  In this paper we construct $n$-valued maps on $k$-dimensional tori, where
$n,k\geq 2$, that are not homotopic to affine $n$-valued maps. This is in high
contrast with the single valued case, where any such map is homotopic to an
affine (even linear) map. We do this by investigating necessary and sufficient
algebraic conditions on certain induced morphisms.
