# On extensions of the core and the anticore of transferable utility games

Working paper

Publisher:

Maastricht University

Year:

2012

We consider several related set extensions of the core and the anticore of games with transferable utility. An eﬃcient allocation is undominated if it cannot be improved, in a speciﬁc way, by sidepayments changing the allocation or the game. The set of all such allocations is called the undom- inated set, and we show that it consists of ﬁnitely many polytopes with a core-like structure. One of these polytopes is the L1-center, consisting of all eﬃcient allocations that minimize the sum of the absolute values of the excesses. The excess Pareto optimal set contains the allocations that are Pareto optimal in the set obtained by ordering the sums of the absolute values of the excesses of coalitions and the absolute values of the excesses of their complements. The L1-center is contained in the excess Pareto optimal set, which in turn is contained in the undominated set. For three-person games all these sets coincide. These three sets also coincide with the core for balanced games and with the anticore for antibalanced games. We study properties of these sets and provide characterizations in terms of balanced collections of coalitions. We also propose a single- valued selection from the excess Pareto optimal set, the min-prenucleolus, which is deﬁned as the prenucleolus of the minimum of a game and its dual.