Allais paradox

The Allais paradox (after Maurice Allais ) is an experimentally observable violation of the axiom of independence (common consequence effect, CCE) of economic decision theory . This means that the addition / removal of common consequences of a decision must not change the preference of the decision maker.

Structure of the experiment

The basic structure of the experiment is that test subjects choose from two lotteries twice in a row:

Choice 1:

${\ displaystyle a = (2500, \ 0 {,} 33; \ 2400, \ 0 {,} 66; \ 0, \ 0 {,} 01)}$

That means: You win 2500 monetary units (MU) with 33% probability , 2400 MU with 66% probability, with 1% probability you get nothing.

${\ displaystyle b = (2400, \ 1)}$

This translates into a secure profit of 2400 GE.

Choice 2

The same people then have to make another selection:

${\ displaystyle a '= (2500, \ 0 {,} 33; \ 0, \ 0 {,} 67)}$ ,

so 2500 GE with 33% probability, otherwise nothing.

${\ displaystyle b '= (2400, \ 0 {,} 34; \ 0, \ 0 {,} 66)}$ ,

So 2400 GE with 34% probability, otherwise nothing.

The four lotteries are summarized in a table:

Probability / Profit	Lottery a	Lottery b	Lottery a '	Lottery b '
0.66	2400	2400	0	0
0.33	2500	2400	2500	2400
0.01	0	2400	0	2400

The decision-making situation between a and b and that between a 'and b' differs only in that the former has a high probability of a profit of 2,400, which the latter does not (but this is not decided at all, it only forms the surrounding'). Otherwise the situations are the same. According to the axiom of independence, the content of the first line should not have any influence on decision-making behavior.

evaluation

The majority of the test persons choose b and a ' in the experiment , i.e. they have the preferences a < b and a'> b ' .

${\ displaystyle b> a}$ means: ${\ displaystyle u (2400)> 0 {,} 33 \ u (2500) +0 {,} 66 \ u (2400) +0 {,} 01 \ u (0)}$

${\ displaystyle a '> b'}$ means: ${\ displaystyle 0 {,} 33 \ u (2500) +0 {,} 67 \ u (0)> 0 {,} 34 \ u (2400) +0 {,} 66 \ u (0)}$

These two inequalities can be transformed into:

${\ displaystyle 0 {,} 34 \ u (2400)> 0 {,} 33 \ u (2500) +0 {,} 01 \ u (0)}$ and

${\ displaystyle 0 {,} 33 \ u (2500) +0 {,} 01 \ u (0)> 0 {,} 34 \ u (2400)}$ ,

two contradicting statements.

This contradiction can be explained by the fact that in the first decision between and the probabilities are in the foreground, whereby these hardly differ in the decision between and and the profits are used as the decisive criterion. ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle a '}$ ${\ displaystyle b '}$

Allais' experiment, published in 1953, is an early example of the use of experimental methods to gain knowledge in economics and contributed to the development of experimental economic research .

Explanation

The psychological explanation of this irrational behavior is provided by the security effect , an aspect of the prospect theory by Amos Tversky and Daniel Kahneman . Kahneman gives the following example:

Choice 1

0.61 chance of winning $ 520,000 or 0.63 chance of winning $ 500,000

Here, most of the respondents choose the first option because the difference of $ 20,000 makes more impression than the probability difference of 0.02. This is also the rational decision for risk-neutral behavior, because the expected values of the two options are 317,200 versus 315,000.

Now you increase the probability of winning by 0.37 with both options and offer the following lotteries:

Choice 2

0.98 chance of winning $ 520,000 or certain win (probability 1) of $ 500,000.

Paradoxically, although the expected benefit of the first option is even greater than before, most respondents now switch to the second option, paradoxically. Now the decision criterion is the certainty of the smaller profit; the security effect takes effect. The expected values of the two options are now 509,600 versus 500,000. Compared to option 1, the expected value of the first option of option 2 has increased by 192,400, that of the second option only by 185,000.

Alternative declarations

The simple heuristic "Take the Best" (see Gerd Gigerenzer ) provides a plausible explanation that does not require the mental calculation of probabilities. "Take the best" can be summarized as follows: take the best criterion and decide - if there is no relevant difference, take the second best, and so on.

Applied to the Allais paradox, this means: When comparing a and b , the probability is used as the criterion, when comparing between a ' and b', on the other hand, the expected profit.

Individual evidence

↑ M. Allais: Le comportement de l'homme rationnel devant le risque: critique of the postulate et l'école Américaine axiom de . In: Econometrica . 21, No. 4, 1953, pp. 503-546.
^ Daniel Kahneman (2011): Thinking, fast and slow , Allen Lane Paperback, ISBN 978-1-846-14606-0 , p. 312 ff.
↑ John D. Lee and Alex Kirlik (2013): The Oxford Handbook of Cognitive Engineering , Oxford University Press, ISBN 978-0-19-975718-3 , p. 495

[1] M. Allais: Le comportement de l'homme rationnel devant le risque: critique of the postulate et l'école Américaine axiom de . In: Econometrica . 21, No. 4, 1953, pp. 503-546.

[2] Daniel Kahneman (2011): Thinking, fast and slow , Allen Lane Paperback, ISBN 978-1-846-14606-0 , p. 312 ff.

[3] John D. Lee and Alex Kirlik (2013): The Oxford Handbook of Cognitive Engineering , Oxford University Press, ISBN 978-0-19-975718-3 , p. 495