Choose between the following two options:

1A. $24,000, with certainty.
1B. 33/34 chance of winning $27,000, and 1/34 chance of winning nothing.

Which seems more intuitively appealing? And which one would you choose in real life?

Now which of these two options would you intuitively prefer, and which would you choose in real life?

2A. 34% chance of winning $24,000, and 66% chance of winning nothing.
2B. 33% chance of winning $27,000, and 67% chance of winning nothing.

The Allais Paradox - as Allais called it, though it's not really a paradox - was one of the first conflicts between decision theory and human reasoning to be experimentally exposed, in 1953. I've modified it slightly for ease of math, but the essential problem is the same: Most people prefer 1A > 1B, and most people prefer 2B > 2A. Indeed, in within-subject comparisons, a majority of subjects express both preferences simultaneously.

This is a problem because the 2s are equal to a one-third chance of playing the 1s. That is, 2A is equivalent to playing gamble 1A with 34% probability, and 2B is equivalent to playing 1B with 34% probability.

Among the axioms used to prove that "consistent" decisionmakers can be viewed as maximizing expected utility, is the Axiom of Independence: If X is strictly preferred to Y, then a probability P of X and (1 - P) of Z should be strictly preferred to P chance of Y and (1 - P) chance of Z.

All the axioms are consequences, as well as antecedents, of a consistent utility function. So it must be possible to prove that the experimental subjects above can't have a consistent utility function over outcomes. And indeed, you can't simultaneously have:

U($24,000) > 33/34 U($27,000) + 1/34 U($0)
0.34 U($24,000) + 0.66 U($0) < 0.33 U($27,000) + 0.67 U($0)

These two equations are algebraically inconsistent, regardless of U, so the Allais Paradox has nothing to do with the diminishing marginal utility of money.

Maurice Allais initially defended the revealed preferences of the experimental subjects - he saw the experiment as exposing a flaw in the conventional ideas of utility, rather than exposing a flaw in human psychology. This was 1953, after all, and the heuristics-and-biases movement wouldn't really get started for another two decades. Allais ...

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