The Von NeumannMorgenstern expected utility axioms formalize a model of rational behavior. However, experiments have shown that people systematically violate some of them.
In 1953, Maurice Allais published a paper regarding a survey he had conducted in 1952, with a hypothetical game. Subjects \"with good training in and knowledge of the theory of probability, so that they could be considered to behave rationally\", routinely violated the expected utility axioms. The game itself and its results have now become famous as the \"Allais Paradox\".

The most famous structure is the following:
Subjects are asked to choose between the following 2 gambles, i.e. which one they would like to participate in if they could:
Gamble A: A 100% chance of receiving $1 million.
Gamble B: A 10% chance of receiving $5 million, an 89% chance of receiving $1 million, and a 1% chance of receiving nothing.
After they have made their choice, they are presented with another 2 gambles and asked to choose between them:
Gamble C: An 11% chance of receiving $1 million, and an 89% chance of receiving nothing.
Gamble D: A 10% chance of receiving $5 million, and a 90% chance of receiving nothing.

This experiment has been conducted many, many times, and most people invariably prefer A to B, and D to C. So why is this a paradox?
The expected value of A is $1 million, while the expected value of B is $1.39 million. By preferring A to B, people are presumably maximizing expected utility, not expected value.
By preferring A to B, we have the following expected utility relationship:
u(1) > 0.1 * u(5) + 0.89 * u(1) + 0.01 * u(0), i.e.
0.11 * u(1) > 0.1 * u(5) + 0.01 * u(0)
Adding 0.89 * u(0) to each side, we get:
0.11 * u(1) + 0.89 * u(0) > 0.1 * u(5) + 0.90 * u(0),
implying that an expected utility maximizer must prefer C to D. Of course, the expected value of C is $110,000, while the expected value of D is $500,000, so if people were maximizing expected value, they should in fact prefer D to C. However, their choice in the first stage is inconsistent with their choice in the second stage, and herein lies the paradox.
From the Von NeumannMorgenstern axoims, the substitution axiom is the one that is clearly violated. The probability of receiving $5 million is the same in both B and D.

In July 2000, Karl Sigmund organized extensive game theoretical experiments on the internet, with the following structure:
Stage 1:
A: A chance of winning 4000 Euro with probability 0.2 (expected value 800 Euro)
B: A chance of winning 3000 Euro with probability 0.25 (expected value 750 Euro)
Stage 2:
C: A chance of winning 4000 Euro with probability 0.8 (expected value 3200 Euro)
D: A chance of winning 3000 Euro with certainty.
In large scale experiments, most players (about 70%) choose 4000 Euro with probability 0.2 (A) rather than 3000 Euro with probability 0.25 (B). And indeed, the expected value 4000 Euro with probability 0.2 (800 Euro) is larger than of 3000 Euro with probability 0.25 (750 Euro). In the second decision, most choose 3000 Euro with certainty (D).
One can easily see the same behavioral pattern from this experiment as well. Many consider this a major flaw in expected utility theory and have attempted to develop alternatives to the theory, in order to explain behavior such as this. See the Advanced Topics section for further discussion on alternatives to expected utility theory.

 Allais, Maurice, (1953), Le Comportement de l\'Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l\'Ecole Americaine, Econometrica
 Allais, Maurice, (1997), An Outline of My Main Contributions to Economic Science, The American Economic Review
 The following reports some experimental results pertaining to such paradox in a particular design.Click here to read the report.
