2.7.3 Suppose that Tom is confronted with the choice between two options: 01, which is simply to receive $1,000,000, or 0 2 , 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 01 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 04. In 0 3 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 0.1 1 and $0 with probability0.89.

Here Tom prefers 0 3 : the “unattractive” option $0 has about the same probability in both O3 and 04, while the positive outcome, although slightly less probable under 0 3 , is much more desirable in O3 that in 04.

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).