# Minimal exact balancedness

Working paper

Issue number:

RM/11/009

Publisher:

Maastricht University

Year:

2011

To verify whether a transferable utility game is exact, one has to check a linear inequal- ity for each exact balanced collection of coalitions. This paper studies the structure and properties of the class of exact balanced collections. Comparing the deﬁnition of exact balanced collections with the deﬁnition of balanced collections, the weight vector of a balanced collection must be positive whereas the weight vector for an exact balanced collection may contain one negative weight. We investigate minimal exact balanced col- lections, and show that only these collections are needed to obtain exactness. The relation between minimality of an exact balanced collection and uniqueness of the corresponding weight vector is analyzed. We show how the class of minimal exact balanced collections can be partitioned into three basic types each of which can be systematically generated.