Game theory ( should be complete within 6hours)

Problem Set 2: Due in class on Tuesday July 14. Solutions to this homework will be posted

right after class, hence no late submissions will be accepted. Group solutions are welcomed

and encouraged. Test 2 on the content of this homework will be given on July 21 at 9:00am

sharp. When turning in the homework put both your name and your class number.

In all problems below a rational preference relation is understood as one that satisfies the axioms of von

Neumann and Morgenstern’s utility theory. When solving problems involving the expected utility theory use

the von Neumann-Morgenstern theorem. In other words, you prove that a preference relation is rational by

showing utility values that satisfy corresponding conditions and you prove that a preference relation is not

rational by showing that no utility values can possibly satisfy these conditions. SOLUTIONS THAT

DON’T USE THIS METHOD WILL NOT BE ACCEPTED !!!

Problem 1 (3p)

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 (3p)

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 (3p)

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 (3p)

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 (Dixit and Skeath p.117) (3p)

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?

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Problem 6 (Dixit and Skeath p.117) (3p)

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?

EXTRA CREDIT PROBLEMS

Problem 7 (1 extra credit point)

Using only 0 and 1 as payoffs construct a 44 game which can be solved through iterated dominance in the maximal possible number of steps.

Problem 8 (2 extra credit points)

(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?

Prove your conclusions.

PS. For part (ii) assume that if you have two strategies defined by the vectors of probabilities (p 1 ,

p 2 , 1-p

1 -p

2 ) and (p

1 *, p

2 *, 1-p

1 *-p

2 *) 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 p 1 + (1-q) p

1 *, q p

2 + (1-q) p

2 *, q (1-p

1 -

p 2 ) + (1-q) 1-p

1 *-p

2 *)).