Lesson · Decision and Rationality · Risk

The Allais Paradox and the Sure-Thing Principle

Most people prefer a sure million to a gamble that might pay five, and also prefer a one-in-ten chance of five million to an eleven-in-a-hundred chance of one. Expected utility scores the gap in each pair by the same expression, so the two preferences together contradict it whatever numbers the utilities take.

Open the interactive lesson Practice

A: 1M for sure B: 89% 1M, 10% 5M, 1% nothing C: 11% 1M, otherwise nothing D: 10% 5M, otherwise nothing EU(A) − EU(B) = EU(C) − EU(D) A and D together contradict EU

One number, tested twice

Core ideas

What this lesson establishes

The paradox is short to state and easy to miss. It asks the same question twice, with the same answer hidden in a different place.

Two choices, one common part

In the first choice the two gambles share an 89% chance of a million. In the second they share an 89% chance of nothing. Take the shared part away and the same two remainders are left to compare.

The sure-thing principle

If two acts give the same outcome in some states, those states should not affect which act you prefer. Expected utility obeys this automatically, because a shared term appears on both sides and cancels.

The pattern A and D

Preferring the sure million and then preferring the one-in-ten shot at five million is the popular response. It lets the shared part change the verdict, and that is the violation.

The material

Four gambles and a term that cancels

First the choices, then the arithmetic that shows they cannot both be answered as most people answer them.

Maurice Allais put two choices to people in 1953. In the first, gamble A pays a million for certain, and gamble B pays five million with probability 10%, a million with probability 89% and nothing with probability 1%. In the second, gamble C pays a million with probability 11% and nothing otherwise, and gamble D pays five million with probability 10% and nothing otherwise. Most people choose A over B, and D over C.

Each choice looks sensible on its own. A gives up a little expected money for certainty, which is just what a concave utility was shown to allow in “Risk Aversion and the Value of Information”, and B risks ending with nothing at all, which is a poor price for a small improvement. In the second choice nobody is certain of anything, so the bigger prize for nearly the same chance is the natural pick.

Lay the gambles out as a table of tickets, numbered 1 to 100. Tickets 1 to 89 are the shared part, ticket 90 is the single ticket that decides the 1% and tickets 91 to 100 decide the 10%. Payoffs are in millions.

gamble 1 to 89 90 91 to 100 A 1 1 1 B 1 0 5 C 0 1 1 D 0 0 5

Read the first column. A and B agree on it, both paying 1, and C and D agree on it too, both paying 0. The only difference between the first choice and the second is what the shared tickets pay. The last two columns are the same in both choices: A and C pay 1 and 1, B and D pay 0 and 5.

Definition · The sure-thing principle

If two acts pay the same in some states, then which of them you prefer should depend only on the states where they differ. Changing what they both pay in the shared states should leave the preference unchanged. The name is Leonard Savage's, from 1954; the principle is older than the name.

Expected utility satisfies this by construction. Write the utilities of nothing, a million and five million as u(0), u(1M) and u(5M). The difference between A and B is computed below, and so is the difference between C and D. The 89% term is common to A and B and cancels; the corresponding term cancels between C and D.

EU(A) − EU(B) = 11/100·u(1M) − 10/100·u(5M) − 1/100·u(0) EU(C) − EU(D) = 11/100·u(1M) − 10/100·u(5M) − 1/100·u(0)

Theorem · The same expression

For any utilities whatever, EU(A) − EU(B) and EU(C) − EU(D) are equal. So preferring A to B makes the first positive, preferring D to C makes the second negative, and one number cannot be both.

The lab enters the utilities u(0) = 0, u(1M) = 10 and u(5M) = 14 and scores each pair. In the first it prints 103/10 for B and 10 for A; in the second 7/5 for D and 11/10 for C. Expected utility picks B and D here. Both gaps are 3/10 in favour of the gamble. Change the utilities and the gaps move together, always by the same amount.

That is the whole result: the rule can prefer A and C, or it can prefer B and D, but it cannot prefer A and D. Preferring A to B needs u(5M) below 11/10·u(1M) when u(0) is zero, and preferring D to C needs it above. The popular pair asks for both.

