Sum of superadditive functions

[jetoso]

Homework Statement

Show that the sum of two superadditive (supermodular) functions is superadditive.

Homework Equations

Let X and Y be partially ordered sets and g(x,y) a real-valued function on XxY. g is supermodular (superadditive) if for x1>=x2 in X and y1>=y2 in Y,
g(x1,y1) + g(x2,y2) >= g(x1,y2) + g(x2,y1)

The Attempt at a Solution

Let g(x,y) and h(x,y) be supermodular functions on XxY. Then the following inequalities hold:
g(x1,y1) + g(x2,y2) >= g(x1,y2) + g(x2,y1)
h(x1,y1) + h(x2,y2) >= h(x1,y2) + h(x2,y1)

Let f(x,y) = g(x,y) + h(x,y), then
[g(x1,y1) + h(x1,y1)] + [g(x2,y2) + h(x2,y2)] >= [g(x1,y2) + h(x1,y2)] + [g(x2,y1) + h(x2,y1)]
implies:
f(x1,y1) + f(x2,y2) >= f(x1,y2) + f(x2,y1)

Thus, the sum of two supermodular functions is supermodular.

 

[NateTG]

It's better to write it out with the f's:
$$f(x_1,y_1)+f(x_2,y_2)=\left(g(x_1,y_1)+h(x_1,y_1)\right)+\left(g(x_2,y_2)+h(x_2,y_2)\right) ... \geq f(x_1,y_2)+f(x_2,y_1)$$

 

[jetoso]

Thank you

Thank you so much.