This article is from the Puzzles FAQ,
by Chris Cole firstname.lastname@example.org and Matthew Daly
email@example.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?
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
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
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