# Allais paradox

The Allais paradox is a paradox that stems from
questions asked in 1951 by the French economist Maurice Allais (1911–2010).^{1} Which of these would you choose: (A) an 89% chance of receiving an unknown
amount and 11% chance of $1 million; or (B) an 89% chance of an unknown
amount (the same amount as in A), a 10% chance of $2.5 million, and a 1%
chance of nothing? Would your choice be the same if the unknown amount were
$1 million, or if it were nothing?

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.

### Reference

1. Allais, M. "Le comportement de l'homme rationnel devant le risque:
Critique des postulats et axiomes de l'école américaine." *Econometrica*,
21: 503-546 (1953).