Topkis Theorem states that: if $f$ is supermodular in $(x,\theta)$, and $D$ is a lattice, then $x^{∗} ( θ ) = \arg\max _{x ∈ D} f ( x , θ )$ is nondecreasing in $θ$.
It is not a simple concept for an undergraduate student from an economic department to understand. After struggling to understand what Theorem states, I am trying to find some examples of $f(x,\theta)$ that implies that $x^{∗} ( θ ) = \arg\max _{x ∈ D} f ( x , θ )$ is not nondecreasing in $θ$.
For example, I'm trying to find a supermodular $f (x,\theta)$ function, not being increase differences in $(x,\theta)$ with $D$ increasing in $\theta$, with $x^{∗} ( θ ) = \arg\max _{x ∈ D} f ( x , θ )$ being non-increasing in $\theta$.
Any help guys?