# The Allais Paradox or the limits of rationality

Let’s play two little games!

First game:

Imagine you have to push one of two buttons:

• If you push the blue button, you are guaranteed to receive \$1 million directly on your bank account
• If you push the red button, you win \$5 million in 8 out of 10 cases and nothing otherwise
Which one do you push?

Second game:

Now, you have to push one of these two buttons:

• If you push the blue button, you win \$1 million in 5 out of 100 cases and nothing otherwise
• If you push the red button, you win \$5 million in 4 out of 100 cases and nothing otherwise

Which one do you push?

Most of us have a certain bias towards pushing the blue button in the first game and the red button in the second.

Why is it a bias?

According to classical financial theories, we are all rational investors, so here is what our reasoning should look like:

In the first game, we are certain to win \$1 million with the blue button, while the red button gives us on average \$4 million (4 000 000 = 8/10 * 5 000 000). A rational agent would therefore rather push the red button.

In the second game, the blue button corresponds to an average of \$50 000 (50 000 = 5/100 * 1 000 000) while the red button corresponds to an average of \$200 000 (200 000 = 4/100 * 5 000 000). A rational agent would therefore rather push the red button in this case also.

However, we’ve seen that most of us choose the blue button in the first game and the red button in the second. Those two preferences are in fact contradictory: we are not consistent!

This bias is called certainty bias. It corresponds to the idea that our brain prefers situations with a sure outcome. That’s why in the first game, we prefer the blue button, which gives us the certainty of a \$1 million gain, while in the second, we are nearly indifferent to both choices, as there doesn’t seem to be a “good” one.

Maurice Allais was one of the first to have the intuition, in the middle of the 20th century, that the economical agent was not rational and developed this paradox to prove it in a theoretical way. This paradox indeed served as a counterexample to the rationality assumption defended by the classical economists.

From this moment, other economists began to challenge this rationality assumption, such as Daniel Ellsberg or, later, Daniel Kahneman who developed a very powerful framework distinguishing two systems to describe our decision-making process. According to this framework, such a bias corresponds to our System 1 – our intuitive, unconscious and effortless way of thinking – taking a faster and more efficient decision and relying on mental shortcuts.

Other experimental results later confirmed this conclusion of Allais’s paradox in various fields such as neurosciences or ethology. As a matter of fact, this sensitivity is not specific to financial situations, and is also observed in primates (Fiorillo et al.,2003) and bees choosing among different flowers for instance (Real, 1991).

## References:

“Le comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école Américaine”, Allais, 1953

https://theses.univ- lyon2.fr/documents/getpart.php?id=lyon2.2007.razgallah_ma&part=204011

“Risk, Ambiguity, and the Savage Axioms”, Daniel Ellsberg, 1961

Thinking, Fast and Slow, Daniel Kahneman, 2012

“Discrete Coding of Reward Probability and Uncertainty by Dopamine Neurons”, Christopher D. Fiorillo, Philippe N. Tobler and Wolfram Schultz, 2003

“Learning foraging tasks by bees: a comparison between social and solitary species”, Reuven Dukas and Leslie Real, 1991