In identical experiments, an Allais paradox occurs when the addition of an independent event influences choice behavior. Consider the choices in the following table (Kahneman and Tversky 1979).

lottery

1 to 33

34

35 to 100

preference

0

18%

82%

0

0

83%

0

17%

In Experiment 1, a choice of and was given, and most participants picked . In Experiment 2, a choice of and was given, and most participants picked .

This observed pattern violates the independence axiom, since in both experiments, the payoff is identical if a ball is picked, while if the event is disregarded, the two experiments are identical.

To see it another way, consider the event to be a black box that is always received if the
random ball value is .
Knowing or not knowing the contents of the black box should not influence behavior.

Allais, M. "Le comportement de l'homme rationnel devant le risque: Critique des postulats et axiomes de l'école américaine."
Econometrica21, 503-546, 1953.Allais, M. "An Outline
of My Main Contributions to Economic Science." Amer. Econ. Rev.87,
3-12, 1997.Fishburn, P. C. Utility
Theory for Decision Making. New York: Wiley, 1970.Kahneman,
D. and Tversky, A. "Prospect Theory: An Analysis of Decision Under Risk."
Econometrica47, 263-292, 1979.Kreps, D. M. Notes
on the Theory of Choice. Boulder, CO: Westview Press, p. 192, 1988.Kulish,
M. "The Independence Axiom: A Survey." May 2002. http://www2.bc.edu/~kulish/papers/indepb.pdf.Savage,
L. J. The
Foundations of Statistics, 2nd ed. New York: Dover, 1972.