  We give an axiomatic characterization of the fixed point index of an
$n$-valued map. For $n$-valued maps on a polyhedron, the fixed point index is
shown to be unique with respect to axioms of homotopy invariance, additivity,
and a splitting property. This uniqueness is used to obtain easy proofs of an
averaging formula and product formula for the index. In the setting of
$n$-valued maps on a manifold, we show that the axioms can be weakened.
