Allais paradox

The Allais paradox is the most prominent example for behavioral inconsistencies related to the von Neumann Morgenstern axiomatic model of choice under uncertainty. The Allais paradox shows that the significant majority of real decision makers orders uncertain prospects in a way that is inconsistent with the postulate that choices are independent of irrelevant alternatives. Basically, it is this postulate that allows to represent preferences over uncertain prospects as a linear functional of the utilities of the basic outcomes, viz. as the expectation of these utilities.

Consider the following choice situation (A) among two lotteries:

  • lottery L1 promises a sure win of $30,
  • lottery L2 is a 80% chance to win $45 (and zero in 20% of the cases).
    Typically, L1 is strictly preferred to L2 (such observed behavior is called a revealed preference).

    Now, consider another choice situation (B):

  • lottery K1 promises a 25% chance of winning $30,
  • lottery K2 is a 20% chance to win $45.
    Here, the typical choice is K2 over K1 although situation B differs from situation A only in that in each lottery, three quarters of the original probability of winning a positive amount are cancelled.

    Assume the typical subject decides among lotteries in the following way. To each of the basic outcomes, a number is assigned that indicates its attractiveness; say u(0)=0, u(45)=1, and u(30)=v (0<v<1). The overall attractiveness of a lottery (compared to another lottery) derives as the sum of the outcomes' elementary attractivenesses, weighted by their respective probabilities. Among two lotteries, the preferred one is that which offers a higher expected level of overall attractiveness. In this way, the decision maker forms a von Neumann Morgenstern utility function over deterministic outcomes, the expectation of which provides the criterion for choosing among uncertain outcomes, such as lotteries. Now in situation A, the revealed preference of L1 over L2 implies u(30) > 0.8 u(45), or v > 0.8; while the revealed preference of K2 over K1 in situation B shows that 1/4 v < 1/5, or v < 0.8.

    In cognitive psychology, this inconsistency is explained as a certainty effect. In situation A, L2 differs from L1 by a winning probability that is 20% lower, just as lottery K2 differs from K1 in situation B (where 4/5 x 25 = 20). Empirically, it seems that cancelling a fixed proportion of winning probability has a higher cognitive impact in a lottery where winning was extremely likely than in a lottery where winning was "a rather unlikely event, anyway."

    By accounting for a misperception of probabilities according to a non-linear weighting function (of the utilities of the elementary outcomes), expected utility can be rescued also in view of the Allais paradox (see prospect theory). The Allais paradox, devised in the 1950's, was the first piece in a series of systematic evidence challenging the traditional concept of von Neumann Morgenstern expected utility, leading to the development of generalized models of ("boundedly rational") choice behavior under uncertainty.

    See also: bounded rationality, certainty effect, cognitive psychology, prospect theory, von Neumann Morgenstern utility function

    Entry by: Jan Vleugels and Joachim Winter

    June 11, 1999
    Direct questions and comments to: Glossary master