Farsighted Coalitional Stability in TU-Games

We study farsighted coalitional stability in the context of TU-games. Chwe (1994, p.318) notes that, in this context, it is difficult to prove nonemptiness of the largest consistent. We show that every TU-game has a nonempty largest consistent set. Moreover, the proof of this result allows to conclude that each TU-game has a farsighted stable set. We go further by providing a characterization of the collection of farsighted stable sets in TU-games. We also show that the farsighted core of a TU-game is empty or equal to the set of imputations of the game. Next, we study the relationships between the core and the largest consistent set in superadditive TU-games and in clan games. In the last section, we explore the stability of the Shapley value in superadditive TU-games. We show that the Shapley value is always a stable imputation. More precisely, if the Shapley value does not belong to the core, then it constitutes a farsighted stable set. We provide a necessary and sufficient condition for a superadditive TU-game to have the Shapley value in the largest consistent set