This short paper shows that the Allais Paradox and the Common Ratio Effect regarded as classic examples of the violation of the Expected Utility Theory Axioms – may be easily explained by assuming that changes in wealth (i.e. gains and losses) are perceived in relative terms. The preference reversal observed in experiments is therefore predictable and the choices shall consequently be assumed to be rational. By contrast, the assumption that wealth changes are perceived in absolute terms leads to the conclusion that the choices violate the axioms underlying Expected Utility Theory, and are therefore irrational. This state of affairs is called the illusion of irrationality.Expected Utility Theory, Relative Utility Function, Allais Paradox, Common Ratio Effect, Prospect Theory

Kontek, Krzysztof

English

Research Papers in Economics

  1 
The Illusion of Irrationality 
 
Krzysztof Kontek 
Artal Investments sp. z o.o. 
ul. Chrościckiego 93/105 
02 414 Warsaw, Poland 
e mail: kontek@artal.com.pl 
kkontek2000@yahoo.com 
tel.:+48 504 16 17 50 
 
Abstract 
  This short paper shows that the Allais Paradox and the Common Ratio Effect – regarded as 
classic examples of the violation of the Expected Utility Theory Axioms – may be easily explained 
by assuming that changes in wealth (i.e. gains and losses) are perceived in relative terms. The 
preference  reversal  observed  in  experiments  is  therefore  predictable  and  the  choices  shall 
consequently  be  assumed  to  be  rational.  By  contrast,  the  assumption  that  wealth  changes  are 
perceived in absolute terms leads to the conclusion that the choices violate the axioms underlying 
Expected Utility Theory, and are therefore irrational. This state of affairs is called the illusion of 
irrationality. 
 
JEL classification: C91, D03, D81, D87 
 
Keywords:  Expected  Utility Theory,  Relative  Utility  Function, Allais  Paradox,  Common  Ratio 
Effect, Prospect Theory, Markowitz’s hypothesis 
 
1. Introduction 
1.1.   The concept of expected utility was first introduced by Bernoulli and Cramer as early as 
the first half of the 18th century, although they did not consider the question of rationality when 
presenting their solutions. Much later, von Neumann and Morgenstern (1944) derived an identical 
solution from axioms which assumed that people made rational decisions. Their Expected Utility 
Theory has four axioms: completeness, transitivity, independence and continuity. If all these axioms 
are satisfied, then the individual is said to be rational and his or her preferences can be represented 
by a utility function. 
1.2.   Expected Utility Theory  has dominated mainstream economic thought since the middle 
of  the  twentieth  century.  Since  NM's  work,  however,  an  increasing  number  of  psychological   2 
experiments  have  pointed  to  the  irrationality  of  human  decision  making,  where  “irrationality” 
means any deviation from the axioms underlying Expected Utility Theory. Most of the arguments 
against  the  Expected  Utility  approach  have  concerned  the  independence  axiom  with  the Allais 
Paradox and the Common Ratio Effect being the main examples of inconsistency of choices. The 
concept of probability weighting has allowed these experimental results to be explained and has led 
to the development of several theories, including Prospect Theory (Kahneman, Tversky, 1979) and 
its Cumulative version (Tversky, Kahneman, 1992).  
1.3.   Kontek (2009) argues that lottery experiments can be explained using the relative utility 
function without recourse to the probability weighting function. This solution, whose shape strongly 
resembles the utility function hypothesized by Markowitz (1952), is based on the assumption that 
people perceive  wealth  changes (i.e.  gains and  losses) in relative rather than absolute terms as 
assumed by Prospect Theory
1. These arguments, however, only apply to simple lotteries. This paper 
goes further and analyzes examples which are regarded to violate the EUT axioms. It will be shown 
that both paradoxes can be easily explained by using the concept of the relative utility function. The 
preference reversal observed in the experiments is therefore only irrational if it is assumed that 
wealth  changes  are  perceived  in  absolute  terms.  This  state  of  affairs  is  called  the  illusion  of 
irrationality.   
 
