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

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.