Editing Supermodular function (section)

== Definition ==
Let <math>(X, \preceq)</math> be a [[Lattice (order)|lattice]]. A real-valued function <math>f: X \rightarrow \mathbb{R}</math> is called '''supermodular''' if 
<math>f(x \vee y) + f(x \wedge y) \geq f(x) + f(y)</math>

for all <math>x, y \in X</math>.<ref>{{Cite book |title=Supermodularity and complementarity |date=1998 |publisher=Princeton University Press |isbn=978-0-691-03244-3 |editor-last=Topkis |editor-first=Donald M. |series=Frontiers of economic research |location=Princeton, N.J}}</ref> 

If the inequality is strict, then <math>f</math> is '''strictly supermodular''' on <math>X</math>. If <math>-f</math> is (strictly) supermodular then ''f'' is called ('''strictly) submodular'''. A function that is both submodular and supermodular is called '''modular'''. This corresponds to the inequality being changed to an equality.

We can also define supermodular functions where the underlying lattice is the vector space <math>\mathbb{R}^n</math>. Then the function  <math>f : \mathbb{R}^n \to \mathbb{R}</math> is '''supermodular''' if  

<math>
f(x \uparrow y) + f(x \downarrow y) \geq f(x) + f(y)
</math>

for all <math>x</math>, <math>y \isin \mathbb{R}^{n}</math>, where <math>x \uparrow y</math> denotes the componentwise maximum and <math>x \downarrow y</math> the componentwise minimum of <math>x</math> and <math>y</math>.

If ''f'' is twice continuously differentiable, then supermodularity is equivalent to the condition<ref>The equivalence between the definition of supermodularity and its calculus formulation is sometimes called [[Topkis's theorem|Topkis' characterization theorem]]. See {{cite journal |first1=Paul |last1=Milgrom |first2=John |last2=Roberts |year=1990 |title=Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities |journal=[[Econometrica]] |volume=58 |issue=6 |pages=1255–1277 [p. 1261] |jstor=2938316 |doi=10.2307/2938316 }}</ref>

:<math> \frac{\partial ^2 f}{\partial z_i\, \partial z_j} \geq 0 \mbox{ for all } i \neq j.</math>