# Is the max of two supermodular functions supermodular?

The supermodularity means for every $x'>x$ and $y'>y$, $$f(x',y') + f(x,y) > f(x',y) + f(x,y').$$

Suppose $f$ and $g$ are supermodular, non-negative and increasing in both arguments. Is the function $$h(x,y) = \max\bigl( f(x,y) , g(x,y) \bigr)$$ supermodular?

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I think the answer is no. Let $f$ be a supermodular, non-negative and increasing (in both arguments) function satisfying $f(0,0)=1, f(0,1)=2, f(1,0)=3$, and $f(1,1)=4.5$. Let $g$ be defined as $g(x,y):=f(y,x)$. Then, $h(x,y)= \max (f(x,y), g(x,y))$ is not supermodular, since it is easy to check that
$h(1,1)+h(0,0)=5.5 < 6=h(0,1)+h(1,0).$
As tony mentioned generally no.but if we assume that $x'=x+1$,$y=y+1$ , $f(x,y)<=g(x,y)$ and $g(x,y)=f(y,x)$ and also $f(a+1,b+1)=f(a+1,b)+f(a,b+1)$ a,b are xory then $h(x,y)=max(f(x,y),g(x,y))$ is also supermodular – Hashem sazegar Jul 21 '10 at 17:40