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 these problems 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.
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:
You value your time at $50 an hour. If you choose to drive to your destination it will cost you $150 and the drive will take you 6 hours with probability 0.9 and 7 hours with probability 0.1. In short, you face a lottery in which you will spend $450 ($150 + cost of 6 hours) with probability 0.9 or $500 ($150 + cost of 7 hours) with probability 0.1. If you fly, it will cost you $300 and your trip will last 2 hours with probability 0.7 and 4 hours with probability 0.3. Here you face a lottery in which you will spend $400 with probability 0.7 and $500 with probability 0.3. Assume that your total assets are $500 and that you have the following utility for the assets that remain after your spending: u($x) = √x (the so called Bernoulli utility.) Please note that the utility function is defined on assets and not expenditures: √x only makes sense when x ≥ 0 while an expenditure of, say, $100, formally is -100.
What should you do? Drive or fly?
Problem 5:
Tom tells you that he is indifferent between $5 and $15 dollars and he prefers $10 over $5 and also $10 over $15. You look at his preferences and make the following argument: If Tom were rational in the sense of the expected utility theory then we know that his utility function must be such that u($5) = u($15) < u($10). Also, his utility of a lottery in which he gets $5 with probability 0.5 and $15 with probability 0.5 is 0.5u($5)+0.5u($15)=u($5)=u($15). But this lottery is equivalent to $10, in terms of its expected value, and thus its utility has to be equal to u($10). But u($10) is larger than u($5), hence we have a contradiction. Thus Tom's preference is not rational in the sense of the expected utility theory. Right? Or, wrong? Explain.
Problem 6:
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? (Specify all values of p, not just selected ones.)
Problem 7:
John prefers a lottery in which he gets 2.5m with probability 0.2, 1m with probability 0.5 and 0 with probability 0.3 over a lottery in which he gets 2.5m with probability 0.15, 1m with probability 0.6 and 0 with probability 0.25. Show that if John is rational then this preference is equivalent to him preferring a lottery in which he gets 2.5m with probability 0.5 and 0 with probability 0.5 to a sure payment of 1m.
Problem 8:
This problem goes back to the topic of the preference theory. Consider a decision maker with a preference relation on a set of alternatives D = {x1,x2, ..., xn,...}. Assume that this decision maker will be willing to exchange xi+1 for xi if and only if xi ¶ xi+1.
Moreover if xi ~ xi+1 he will exchange without giving up or gaining any utility. If, however, xi xi+1 then he will pay for the exchange amount Δ that will reduce his total utility from T to T-Δ. A money pump can be defined as a sequence of exchanges E1, E2, .... in which the decision maker loses all of his utility T. Define a cycle to be any finite set D = {x1, ..., xn} of alternatives in D such that xi ¶ xi+1 for all i=1,...,n and xn ¶ x1 and at least one of these relations is a strict preference, i.e., there exists i, 1 ≤ i ≤ n such that xi < xi+1 or xn < x1. For a finite D (the case of the preference theory) prove that
(i) A money pump exists if and only if there is a cycle.
(ii) A cycle exists if and only if is not rational.
Problem 9:
(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*)).