Allais Paradox

The Allais Paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.

In identical experiments, an Allais paradox occurs when the addition of an independent event influences choice behavior. Consider the choices in the following table (Kahneman and Tversky 1979).

lottery	1 to 33	34	35 to 100	preference
		0		18%
				82%
		0	0	83%
			0	17%

In Experiment 1, a choice of and was given, and most participants picked . In Experiment 2, a choice of and was given, and most participants picked .

This observed pattern violates the independence axiom, since in both experiments, the payoff is identical if a ball is picked, while if the event is disregarded, the two experiments are identical.

To see it another way, consider the event to be a black box that is always received if the random ball value is . Knowing or not knowing the contents of the black box should not influence behavior.

Mathematical proof of inconsistency

Using the values above and a utility function of u(W), where W is wealth, we can demonstrate exactly how the paradox manifests.

Because the typical individual prefers 1A to 1B and 2B to 2A, we can write conclude that the expected utilities of the preferred is greater than the expected utilities of the second choices, or,

Experiment 1

Experiment 2

We can rewrite the latter equation (Experiment 2) as

which contradicts the first bet (Experiment 1), which shows the player prefers the sure thing over the gamble

Most people don’t like risk and so prefer the better chance of winning $1 million in option A. This choice is firm when the unknown amount is $1 million, but seems to waver as the amount falls to nothing. In the latter case, the risk-averse person favors B because there isn’t much difference between 10% and 11%, but there’s a big difference between $1 million and $2.5 million. Thus the choice between A and B depends on the unknown amount, even though it is the same unknown amount independent of the choice. This flies in the face of the so-called independence axiom, that rational choice between two alternatives should depend only on how those two alternatives differ. Yet, if the amounts involved in the problem are reduced to tens of dollars instead of millions of dollars, people’s behavior tends to fall back in line with the axioms of rational choice. In this case, people tend to choose option B regardless of the unknown amount. Perhaps when presented with such huge numbers, people begin to calculate qualitatively. For example, if the unknown amount is $1 million the options are essentially (A) a fortune guaranteed or (B) a fortune almost guaranteed with a small chance of a bigger fortune and a tiny chance of nothing. Choice A is then rational. However, if the unknown amount is nothing, the options are (A) a small chance of a fortune ($1 million) and a large chance of nothing, and (B) a small chance of a larger fortune ($2.5 million) and a large chance of nothing. In this case, the choice of B is rational. Thus, the Allais paradox stems from our limited ability to calculate rationally with such unusual quantities.