One of the best known challenges to the Independence Axiom was set forth by Maurice Allais in the " Allais Paradox". Let the possible outcomes be x 1 = $0, x 2 = $1 million and x 3 = $12 million. Consider the following pair of lotteries:

f 1: $1 million with certainty
f 2: $0 with 1% chance, $1 million with 89% chance, $12 million with 10% chance,

Faced with these possibilities, most people will select the first lottery. Therefore, f 1 is preferred to f 2, or f 1>f 2. Now consider the next pair of lotteries:

f 3: $0 with 89% chance and $1 million with 11% chance
f 4: $0 with 90% chance and $12 million with 10% chance

Faced with these possibilities, most people will prefer f 4 to f 3, or f 4>f 3.

However, notice that:

(.5)f 1 + (.5)f 4 = (.5)($1 million) + (.45)($0) + (.05)($12 million)
while
(.5)f 2 + (.5)f 3 = [(.5)(.01) + (.5)(.89)]($0) + [(.5)(.89) + (.5)(.11)]($1 million) +
(.5)(.1)($12 million)
= (.5)($1 million) + (.45)($0) + (.05)($12 million)

Therefore,

(.5)f 1 + (.5)f 4 = (.5)f 2 + (.5)f 3

But this contradicts our two inequalities: f 1>f 2 and f 4>f 3.
Apparently, even Leonard Savage, when confronted by Allais's example, made this contradictory choice. Thus, they must be violating the independence axiom of expected utility. This is the "Allais Paradox".

It has been hypothesized that these contradictory choices imply what is called a "fanning out" of indifference curves. Specifically, assume the indifference curves are linear but not parallel so they “fan out”. If we can somehow allow the fanning of indifference curves, then Allais's Paradox would no longer be that paradoxical.

However, what guarantees this "fanning out"? One suggestion was that the expected utility decomposition was incorrect. The utility of a particular lottery p is not U(p) = E(u; p) = å x Î X p(x)u(x). Rather, U(p) = ¦ [E(u; p), var(u; p)], so that the utility of a lottery is not only a function of the expected utility E(u; p) but also incorporates the variance of the elementary utilities var(u; p). (One can incorporate additional moments as well).

Allais's "fanning out" hypothesis would also yield what Kahneman and Tversky have called the "common consequence" effect. The common consequence effect can be understood by appealing to the independence axiom which, recall, claims that if p > h q, then for any b Î [0, 1] and r Î D (X), then b p + (1- b )r > h b q + (1- b )r. In short, the possibility of a new lottery r should not affect preferences between the old lotteries p and q. However, the common consequence effect argues that the inclusion of r will affect one's preferences between p and q. Intuitively, p and q now become "consolation prizes" if r does not happen. The short way of describing the common consequence effect, then, is that if the prize in r is great, then the agent becomes "more" risk-averse and thus modifies his preferences between p and q so that he takes less risky choices. The idea is that if r offers indeed a great prize, then if one does not get it, then one will be very disappointed ("cursing one's bad luck") - and the greater the prize r offered, the greater the disappointment in the case one does not get it. Intuitively, the common consequence effect argues that getting $50 as a consolation prize in a multi-million dollar lottery one has lost is probably less exhilarating than finding $50 on the street. Consequently, in order to compensate for the potential disappointment, an agent will be less willing to take on risks as an alternative - as that would only worsen the burden. In contrast, if r is not that good, then one might be more willing to take on risks.