Nir Dagan / Teaching

Exercises in Bargaining

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.

1.1 Why would the rich man also be indifferent between having the trigger pulled when the revolver contains four bullets and paying $10,000,000 to have one of the bullets removed before the trigger is pulled? (Assume that his preferences satisfy the expected utility axioms.
1.2 Why would the rich man not be willing to pay as much as $10,000,000 to have one bullet removed from a revolver containing only one bullet?

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.

2.1. Calculate and draw the set of possible pairs of utilities that the players can get assuming that they may also divide amounts smaller than 1 dollar. That is, x1+x2≤1.
2.2. Assume now that the agents can also conduct lotteries over the money they divide. Does their set of possible utilities change?
2.3. Assume now that instead of player 1, another player (player 3) with the utility function u3(x3)=x32 is playing with player 2. Answer 1.1. and 1.2. for the new problem.
2.4. What is the maximum utility player 3 can get whenever player 2 receives the utility of a half. What is the agreement that gives these utilities?

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.

6.1 Are all the players risk averse?
6.2 Can you tell which player is more risk averse than which? Prove your answer.
6.3 If players 1 and 2 would bargain over a dollar, who would receive a larger share? Base your answer on the Nash bargaining solution.
6.4 Is the set of possible utility pairs derived from dividing a dollar (without lotteries) between players 1 and 4 convex?