In Gamble 1 you have a 99% chance of winning a trip to Venice and a 1% chance of winning tickets to a movie about Venice; and in Gamble 2, you have a 99% of winning the same trip to Venice and a 1% chance of not winning anything
A. Suppose you very much like Venice, and, were you to be asked to rank the three possible outcomes, you would rank the trip to Venice first, the tickets to the movie about Venice second, and having nothing third.
(a) Assume that you can create a consumption index such that getting nothing is denoted as 0 consumption, getting the tickets to the movie is x1 > 0 and getting the trip is x2 > x1. Denote the expected value of Gamble 1 by E (G i) and the expected value of Gamble 2 by E (G2). Which is higher?
(b) On a graph with x on the horizontal axis and utility on the vertical, illustrate a consumption / utility relationship that exhibits risk aversion.
(c) In your graph, illustrate the expected utility you receive from Gamble 1 and from Gamble 2. Which gamble will you choose to participate in?
(d) Next, suppose tastes are risk neutral instead. Re-draw your graph and illustrate again which gamble you would choose. (Hint: Be careful to accurately differentiate between the expected values of the two gambles.)
(e) It turns out (for reasons that become clearer in part B) that, risk aversion (or neutrality) is irrelevant for how individuals whose behavior is explained by expected utility theory will choose among these gambles. In a separate graph, illustrate the consumption/utility relationship again, but this time assume risk loving. Illustrate in the graph how your choice over the two gambles might still be the same as in parts (c) and (d). Can you think of why it in fact has to be the same?
(f) It turns out that many people, when faced with a choice of these two gambles, end up choosing Gamble2. Assuming that such people would indeed rank the three outcomes the way we have, is there any way that such a choice can be explained using expected utility theory (taking as given that the choice implied by expected utility theory does not depend on risk aversion?)
(g) This example is known as Machina's Paradox. One explanation for it (i.e. for the fact that many people choose Gamble 2 over Gamble 1) is that expected utility theory does not take into account regret. Can you think of how this might explain people’s paradoxical choice of Gamble 2 over Gamble 1?
B. Assume again, as in part A, that individuals prefer a trip to Venice to the movie ticket, and they prefer the movie ticket to getting nothing. Furthermore, suppose there exists a function u that assigns u2 as the utility of getting the trip, u1 as the utility of getting the movie ticket and u0 as the utility of getting nothing, and suppose that this function u allows us to represent tastes over risky pairs of outcomes using an expected utility function.
(a) What inequality defines the relationship between u1 and u0?
(b) Now multiply both sides of your inequality from (a) by 0.01, and then add 0.99u2 to both sides. What inequality do you now have?
(c) Relate the inequality you derived in (b) to the expected utility of the two gambles in this example. What gamble does expected utility theory predict a person will choose (assuming the outcomes are ranked as we have ranked them)?
(d) When we typically think of a “gamble”, we are thinking of different outcomes that will happen with different probabilities. But we can also think of “degenerate” gambles — i.e. gambles where one outcome happens with certainty. Define the following three such “gambles”: Gamble A results in the trip to Venice with probability of 100%; Gamble B results in the movie ticket with probability of 100%; and Gamble C results in nothing with probability of 100%. How are these degenerate “gambles” ranked by someone who prefers the trip to the ticket to nothing?
(e) Using the notion of mixed gambles introduced in Appendix 1, define Gambles 1 and 2 as mixed gambles over the degenerate “gambles” we have just defined in (d). Explain how the IndependenceAxiomfromAppendix1 implies thatGamble1 must be preferred to Gamble2.
(f) True or False: When individuals who rank the outcomes the way we have assumed choose Gamble 2 over Gamble 1, expected utility theory fails because the independence axiom is violated.
(g) Would the paradox disappear if we assumed state-dependent tastes? (Hint: As with the Allais paradox in Appendix 2, the answer is no.)