Weighted values and the core in NTU games. (English) Zbl 1415.91031
Summary: D. Monderer et al. [Int. J. Game Theory 21, No. 1, 27–39 (1992; Zbl 0778.90094)] proved that the core is included in the set of the weighted Shapley values in TU games. The purpose of this paper is to extend this result to NTU games. We first show that the core is included in the closure of the positively weighted egalitarian solutions introduced by E. Kalai and D. Samet [Int. J. Game Theory 16, 205–222 (1987; Zbl 0633.90100)]. Next, we show that the weighted version of the Shapley NTU value by L. S. Shapley [Colloques Int. Centre nat. Rech. Sci. 171, 251–263 (1969; Zbl 0218.90088)] does not always include the core. These results indicate that, in view of the relationship to the core, the egalitarian solution is a more desirable extension of the weighted Shapley value to NTU games. As a byproduct of our approach, we also clarify the relationship between the core and marginal contributions in NTU games. We show that, if the attainable payoff for the grand coalition is represented as a closed-half space, then any element of the core is attainable as the expected value of marginal contributions.
MSC:
| 91A12 | Cooperative games |
References:
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