What is said for the popular pair is not foolish. Certainty is a state of mind as well as a payoff: a chooser who ends up with nothing after turning down a sure million may feel a regret that nothing in the second choice can produce. On this view the shared 89% of tickets is not the same in the two choices, because in the first it contributes certainty. A defender must then say what the outcome is: if regret is part of it, the payoff table should list it, and the choices are no longer about the same payoffs.

The reader has two ways out. One keeps the table as stated and gives up the sure-thing principle, which means giving up expected utility as the measure of these acts. The other keeps the principle and revises one of the two choices. The lab computes what each costs, not which to pay.

Interactive

A decision table under every rule

the same payoffs, eight ways of choosing, each computed exactly

Move the utilities and watch both gaps move together

The payoffs are utilities, with u(0) = 0, u(1M) = 10 and u(5M) = 14. Score the first pair, then the second, and compare each winning score with the other act's. Now change the 14 to 21/2 in both presets. Gamble A beats B in the first, and C beats D in the second: the preference flips in both places at once, which is the point. No value of the 14 produces A with D.

Worked example

Acts, separated by commas

States, separated by commas

Payoffs: one row per act, rows separated by “;”

State probabilities (empty for ignorance)

Probabilities given each act, “act: p1 p2; …”

Forbidden acts

Decision rule

Choice—

The rule’s score—

Value of perfect information—

Expected utilities tie at—

Method

Testing a pair of choices against expected utility

Four steps, and none needs a number for any utility.

Tabulate on common tickets

Give the gambles the same states, one column for what they share and one for each place they differ. Without common states the shared part is hidden.

Write each gap

Write EU of the first act minus EU of the second as a sum over the states, with the probabilities and utilities left as symbols.

Cancel the shared columns

In each pair, the states where the two acts pay the same give equal terms, which subtract to zero.

Compare what remains

If the two remainders are the same expression, the two choices must go the same way under expected utility, and a pattern that splits them contradicts it.

Same gap, opposite signs

Score 0 / 0

Choose the answer you can justify.

Both gaps, from one set of utilities

Take u(0) = 0, u(1M) = 10 and u(5M) = 14, with the tickets split 89, 1 and 10 in a hundred.

EU(A) = 10 EU(B) = 89/100·10 + 1/100·0 + 10/100·14 = 103/10 EU(A) - EU(B) = -3/10 EU(C) = 89/100·0 + 1/100·10 + 10/100·10 = 11/10 EU(D) = 89/100·0 + 1/100·0 + 10/100·14 = 7/5 EU(C) - EU(D) = -3/10

The two gaps print the same fraction. These utilities favour B and D; lower the five-million utility far enough and they favour A and C. Preferring A and D is choosing both signs of one number.

Common mistakes

Thinking a sure thing always earns a premium under expected utilityUnder the rule a sure million is worth u(1M) and nothing more. With u(0) = 0, u(1M) = 10 and u(5M) = 14 the gamble B scores 103/10 against 10 for A, so the sure thing loses. It wins only if u(5M) is below 11 on this scale, and then it must also win the second choice for C over D. Certainty carries no bonus of its own; that is the thing the popular pattern adds.

Judging each choice aloneTaken alone, A over B fits expected utility for some utilities, and so does D over C. The contradiction lives in the pair. Checking the choices one at a time will find nothing wrong, which is why the pattern persisted for so long.

Concluding that the people are simply irrationalThe argument shows the pair is inconsistent with the rule, not that anyone has made an error. A defender may say the rule leaves out something outcomes carry, such as regret. Whether to revise the choices or the rule is the open question.

Check your understanding

✓

Finish when you can show that a pair of choices breaks expected utility.

Given the four gambles, you should be able to tabulate them on common tickets, write both gaps, show that they are the same expression, and say what accepting the popular pattern costs.

Note: The utilities are the same assumption as everywhere else on this course: a number somebody chose. The argument does not depend on them, which is why the paradox is stated for any utilities. The reasons people choose as they do are not computed here.