Nash equilibria of quasisupermodular games

We prove three results on the existence and the structure of Nash equilibria for quasisupermodular games. They all require certain continuity as assumption. The first result is purely order-theoretic, where we assume a continuity introduced by Shannon. The second continuity hypothesis is a mixture in the following sense. We require another order-theoretic continuity defined by Milgrom and Roberts, which we show to be only a half of Shannon's condition. As compensation, we also require a weak topological continuity proposed by Tian and Zhou. The last continuity supposition is purely topological, which is a slightly stronger continuity due to Tian and Zhou. The last theorem simultaneously generalizes Zhou's theorem on supermodular games and Calciano's theorem on quasisupermodular games.