The Nucleolus and the $\tau$-value of Interval Games

Interval cooperative games model situations with cooperation in which the agents do not know for certain their coalitional payoffs, they know only bounds for the payoffs. Cooperative interval games have been introduced and studied in (Alparslan Gök, Branzei and Tijs, 2008, Branzei, Tijs and Alparslan Gök, 2008). Each interval game is defined by two cooperative games – the lower and the upper games – whose characteristic function values are bounds of the coalitional payoffs. Solutions for interval games are defined also in interval form. A TU game value $\varphi$ generates the interval value for the corresponding class of interval games if the value of the upper game dominates the value of the lower game. In (Alparslan Gök, Miquel and Tijs, 2009) it was shown how some monotonicity properties of some TU game values provide existence of the corresponding interval values for the class of convex interval games. However, the nucleolus and the $\tau$-value on this class do not possess such properties. Thus, in this paper the nucleolus for the interval values is defined as the result of the lexicographic minimization of the joint excess vector for upper and lower games. Its existence has been proved. The existence of the $\tau$-value is proved on the subclass of convex interval game generated by totally positive upper and lower games.