The Allais Paradox

Jonah Lehrer in Wired (via Delong):

Suppose somebody offered you a choice between two different vacations. Vacation number one gives you a 50 percent chance of winning a three-week tour of England, France and Italy. Vacation number two offers you a one-week tour of England for sure.

Not surprisingly, the vast majority of people (typically over 80 percent) prefer the one-week tour of England. We almost always choose certainty over risk, and are willing to trade two weeks of vacation for the guarantee of a one-week vacation. A sure thing just seems better than a gamble that might leave us with nothing. But how about this wager:

Vacation number one offers you a 5 percent chance of winning a three week tour of England, France and Italy. Vacation number two gives you a 10 percent chance of winning a one week tour of England.

In this case, most people choose the three-week trip. We figure both vacations are unlikely to happen, so we might as well go for broke on the grand European tour. (People act the same way with lotteries: we typically buy the ticket for the biggest possible prize, regardless of the odds.)

Allais presciently realized that this very popular set of decisions – almost everybody made them – violated the rational assumptions of economics. Instead of making decisions that could be predicted by a few mathematical equations, people acted with frustrating inconsistency. After all, both questions involve 50 percent reductions in probability (from 100 percent to 50 percent, and from 10 percent to 5 percent), and yet generated completely opposite responses. Our choices seemed incoherent.

The Allais paradox was mostly ignored for the next two decades. But then, in the early 1970s, two Israeli psychologists, Daniel Kahneman and Amos Tversky, read about the paradox and were instantly intrigued: they wanted to know why people didn’t respond to probabilities in a linear manner. Based upon their conversations with each other, it seemed obvious that people perceived a smaller difference between probabilities of 1 percent and 2 percent than between 0 percent and 1 percent, or between 99 percent and 100 percent. In other words, all changes in risk are not created equal. As Allais had observed decades before, we value complete certainty an inordinate amount.

But why was certainty so attractive?

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