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The Allais paradox arises when comparing participants' choices in two different experiments, each of which consists of a choice between two gambles, A and B. The payoffs for each gamble in each experiment are as follows:

Experiment 1:
A. I will give you $1,000,000 for certain
B. You will take part in a gamble in which you have
10% chance of winning $5,000,000
89% chance of winning $1,000,000
and 1% chance of winning $0

Experiment 2:
A. A 11% chance of winning $1,000,000
and 89% chance of winning $0
B. A 10% chance of winning $5,000,000
and 90% chance of winning $0

People commonly choose A for experiment 1 and B for experiment 2.

Suppose the gamble takes the form of drawing one numbered ticket from a barrel contains 100 tickets.

Experiment 1:

Experiment 2:

Option A1 and B1 are distinguished only by their rewards on tickets 1-11 and option A2 and B2 are distinguished by the same tickets, and in both cases the distinction is the same. In short the choice in experiment 1 and experiment 2, so if you choose A1 in the experiment 1, you should choose A2 in the second; or B1 in the first and B2 in the second.

It is an assumption in decision theory and economic theory that people behave rationally, but is this the same as consistency? Consistency means a decision maker who is consistent reached the same estimate for the recurrence of some event whatever method of assessment is used.

The main point Allais wished to make is that the independence axiom of expected Utility theory may not be a valid axiom. The independence axiom states that two identical outcomes within a gamble should be treated as irrelevant to the analysis of the gamble as a whole.

However, this overlooks the notion of complementarities, the fact your choice in one part of a gamble may depend on the possible outcome in the other part of the gamble. In the above choice, 1B, there is a 1% chance of getting nothing. Yet, this 1% chance of getting...
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