Economist Steve Landsburg has explained the Allais paradox.(Update: His latest is here.) It is an argument that people make choices that are contrary to what is considered rational under common economic models. Allais won the Bank of Sweden prize (aka Nobel for economics) for the work, and died recently, and was also known for anti-relativity views.
Here is a variant of the paradox. Suppose I give you $10 if a coin toss gives heads. Before tossing the coin, I offer you two choices for additional money:
A. I give you another $10 is that coin toss is tails.
B. I give you $5 regardless.
Apparently rational utility theory says that your choices should be independent of what you already have, so you should just compare A and B. Both choices give you $5 on average if we did this many times, but B is a sure thing and should therefore be preferred by the rational person.
But people would normally prefer A. Choice A, combined with the previous offer, implies that you will get a sure $10. Under Choice B you could end up with either $5 or $15, depending on the coin toss. It is an unnecessary gamble.
I do not think that this proves that people are irrational. It only shows that economic models are deficient if they assume that people do not change their preferences after acquiring assets. A man might prefer a dog to a cat, but once he has a dog, he might not want another one. In the above case, one of the choices is correlated with what you have already been offered.
I think Landsburg disagrees with me, so you might want to read his analysis and decide for yourself.
Update: Apparently the object was to find a counterexample to the Independence Axiom in the Von Neumann–Morgenstern utility theorem. The above example does not quite do it. But there are still situations where people make choices contrary to the axioms.