Allais paradox
apparent violation of the predictions of expected utility theory
The Allais paradox is a choice problem designed by Maurice Allais in 1953 to show an inconsistency of actual observed choices with the predictions of expected utility theory. The Allais paradox demonstrates that individuals rarely make rational decisions consistently when required to do so immediately. The independence axiom of expected utility theory, which requires that the preferences of an individual should not change when altering two lotteries by equal proportions, was proven to be violated by the paradox.
Statement of the problem
The Allais paradox arises when comparing participants' choices in two different experiments, each of which consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows:
Several studies involving hypothetical and small monetary payoffs, and recently involving health outcomes, have supported the assertion that when presented with a choice between 1A and 1B, most people would choose 1A. Likewise, when presented with a choice between 2A and 2B, most people would choose 2B.
Allais further asserted that both of these preferences could be reasonable. Both B gambles have a 1% chance of getting a million dollars less than the corresponding A gamble, and a 10% chance of getting 4 million dollars more than the corresponding A gamble. If a person values money linearly, such that the next million dollars is always worth just as much to them as the previous million, then they should prefer both B gambles. But if a person values the first million more than 36 times as much as each of the next four millions, then they should prefer both A gambles.
However, there is no way of assigning values to money that enables a preference for 1A over 1B, and a preference for 2B over 2A, to simultaneously be compatible with expected utility theory. According to expected utility theory, the person should choose either 1A and 2A or 1B and 2B.
The inconsistency stems from the fact that in expected utility theory, equal outcomes added to each of two choices (e.g. the 89% chance of $1 million for both 1A and 1B over 2A and 2B) should have no effect on the relative desirability of one gamble over the other; equal outcomes should "cancel out".
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