In this paper, we consider a non-cooperative n-person game in the strategic form. As is well known, the game has a mixed-strategy Nash equilibrium. While, it does not always have a pure-strategy Nash equilibrium. Wherein, Iimura (2003), and Sato and Kawasaki (to appear in Taiwanese J Math) provided a sufficient condition for the game to have a pure-strategy Nash equilibrium. This paper has two aims. The first is to extend the authors’ sufficient condition. The second is to give a necessary condition for the existence of a pure-strategy Nash equilibrium in the case of two persons. In both sections, monotonicity of the best response correspondences plays the central role. Kyushu University 21st Century COE Program Development of Dynamic Mathematics with High Functionality 九州大学21世紀COEプログラム「機能数理学の構築と展開」