Game Theory: 4th Edition by Owen Guillermo

Author:Owen, Guillermo [Owen, Guillermo]
Language: eng
Format: epub
Publisher: Emerald Group Publishing Limited
Published: 2013-05-08T00:00:00+00:00

Proof. Necessity follows from Theorem 10.5.1 and the fact that every balancing vector must satisfy (10.19)–(10.20). To show that the condition is sufficient, we note that the maximum of program (10.18)–(10.20), which is known to exist, is to be found at an extreme point. This extreme point is the balancing vector of some minimal balanced collection. Thus, if (10.23) holds for all such vectors, the maximum of (10.18)–(10.20) must be less than or equal to .

Theorem 10.5.10 gives us a system of inequalities which must be satisfied if a game has a nonempty core. Now for superadditive games, the condition (10.23) will certainly hold if is a partition. Thus we need concern us only with minimal balanced collections which are not partitions.

10.5.11 Example. Let . Apart from the partitions there is only one minimal balanced collection, namely

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