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 6:

The game known as the battle of the Bismarck Sea is a model of an actual naval engagement between the US and Japan in World War II. In 1943, a Japanese admiral was ordered to move a convoy of ships to New Guinea; he had to choose between a rainy northern route and a sunnier southern route, both of which required 3 days sailing time. The Americans knew that the convoy would sail and wanted to send bombers after it, but they didn't know which route it would take. The Americans had to send reconnaissance planes to scout for the convoy, but they had only enough reconnaissance planes to explore one route at a time. Both the Japanese and the Americans had to make their decisions with no knowledge of the plans being made by the other side.

If the convoy was on route explored by the Americans first, they could send bombers right away; if not, they lost a day of bombing. Poor weather on the northern route would also hamper bombing. If the Americans explored the northern route and found the Japanese right away, they could expect only 2 (out of 3) bombing days; if they explored the northern route and found that the Japanese had gone south, they could also expect 2 days of bombing. If the Americans chose to explore the southern route first, they could expect 3 full days of bombing if they found the Japanese right away but only one day of bombing if they found that the Japanese had gone north. For payoffs use the days of bombing, positive number for Americans and negative one for Japanese.

(i) Construct a game with (ordinal) payoffs that corresponds to this situation.

(ii) Can you solve the game through iterated dominance? Why, why not?

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) Is this preference relation rational in the sense defined by the expected utility theory?

Prove your conclusions.

PS. For part (ii) assume that if you have two strategies defined by the vectors of probabilities (p₁,p₂, 1-p₁-p₂) and (p₁*, p₂*, 1-p₁*-p₂*) which give you the probabilities of (lose, win, draw) respectively then a lottery that gives you the first strategy with probability q and the second with probability 1-q is equivalent to the following strategy (q p1+ (1-q) p₁*, q p₂+ (1-q) p₂*, q (1-p₁-p₂) + (1-q) 1-p₁*-p₂*)).