The Allais paradox was developed by Maurice Allais in his paper “Le Comportement de l’homme rationnel devant le risque: critique des postulats et axiomes de l’école américaine”, 1953 and it describes the empirically demonstrated fact that individuals’ decisions can be inconsistent with expected utility theory. This paradox is usually explained with Allais experiment (you may try it yourself), which is:
An individual is asked to choose one between the following gambles:
Gamble A: 100% chance of receiving 100 millions
Gamble B: 10% chance of receiving 500 millions,
89% chance of receiving 100 millions,
1% chance of receiving nothing
And another amongst the following:
Gamble C: 11% chance of receiving 100 millions,
89% chance of receiving nothing
Gamble D: 10% chance of receiving 500 millions,
90% chance of receiving nothing
If expected utility axiomatic was applied, the preference A > B should imply that C > D. However, the experiment shows that most rational individuals would choose so that A > B but C < D, even though it can be easily seen that the expected value of each gamble is a=100, b=139, c=11 and d=50.
In the first gamble the less risky choice is preferred over a higher expected utility, while in the second gamble a higher expected utility is preferred over a less risky choice.
This paradox relies in the fact that in certain types of gambling, although people usually prefer certainty to uncertainty, if they are approached differently, they will prefer the uncertainty that was previously rejected. Therefore we can extract a series of conclusions. First, as we have seen, in the real world expected utility theory does not really applied, therefore its many critiques and alternative approaches. Second, people tend to give an extra value to total absence of risk on the contrary to remote and highly uncertain risk. Third and last, when the differences in probabilities are very small, people tend to ignore them and the expected utility is the alternative.