Allais Paradox, explained.
The Allais paradox compares lottery choices that can reveal a tension with the independence axiom of expected utility theory, often involving a special preference for certainty.
Why it happens
The same common outcome is removed or changed across two choice problems. Under the independence axiom, that shared component should not reverse the ranking of the remaining alternatives. Some preference patterns reverse it anyway.
Maurice Allais used linked lottery choices to challenge the independence axiom of expected utility. Many people choose certainty in one pair and a larger long-shot in the other.
Read the result
Make both choices before evaluating the paired result. Look at how common probability components change, not just at the largest prize or the expected monetary payoff in isolation.
A worked example
Comparing two pairs of lotteries
In the first pair, one option has a guaranteed payoff while another introduces a small chance of receiving nothing.
In the second pair, a shared outcome is changed so that both options are uncertain.
A preference reversal can indicate that certainty received a special weight inconsistent with the stated independence comparison.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Pair 1: A pays the base prize for sure; B pays 5×base with 10%, base with 89%, zero with 1%. Pair 2: C pays base with 11% and zero otherwise; D pays 5×base with 10% and zero otherwise. The game draws seeded uniform outcomes and shows both expected payoffs. A with D is the classic independence-axiom conflict.
Where this idea is useful
A practical use
A guaranteed benefit and a high-upside gamble may be evaluated differently when a common outcome is removed from both choices.
A common misconception
“Choosing something other than the highest expected payout proves irrationality.”
Expected utility allows attitudes toward risk. The paradox concerns a specific consistency axiom across choices, not merely maximizing average money.
What this explanation leaves out
- One answer cannot diagnose a person's rationality. Stakes here are fictional, and risk attitudes are not estimated from a single trial.
Does this experiment tell me which lottery to choose?
It helps you inspect the consistency and assumptions behind your preferences. It does not establish one correct preference for every person or reproduce an entire theory of decision-making.
Would your preference survive replacing a shared outcome in both options?
Associated thinkers
Further reading
Explore the original research or the teaching reference behind this experiment.