Due November 2, 1999.
1. A rich man is forced to play a version of Russian Roulette: The cylinder of a six-shooter revolver containing two bullets is spun and the barrel is then pointed at his head. He is now offered the opportunity of paying money to have the two bullets removed before the trigger is fired. It turns out that the payment can be made as high as $10,000,000 before he becomes indifferent between paying and taking the risk of getting shot.
2. There are two players who may divide 1 dollar between them. The (von Neumann-Morgenstern) utility function of player 1 is u1(x1)=x10.5 and of player 2 is u2(x2)=x2.
3. Consider all pairs of players of question 1 above. Assume that if they do not reach an agreement both get 0 dollars. Calculate, for the two different cases, the utilities the players will get according to the Nash solution. How much money each player gets?
4. There are to possible events R (rain) and S (sun). The lottery that gives the decision maker x(R) dollars if it will rain tomorrow and x(S) dollars if it will be sunny is donated by [x(R),x(S)]. Assume that x(R) and x(S) cannot be negative. The preferences of the decision maker on lotteries is presented by the function V[x(R),x(S)]=[x(R)]0.75[x(S)]0.25. Can his preferences be presented by subjective probabilities and a von Neumann-Morgenstern utility function? If so what are the probabilities and the function?
5. The dictatorial solution is defined as follows:
s*1 = argmax {s1 | s2≥d2, s∈S}
s*2 = argmax {s2 | s1=s*1, s∈S}
Show that the dictatorial solution w.r.t player 1 satisfies Pareto efficiency and IIA.
6. There are four players. The (von Neumann-Morgenstern) utility function of player 1 is u1(x1)=x1a, of player 2 is u2(x2)=x2b, of player 3 is u3(x3)=ln(x3+1), and of player 4 is u4(x4)=x42, where 1>a>b>0.