E. [60 points, equally divided between the parts] One of the difficulties with using supermodularity is that taking monotonic transformations can change a supermodular function into a submodular function. The following weakens supermodularity in a fashion that is immune to monotonic transformations for X R and R, a function (x, ) 7 f(x, ) is quasi-supermodular if for all x 0 > x and all 0 > , [f(x 0 , ) f(x, ) > 0] implies [f(x 0 , 0 ) f(x, 0 ) > 0] and [f(x 0 , ) f(x, ) 0] implies [f(x 0 , 0 ) f(x, 0 ) 0]. Let X = [10, 15] and = [10, 15], and consider the problems maxxX f(x, ) = max10x15 f(x, ) where f(x, ) = x and let x () be the solutions as a function of , 10 15.

E.1. Show that f(, ) is strictly supermodular and that the solutions, x (), are weakly increasing.