Allais paradox

A paradox in decision theory. Suppose you are offered a choice between two gambles:

Gamble 1: $500,000 with probability of 1

Gamble 2: $2,500,000 with probability 0.1; $500,000 with probability 0.89; $0, with probability 0.01.

Many people will prefer the first option. Now suppose you are offered a choice between another two gambles:

Gamble 3: $500,000 with probability 0.11; $0 with probability 0.89

Gamble 4: $2,500,000 with probability 0.1; $0 with probability 0.9.

Many people will take the second. The problem is that this pair of preferences is not consistent with any utility function. For from the first choice we have it that

U($500,000) > 0.1U($2,500,000) + 0.89U($500,000) + 0.01U($0)

and from the second choice we have it that

0.1U($2,500,000) + 0.9U($0) > 0.11U($500,000) + 0.89U($0)

but these are inconsistent: by the first equation

0.11U($500,000) – 0.01U($0) > 0.1U($2,500,000)

but by the second equation the inequality is reversed. Although the paradox can be used to attack the sure thing principle, another approach is to use it to educate choices, so that if one genuinely prefers gamble 1 to gamble 2, one learns to reverse the initial feeling that gamble 4 is a better choice than gamble 3.