции $\alpha :2^{N}\setminus \{\varnothing \}\to [0,1],$ где
$\forall i\in N:\sum _{S\in 2^{N}:\;i\in S}\alpha (S)=1$
выполнено следующее условие:

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

Литература

Бондарева О.Н. Некоторые применения методов линейного программирования к теории кооперативных игр // Проблемы кибернетики. Выпуск 10. — М.: Государственное издательство физико-математической литературы, 1963. — С. 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. On balanced sets and cores (англ.) // Naval Research Logistics Quarterly^[англ.] : journal. — 1967. — Vol. 14. — P. 453—460. — doi:10.1002/nav.3800140404.

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