Positivity 12 (2008), 613–629
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040613-17, published online May 27, 2008
On the Aumann–Shapley value
Achille Basile, Camillo Costantini and Paolo Vitolo
Abstract. Let L be a D-lattice, i.e. a lattice ordered eﬀect algebra, and let
BV be the Banach space of all real-valued functions of bounded variation on
L (vanishing at 0) endowed with the variation norm. We prove the existence
of a continuous Aumann–Shapley value ϕ on bv
NA, the subspace of BV
spanned by all functions of the form f ◦ µ,whereµ: L → [0, 1] is a non-
atomic σ-additive modular measure and f :[0, 1] → R is of bounded variation
and continuous at 0 and at 1.
Mathematics Subject Classiﬁcation (2000). 28A12, 06C15, 91A12.
Keywords. Aumann–Shapley value, eﬀect algebra, D-lattice, modular
Let C be an algebra of subsets of a given set I,andQ a space of functions of
bounded variation on C. In the book , R. J. Aumann (a Nobel laureate in
Economics, 2005) and L. S. Shapley deﬁne a value on Q as a positive linear operator
φ from Q to the additive set-functions on C, which is symmetric in a natural sense
and has the property that (φ(v)) (I)=v(I) for every v ∈ Q.
In [2, Theorem A] it is proved that, if BV denotes the space of all games (i.e.
set-functions) of bounded variation on C endowed with the variation norm, there
exists a continuous value on the subspace bv
NA of BV spanned by all functions
of the form f ◦ µ, where µ : C → [0, 1] is a non-atomic σ-additive measure and
f :[0, 1] → R is of bounded variation and continuous at 0 and at 1.
In the book , studying games with fuzzy coalitions, D. Butnariu and
E. P. Klement extend some of the results of  to the case where C is a T
of fuzzy sets. Therefore, it has been natural for us to investigate whether further
generalizations are possible, by assuming C to be an even more comprehensive
structure than that considered in .
Here and in the previous paper , a generalization of that kind is
accomplished in case the original algebra of subsets C is replaced by a σ-complete