A fundamental result on Nash equilibria of games with strategic
complementarities is that for a quasisupermodular game, all Nash
equilibria form a nonempty complete lattice. In particular, there exist
largest and least Nash equilibria. Although not a quasisupermodular
game, a pure Bertrand game also exhibits strategic complementarities.
We show that a generalized Bertrand game has a least Nash equilibrium.
We also consider an extension of quasisupermodular games and prove
that the set of Nash equilibria is a nonempty complete lattice. Loosely,
our observation is that a half of the single crossing property suffices
for such structure results on Nash equilibria.