1. Consider the set of lotteries on the set of monetary prizes Z = (0, 1, 5}, this set 2(Z) can be identified with the three-dimensional unit simplex, i.e., 2(Z) = A(Z) = ((p1. p2. p3) E R310 S pi S l,i = 1,2,3, and p1 + p2 + p3 = 1}. (a) Consider the following three lotteries on A(Z): (1/3, 1/3, 1/3), (1/4, 1/4, 1/2), (1/5,3/5, 1/5). Does any lottery first-order stochastically dominate another? (b) For any p E 4(Z), let Drosd(p) $ 4(Z) be the set of lotteries that first-order stochastically dominate p. Show that Drosd(p) is convex and closed. (c) The Marshack-Machina Triangle, AM, is a useful graphical apparatus to analyze graphically the set of lotteries over a three-element prize set. It is the projection of the simplex 4(Z) to the two-dimensional Euclidean space and is defined as follows: AM = ((PI. P3) ER |(P1. P2. P3) E A(Z); = ((P1. P3) E R210