Bargaining property of nucleolus and τ-value in a class of TU-games

First, we introduce pairwise-bargained consistency with a reference point, and use as reference points the maxmin and the minmax value within pure strategies of a certain constant-sum bimatrix game, and also the game value within mixed strategies of it. Second, we show that the pairwise-bargained consistency with reference point being the maxmin or the minmax value determines the nucleolus in some class of transferable utility games. (This result is known in the bankruptcy games and the pseudoconcave games with respect to supersets of the managers.) This class of games whose element we call a pseudoconcave game with respect to essential coalitions, of course, includes the bankruptcy games and the pseudoconcave games with respect to supersets of the managers. It is proved that this class of games is exactly the same as the class of games which have a nonempty core that is determined only by one-person and (n − 1)-person coalition constraints. And we give a sufficient condition which guarantees that the bargaining set coincides with the core in this class of games. Third, we interpret the τ-value of a quasibalanced transferable utility game by the pairwise-bargained consistency with reference point being the game value. Finally, by combining the second and the third results, if a transferable utility game in this class is also semiconvex, then the nucleolus and the τ-value are characterized by the pairwise-bargained consistency with different reference points which are given by the associated bimatrix game.