Daniel evaluates risky alternatives based on prospect theory. For positive values of X (up to $10,000), his valuation function is V(X) = 20,000X - X2. For negative values of X (closer to zero than - $10,000), it's V(X) = 30,000X - 1.5X2. Daniel's probability weighting function is Z(P) = 0.1 - 0.8P. Do these functions fit the assumptions of prospect theory? Determine whether Daniel would take the following two gambles (where in each case the alternative is zero for sure):
(a) Win $21 with 50 percent probability; lose $25 with 50 percent probability;
(b) Win $20.98 with 10 percent probability, win $20.99 with 10 percent probability, win $21.00 with 10 percent probability, win $21.01 with 10 percent probability, win $21.02 with 10 percent probability, and lose $25 with 50 percent probability. Also calculate the expected gain or loss (in dollars) for each of these gambles. Is it reasonable for Daniel to treat them differently? Why or why not? What features of prospect theory causes him to treat them differently?