The definition of submodularity led me to define a type of function on numbers (I suppose ordered rings, but consider the reals for now) as $f(x) + f(y) \ge f(x+y) + f(xy)$. Such functions exist: for example, the constant functions.
I can infer some basic properties of these functions: plugging in $x = 1$ and $y = c-1$ yields $f(1) \ge f(c)$ so 1 is a global maximum; similarly, $x = 2$ and $y = 2$ yields $f(2) \ge f(4)$. Another property of interest is that if $f$ is differentiable at $x$ and 0, $y = \epsilon$ gives $f(x+\epsilon) - f(x) \le f(\epsilon) - f(x\epsilon)$ and letting $\epsilon \rightarrow 0$ yields $f'(x) \le (1-x)f'(0)$. (edit: thanks kingW3)
Can anyone characterize such functions more completely? Are there any such functions beside the constant ones? I am having difficulty coming up with one.