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Sum of superadditive functions

  1. Feb 16, 2007 #1
    1. The problem statement, all variables and given/known data
    Show that the sum of two superadditive (supermodular) functions is superadditive.


    2. Relevant 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)


    3. 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.
     
  2. jcsd
  3. Feb 16, 2007 #2

    NateTG

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    Science Advisor
    Homework Helper

    It's better to write it out with the f's:
    [tex]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)[/tex]
     
  4. Feb 16, 2007 #3
    Thank you

    Thank you so much.
     
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