On the symmetry axiom for values of nonatomic games
A game on the unit interval I of players, is a set function v that assigns to a (measurable) coalition \(B\subset I\) a number v(B). A value on a linear space V of games is a mapping that assigns to each \(v\in V\) a measure \(\phi\) (v) on I, and such that \(\phi\) is linear on V, symmetric (i.e. \(\phi (v\circ \pi)=\phi \pi (v)\) with \(\pi\) a permutation on I, and \(v\circ \pi\) and \(\phi_{\pi}\) are defined in a natural way) and \(\phi (I)=v(I)\). Existence of a value does not always hold, and for instance, Aumann and Shapley showed that there is no value on the space of games v with bounded variation. The present paper examines the existence of a relaxed version of a value, called H-value and it is such that the symmetry property is relaxed: Instead of requiring it to hold for all permutations \(\pi\), it is required to hold for a subgroup H of permutations. Existence of an H-value depends now on the properties of the subgroups H. The paper proves that if every finitely generated subgroup of H is valuable, then so is H, and that every almost compact group is valuable. Consequences are derived for subspaces of the space of games with bounded variation.
scientific article; zbMATH DE number 4162696