Geodesic
On Quasi-cores, the Shapley Value and the Semivalues
Contributions to game theory and management, Tome 1 (2007), pp. 107-122

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The aim of the present paper is that of introducing a new concept of coalitional rationality for values of cooperative TU games, called $w$-coalitional rationality, such that this becomes the usual coalitional rationality in the case of an efficient value. As amotivation for our new concept, the class of Semivalues, which are in general non efficient values, was considered, and we proved necessary and sufficient conditions for the $w$-coalitional rationality of Semivalues. Itis well known that the only efficient Semivalue is the Shapley value. The basic idea to be followed here is the fact that the Semivalues are not connected to the core, because the efficiency is missing, but may be connected to some quasi-core. (in particular, to the Shapley–Shubik weak $\varepsilon$-core, from which the $w$ is borrowed). Shapley and Shubik (1966) have introduced the quasi-cores in connection with the market games and have discussed the non-emptiness of two types of quasi-cores (see the paper by Kannai in Handbook of Game Theory, vol. I, 1992).
Keywords: Coalitional rationality, Shapley value, Semivalues, Average per capita formulas, Quasi-cores. 
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Irinel Dragan. On Quasi-cores, the Shapley Value and the Semivalues. Contributions to game theory and management, Tome 1 (2007), pp. 107-122. http://geodesic.mathdoc.fr/item/CGTM_2007_1_a7/