Allais' paradox. Suppose you are given the choice between A and B:
A: 100% chance to win £1 billion
B: 89% chance to win £1 billion, 10% chance to win £5 billion, 1% chance to win £0
Record your choice: A or B.
Now suppose you are given a separate choice

The mathematical proof of the inconsistency in the choice of A vs. B and C vs. D can be shown as follows. Let X1 denote the expected utility of choice A and X2 denote the expected utility of choice B. Similarly, let Y1 denote the expected utility of choice C and Y2 denote the expected utility of choice D. The expected utility of each choice can be calculated as follows: X1 = 100% x £1 billion X2 = 89% x £1 billion + 10% x £5 billion + 1% x £0 = £8.9 billion Y1 = 11% x £1 billion + 89% x £0 = £110 million Y2 = 10% x £5 billion + 90% x £0 = £0 There are two ways of making a rational choice: the maximin rule and the maximax rule. The maximin rule states that one should choose the option with the highest minimum expected utility (i.e. the option with the worst case outcome). In this case it is choice A with an expected utility of £1 billion. The maximax rule states that one should choose the option with the highest maximum expected utility (i.e. the option with the best case outcome). In this case it is choice B with an expected utility of £8.9 billion. Therefore, if one chooses A in