Geodesic
A New Characterization of the Pre-Kernel for TU Games Through its Indirect Function and its Application to Determine the Nucleolus for Three Subclasses of TU Games
Contributions to game theory and management, Tome 6 (2013), pp. 200-210

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The main goal is twofold. Thanks to the so-called indirect function known as the dual representation of the characteristic function of a coalitional TU game, we derive a new characterization of the pre-kernel of the coalitional game using the evaluation of its indirect function on the tails of pairwise bargaining ranges arising from a given payoff vector. Secondly, we study three subclasses of coalitional games of which its indirect function has an explicit formula and show the applicability of the determination of the pre-kernel (nucleolus) for such types of games using the indirect function. Three such subclasses of games concern the $1$-convex and $2$-convex $n$ person games and clan games. A clan game with the clan to be s singleton is known as a big boss game.
Keywords: dual representation, indirect function, pre-kernel, $1$- and $2$-convex $n$ person games, clan games, big boss games. 
@article{CGTM_2013_6_a13,
     author = {Dongshuang Hou and Theo Driessen and Antoni Meseguer-Artola and Bolg\'arka Mosoni},
     title = {A {New} {Characterization} of the {Pre-Kernel} for {TU} {Games} {Through} its {Indirect} {Function} and its {Application} to {Determine} the {Nucleolus} for {Three} {Subclasses} of {TU} {Games}},
     journal = {Contributions to game theory and management},
     pages = {200--210},
     publisher = {mathdoc},
     volume = {6},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CGTM_2013_6_a13/}
}
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Dongshuang Hou; Theo Driessen; Antoni Meseguer-Artola; Bolgárka Mosoni. A New Characterization of the Pre-Kernel for TU Games Through its Indirect Function and its Application to Determine the Nucleolus for Three Subclasses of TU Games. Contributions to game theory and management, Tome 6 (2013), pp. 200-210. http://geodesic.mathdoc.fr/item/CGTM_2013_6_a13/