  Given a multifunction from $X$ to the $k-$fold symmetric product $Sym_k(X)$,
we use the Dold-Thom Theorem to establish a homological selection Theorem. This
is used to establish existence of Nash equilibria. Cost functions in problems
concerning the existence of Nash Equilibria are traditionally multilinear in
the mixed strategies. The main aim of this paper is to relax the hypothesis of
multilinearity. We use basic intersection theory, Poincar\'e Duality in
addition to the Dold-Thom Theorem.
