1 Expected Utility Theory Consider the following experiment that involves asking a person two questions. Question 1: Choose between two lotteries:

A. Win $6000 with 80% chance and win nothing with 20% chance;

B.Win $3000 with certainty.

Question 2: Choose between two lotteries:

C. Win $6000 with 20% chance and win nothing with 80% chance;

D. Win $3000 with 25% chance and win nothing with 75% chance.

Suppose 100 people were asked these two questions. For Question 1, 20 people chose A and 80 chose B; for Question 2, 65 chose C and 35 chose D. So at least 80% 65% = 52% people preferred B over A and at the same time preferred C over D. Show that this preference relation cannot be represented by any utility function that has the expected utility property.

2 Subjective Utility Theory Consider same the experiment in Problem 1. Suppose Tom has a pessimistic view about uncertain outcomes in such a way that he would always mix every gamble with a gamble that only involves the worst-case outcome of the original gamble, i.e. he would construct a new compound gamble out of the original gamble g and the worst-case outcome gamble g(worst) with certainty as (p *g(worst),(1 p) g). Also assume Tom's utility for sure outcome x is u(x) = x^2 and his utility over gambles satisfies the expected utility property over his subjective probability distributions. Can you find such a pessimistic subjective view (i.e. p) so that Tom will prefer B to A and C to D?

3 Robust Utility Theory (20 pts) Again, consider the experiment in Problem 1. Now suppose the chances of winning $6000 in lotteries A and C are unknown to Tom. Other chances remain the same as in Problem 1 and Tom follows these known objective probabilities when making his decision. For the unknown probabilities, Tom believes that the chance of winning $6000 in A is quite high between 70% and 90%, whereas for lottery C, Tom has a bad feeling that the chance of winning $6000 is quite low between 10% and 25%. Tom still needs to make a choice between A and B, and between C and D. He can only rely on the robust expected utility theory by looking at the worst-case expected utility for each choice. In this case, what will Tom choose for Question 1 and Question 2? His utility for sure outcome is still u(x) = x^2.