The Allais Paradox: a paper tiger
How one of the most common arguments against utilitarianism fails
The Allais paradox is a choice problem designed by Maurice Allais (1953) to show an inconsistency of actual observed choices with the predictions of expected utility theory.
— Wikipedia
Suppose you ask someone to choose between two lotteries:
a. 11% chance of 1M, 89% chance of X
b. 10% chance of 5M, 1% chance of 0, 89% chance of X
Regardless of which choice they make, virtually anyone you ask will recognize that the value of X should be irrelevant to the decision. You get an 89% chance of X regardless, so it’s not a differentiator between the two options. This is central to the independence axiom of expected utility theory.
But now suppose you specify a value of one million dollars, like so:
1a. 11% chance of 1M, 89% chance of 1M
1b. 10% chance of 5M, 1% chance of 0, 89% chance of 1M
This simplifies to:
Experiment 1
1a. Guarantee of 1M
1b. 10% chance of 5M, 1% chance of 0, 89% chance of 1M
Given this choice, most people will choose 1a.
But now suppose we set X to 5M instead of 1M. We get:
Experiment 2
2a. 11% chance of 1M, 89% chance of 5M
2b. 99% chance of 5M, 1% chance of 0
Here, most people choose 2b.
So we have an apparent violation of the independence axiom of expected utility theory.
Rebuttal
How do we explain this contradictory behavior without rejecting the notion that people are utilitarian? Easy. As we said from the start, both experiments can be generalized by using the variable X. And we find that if phrased that way, people are consistent. That is, if you ask their choice, they almost never object that “it depends on the value of X”.
The problem with Allais’s phrasing is that he doesn’t tell the test subjects about the equivalence as a generalized formula with the X variable. And it’s very difficult for people to come to that realization on their own without pretty strong math expertise. Indeed, you can go out of your way to ask most people how the two experiments can be generalized, and I’ll bet you fewer than 30% of adults will be able to readily find that equivalence on their own.
Individual assessments of the two experiments are not rigorous. People are vulnerable to built in mental shortcuts conferred to us by eons of biological evolution, in which accuracy is in competition with time. Take too long deciding between fight or flight, and you’re lion food.
