Suppose that Tom is confronted with the choice between two options: O1, which is simply to receive $1,000,000, or O2, which is to receive $5,000,000 with probability 0.1, receive $1,000,000 with probability 0.89, and receive $0 with the remaining probability 0.01. After some deliberation, Tom decides that O1 is better, mostly because the outcome $0, unlikely as it may be, is very unattractive.

Tom is also confronted with a choice between two other options, O3 and O4. In O3, he would receive $5,000,000 with probability 0.1 and $0 with probability 0.9. In O4, he would receive $1,000,000 with probability 0.11 and $0 with probability 0.89. Here Tom prefers O3: the “unattractive” option $0 has about the same probability in both O3 and O4, while the positive outcome, although slightly less probable under O3, is much more desirable in O3 that in O4. Show that these preferences of Tom are not compatible with the assumption that he has utilities A, B, and C of $5,000,000, $1,000,000, and $0, such that A>B>C (This is known as Allais’ paradox; Allais, 1953).