# How important is the Allais paradox

## Allais paradox

The Allais paradox tries to refute the independence principle and thus the reasonableness of the decision rule "Maximize the income value of the risk benefit", ie the Bernoulli principle. Allais constructs an example for this, in which a test person has to make a double decision, so that a contradiction to the Bernoulli principle arises due to the order of preference that has been made. Allais believes that he can thus prove that the Bernoulli principle should not be viewed as a condition of rational action. The Allais example is not conclusive in itself. The additional risk aversion that the test subject shows in the double decision can be interpreted as an expression of rational action in the presence of subjective credibility in maximizing the benefit.

An example presented by Maurice ALLAIS (born 1911), which questions the plausibility of the expected utility maximization as an axiomatic formulation of rational individual decisions in the case of risk and uncertainty. ALLAIS designs the following alternatives: Situation S 1: A: 100% chance of

$ 1,000,000 or B: 10% chance of

$ 5,000,000 89% chance of

$ 1,000,000 I% chance of $ 0 Situation S2: X: 11% chance of

$ 1,000,000 89% chance of $ 0 or Y: 10% chance of

5. 000,000 $ 90% chance of $ 0 An individual acting in accordance with expected utility maximization would always have to express the preference X> Y in S2 if he preferred A> B in Si (or Y> X if B> A) . Experimentally it was found that more than half of the individuals prefer Y> X and A> B at the same time. However, this implies a decision calculus in which the probabilities are included in a non-linear way, while with expected utility maximization, i.e. max Iu (r;) p;, the probabilities are p-linearly included (decision theory). ALLAIS \ '»Paradoxon« indicates that linearity in the probabilities, at least in the range close to certainty (p = 100%), is problematic as an axiom and is impermissible as a behavioral assumption because of the »preference for certainty« relevant here. In numerous experiments, various types of systematic violations of the axioms of the expected utility hypothesis that have other causes have now been identified. This has led to a discussion about alternative decision theories in the event of uncertainty. R.S. Literature: Machina, M. (1987)

An example presented by Maurice ALLAIS (born 1911), which questions the plausibility of the expected utility maximization as an axiomatic formulation of rational individual decisions in the case of risk and uncertainty. ALLAIS designs the following alternatives: Situation S 1: A: 100% chance of

$ 1,000,000 or B: 10% chance of

$ 5,000,000 89% chance of

$ 1,000,000 I% chance of $ 0 Situation S2: X: 11% chance of

$ 1,000,000 89% chance of $ 0 or Y: 10% chance of

5. 000,000 $ 90% chance of $ 0 An individual acting in accordance with expected utility maximization would always have to express the preference X> Y in S2 if he preferred A> B in Si (or Y> X if B> A) . Experimentally it was found that more than half of the individuals prefer Y> X and A> B at the same time. However, this implies a decision calculus in which the probabilities are included in a non-linear way, while with expected utility maximization, i.e. max Iu (r;) p;, the probabilities are p-linearly included (decision theory). ALLAIS \ '»Paradoxon« indicates that linearity in the probabilities, at least in the range close to certainty (p = 100%), is problematic as an axiom and is impermissible as a behavioral assumption because of the »preference for certainty« relevant here. In numerous experiments, various types of systematic violations of the axioms of the expected utility hypothesis that have other causes have now been identified. This has led to a discussion about alternative decision theories in the event of uncertainty. R.S. Literature: Machina, M. (1987)

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