Allais Paradox

Expected Utility Theory

In the 18th century, Daniel Bernoulli proposed the expected utility theory. Bernoulli suggested evaluating a risky prospect by multiplying the probability of each possible outcome by the utility of the outcome (a numerical estimate of the value of the outcome to the decision maker). Summing these values over all possible outcomes produces the expected utility of an option, and the option with the highest expected utility is chosen. In the first half of the 20th century, Frank Ramsey, John von Neumann and Oskar Morgenstern, and Leonard J. Savage demonstrated that expected utility theory can be derived from a small set of intuitive axioms and that it is the decision strategy that maximizes utility when used over time.

The Independence Axiom

One of the axioms from which expected utility theory is derived is the independence axiom, which says that any outcome common to two options is irrelevant when choosing between them. For example, consider the following gambles:

When choosing between ticket A and ticket B, all that matters is whether the decision maker prefers Rome or Paris. According to the independence axiom, how the decision maker feels about New York is irrelevant, because he or she will win a trip to New York if the coin flip is tails regardless of which lottery is chosen. Because of this, any prize could be substituted for the trip to New York without changing the decision.

The Allais Paradox

The Allais paradox is a violation of the independence axiom. Although the paradox can be demonstrated with a variety of payouts and probabilities, the following is one of the most commonly seen presentations. Consider the following two gambles: