Problem 1 Suppose you have asked your friend Peter if he prefers a sure payment of $20 or a lottery in which he gets $15 with probability 0.5 and $10 with probability 0.5. Is it rational for Peter to prefer the sure payment over the lottery? Is it rational to prefer the lottery over the sure payment? Is it rational to be indifferent between the lottery and the sure payment? Would your answer be any different had I asked you the same question but with A substituted for $20, B for $15 and C for $10? What is the general lesson to learn from this exercise?
Problem 2 George tells you that he prefers more money over less. George also tells you about his preference between a lottery in which he gets $30 with probability 0.9 and 0 with probability 0.1 and a sure payment of $20. Assume that George is rational. Is it possible for him to prefer the lottery over the sure payment? Is it possible to prefer the sure payment over the lottery? Is it possible for him to be indifferent between the sure payment and the lottery? What is the general lesson to learn from this exercise?
Problem 3
Paul told you that he is indifferent between a lottery in which he gets A with probability 0.8 and C with probability 0.2 and a lottery in which he gets A with probability 0.5 and B with probability 0.5. Paul told you also that he prefers a lottery in which he gets A with probability 0.3 and C with probability 0.7 over a lottery in which he gets B with probability 0.5 and C with probability 0.5. Is Paul's preference relation rational?
Problem 4
Tom prefers A over B and B over C. Also, Tom is indifferent between a lottery in which he gets C with probability p and A with probability 1-p and a lottery in which he gets B with probability p and C with probability 1-p. The value of p in both lotteries is the same. For what values of p would Tom's preferences be rational in the sense of von Neumann- Morgenstern's expected utility theory?
Problem 5:
An old lady is looking for help crossing the street. Only one person is needed to help her; more are okay but no better than one. You and I are the two people in the vicinity who can help; we have to choose simultaneously whether to do so. Each of us will gain (get pleasure) 3 "utiles" from her success, no matter who helps her. But each one who goes to help will bear a cost of 1 utile, this being the utility of our time taken up in helping. With no cost incurred and no pleasure derived our payoff is 0. Set this up as a normal form game.
Can you solve the game through iterated dominance?
Problem 7:
Using only 0 and 1 as payoffs construct a 4 x 4 game which can be solved through iterated dominance in the maximal possible number of steps.
Problem 8:
(after Kreps 1988) Assume that the President has the following preferences over any two strategies S and S* on how to conduct a war: When choosing between S and S* prefer S if and only if (1) it gives a lower probability of losing or (2) in case they both give the same probability of losing, when S gives a higher probability of winning. Suppose that we have three possible outcomes of a war: win, lose and draw. A strategy is understood as a probability distribution on the three possible outcomes.
(i) Is this preference relation rational in the sense defined by the preference theory?
(ii) (ii) Is this preference relation rational in the sense defined by the expected utility theory?