Child covering its face whilst someone holds red and yellow peppers

The Allais Paradox: Explained

It is a Tuesday afternoon. George, exhausted from a tough football match, trudges into the local deli. He looks up at the display and gasps: all but two burgers are out of stock – a turkey, and a chicken and pesto! George ponders which one he would most enjoy and snatches the turkey. Just as he is about to pass it to the shopkeeper, he notices a freshly prepared batch of tomato and mozzarella sandwiches are being put on the display. After a brief hesitation, he changes his mind. He puts down the turkey and picks up the chicken and pesto burger instead.

In this imagined scenario, George, a half-witted student who has not yet been enlightened by economic theory, acted irrationally. He changed his decision based on information that was irrelevant: the fact that there was a third sandwich that he did not pick.

Assuming that people are rational when given a choice between A and B, they will pick the option that they regard as having the most utility: the one that is more valuable to them for the money that they spend on it. If an option C (in our case, the tomato and mozzarella sandwich) presents itself, it should not incite a switch from A to B or vice versa, because each option’s utility had not changed.

Historically, most economists believed that the general populous is more rational than George when it came to preferences with uncertain outcomes (the vast majority of choices that we make). Until recently, the reigning theory of decision making was expected utility theory, developed by Daniel Bernoulli in 1738. It states that people make choices that maximise their expected utility- utility of an option multiplied by the probability of it occurring. The conclusion was that people act rationally at least most of the time. This is significant, because if people act rationally, then markets act rationally. If markets act rationally, then the state should have minimal involvement in the economy, which was for decades the advice most economists gave to politicians.

In 1953, Maurice Allais, a French economist, presented one of the most substantial arguments against expected utility theory to date. It became known as the Allais Paradox and is outlined below for you to try on yourself.


Stage 1. Would you rather get…

A) $5 million for sure

Or

B) An 89% chance of winning $5 million
A 10% chance of winning $15 million
A 1% chance of winning nothing

The majority of people pick A: the certainty of earning $5 million over a slim chance of getting even more. Now consider the second stage of the paradox.

Stage 2. Would you rather get…

C) An 11% chance of winning $5 million
An 89% chance of winning nothing

Or

D) A 10% chance of winning $15 million
A 90% chance of winning nothing


This time, almost everyone votes for D. Understandably, the more substantial payout outweighs the slightly higher risk of ending up with nothing. On the face of it, in both stages people act rationally; they sensibly judge the payout of their decisions against the odds involved, trying to maximise their expected utility. In actuality, in first choosing A and then D, they violate expected utility theory. They act irrationally. Consider the following proof:


Let u stand for utility

In stage 1, we determined that u(A) > u(B)

∴ 1u(5 million) > 0.89u(5 million) + 0.1u(15 million) + 0.01u(0)

0.11u(5 million) > 0.1u(15 million) + 0.01u(0)

In stage 2, we determined that u(D) > u(C)

∴ 0.1u(15 million) + 0.9u(0) > 0.11u(5 million) + 0.89u(0)

0.11u(5 million) < 0.1u(15 million) + 0.01u(0)

This is the reverse of the conclusion we arrived at earlier.


Similarly to George, our choices were altered by seemingly irrelevant information. Stage 2 is the same as stage 1 except for an 89% reduction in the chance of winning $5 million in both options. The expected utility of 5 million dollars and 15 million dollars had not changed, yet our choices were incoherent.

In the 1970s, Amos Tversky and Daniel Kahneman, two psychologists who throughout their careers had upended many perceptions about how the human mind functions, developed a theory that explained our behaviour when presented with the Allais Paradox. They realised that we are more sensitive to the difference between 100% chance of winning at least $5 million (A) and 99% (B) than we are to the difference between 11% (C) and 10% (D). That is why we opt for A and then D.

Or, to put it simply, humans tend to overvalue complete certainty.

“In human decision making, losses loom larger than gains.” – Amos Tversky

What Tversky and Kahneman found was that people think of different outcomes in terms of relative changes rather than absolute changes. If they choose B and end up with nothing, they regard this outcome as a loss of $5 million, despite the fact that there are no richer or poorer than they were before. This happens because A offered complete certainty in winning. If they choose D and end up with nothing, they do not regard it as a loss because the odds presented by C were not much better. People are more sensitive to losses than to gains, making them risk averse.

This discovery, sparked by the Allais Paradox, helped Kahneman win a Nobel Prize in Economics in 2002. Even more significantly, it contributed to the foundation of the new and exciting field of behavioural economics.

This piece has been updated to improve accessibility for a wider audience.

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