A Note on Nash Equilibrium and Fixed Point Theorems

We give a proof of the existence of a Nash equilibrium forn-personnormal form games when each player's utility function is continuousw.r.t. strategy profiles, and concave and differentiable w.r.t. his ownstrategy. The proof uses only elementary mathematical tools such asmathematical induction. We show that this equilibrium existence re-sult is sufficiently general to imply the Brouwer Fixed Point Theorem.The Kakutani Fixed Point Theorem is obtained as a corollary by usingstandard techniques.