This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

The Allais Paradox involves the choice between two alternatives:

A. 89% chance of an unknown amount 10% chance of $1 million 1% chance of $1 million B. 89% chance of an unknown amount (the same amount as in A) 10% chance of $2.5 million 1% chance of nothing

What is the rational choice? Does this choice remain the same if the

unknown amount is $1 million? If it is nothing?

decision/allais.s

This is "Allais' Paradox".

Which choice is rational depends upon the subjective value of money.

Many people are risk averse, and prefer the better chance of $1

million of 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 is not much

difference between 10% and 11%, but there is a big difference between

$1 million and $2.5 million.

Thus the choice between A and B depends upon the unknown amount, even

though it is the same unknown amount independent of the choice. This

violates the "independence axiom" that rational choice between two

alternatives should depend only upon how those two alternatives

differ.

However, 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. 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:

A. a fortune, guaranteed B. a fortune, almost guaranteed a tiny chance of nothing

Then the choice of A is rational. However, if the unknown amount is

nothing, the options are:

A. small chance of a fortune ($1 million) large chance of nothing B. small chance of a larger fortune ($2.5 million) large chance of nothing

In this case, the choice of B is rational. The Allais Paradox then

results from the limited ability to rationally calculate with such

unusual quantities. The brain is not a calculator and rational

calculations may rely on things like training, experience, and

analogy, none of which would be help in this case. This hypothesis

could be tested by studying the correlation between paradoxical

behavior and "unusualness" of the amounts involved.

If this explanation is correct, then the Paradox amounts to little

more than the observation that the brain is an imperfect rational

engine.

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