Summary: We consider a non-cooperative \(n\)-person game in the strategic form. As is well known, the game has a mixed-strategy Nash equilibrium. However, it does not always have a pure-strategy Nash equilibrium. Wherein, D. M. Topkis [SIAM J. Control Optimizat. 17, 773–787 (1979; Zbl 0433.90091)], T. Iimura [J. Math. Econ. 39, No. 7, 725–742 (2003; Zbl 1055.91061)], and J.-I. Sato and H. Kawasaki [Taiwanese J. Math. 13, No. 2A, 431–440 (2009; Zbl 1196.47041)] provided a sufficient condition for the game to have a pure-strategy Nash equilibrium. However, they did not consider necessary conditions. This paper has two aims. The first is to extend the authors’ sufficient condition, which is based on monotonicity of the best responses. The second is to show that the existence of a pure-strategy Nash equilibrium implies the monotonicity of the best responses or an isolation of the equilibrium.