2. Common Ratio Effect and Allais Paradox.       
2.1. The Common Ratio Effect will be considered first. This was presented by Prospect 
Theory as follows: 
Problem 1: Choose between  
A: 4,000 with a probability of 0.80  or  B: 3,000 with a probability of 1.00 
  0 with a probability of 0.20   
Problem 2: Choose between 
C: 4,000 with a probability of 0.20  or  D: 3,000 with a probability of 0.25 
  0 with a probability of 0.80      0 with a probability of 0.75 
  Experimental results consistently reveal that most people choose option B in Problem 1 and 
option C in Problem 2. Expected Utility Theory, per contra, predicts that people would choose 
either A and C or B and D, as the probabilities of winning the main prize in the second pair of 
choices differ by a common ratio factor of 4 compared with the first pair. The graphical solution to 
this problem using the relative utility function is presented in Fig.1.  
The shape of both the red and blue curves are the same as they are rescaled relative utility 
                                                 
1 This observation, which results from Weber’s Law is not new and was noticed by Kahneman and Tversky (1984) as 
well: The topical organization of mental accounts leads people to evaluate gains and losses in relative rather than in 
absolute terms (emphasis added). This was, however, not implemented in PT and CPT.   3 
functions
2. The difference is that the range of the red curve is [0, 4000] as this corresponds with 
options A and C, which have a maximum outcome of 4,000, whereas the range of the blue curve is 
[0, 3000] as this corresponds with options B and D, which have a maximum outcome of 3,000. 
Fig. 1. The Common Ration Effect explained using the relative utility function. 
 
The certainty equivalents of each option can be easily found on the x axis as presented in the 
figure. There is obviously no inconsistency in choices between options having the same maximum 
outcome (e.g. between options A and C or B and D). The preference reversal only appear when 
options  having  different  maximum  outcomes  are  compared.  For  greater  probabilities,  option  B 
prevails over option A, but for lower probabilities, C is a better option than D. As presented this 
effect is easily predicted and explained by assuming that people perceive wealth changes in relative 
terms.  
2.2. The second classical problem considered here is the Allais Paradox, which is presented 
by Prospect Theory as follows: 
Problem 3: Choose between 
A: 2,500 with a probability of 0.33    B: 2,400 with a probability of 1.00 
     2,400 with a probability of 0.66 
0 with a probability of 0.01 
Problem 4: Choose between 
C: 2,500 with a probability of 0.33    D: 2,400 with a probability of 0.34 
  0 with a probability of 0.67           0 with a probability of 0.66 
                                                 
2The parts for gains are presented.   4 
This problem is similar to the Common Ratio Effect except that here both pairs differ by an 
outcome of 2,400 with a probability of 0.66. Once again, the experiments show a similar pattern in 
that most people choose option B in Problem 3 and option C in Problem 4. The graphical solution to 
this problem using the relative utility function is presented in the Fig. 2. 
 
Fig. 2. The Allais Paradox explained using the relative utility function. The diagrams on the right show the 
details of the main diagram.  
 
The red and blue curves have the same shape, as was the case with the Common Ratio 
Effect. Similarly, they differ in that the range of the red curve is [0, 2500] as this corresponds with 
options A and C, which have a maximum outcome of 2,500, whereas the range of the blue curve is 
[0, 2400] as this corresponds with options B and D, which have a maximum outcome of 2,400. The 
certainty equivalents of each option are found as presented in the figure. In order to simplify the 
analysis, the two option A outcomes can be combined to create (2500, 0.99), which is a better 
outcome. Even though its certainty equivalent is lower than that of the certain outcome of 2400. The 
preference  reversal  appears  when  comparing  options  having  different  maximum  outcomes.  For 
greater probabilities option B prevails over A, but for lower probabilities option C is better than D.  
It follows that this reversal is wholly predictable and can be explained using the relative utility 
approach.  
 
3. Conclusions 
   Expected Utility Theory, based as it is on the assumption that wealth changes are perceived 
in absolute terms, can neither predict nor explain the preference reversals observed in the Common 
Ratio and Allais paradoxes. This leads to the conclusion that people are irrational. However, once it   5 
is assumed that wealth changes are perceived in relative terms, the preference reversals are fully 
predictable and there is no reason to regard people as irrational. As the perception of wealth changes 
in relative terms was stated even by Kahneman and Tversky, people's behavior should be regarded 
as rational and the relative utility function should be used for its description. It follows that the 
stated irrationality is merely a result of using an improper model. This is what we call the illusion of 
irrationality.  
 
References 
Allais, M., (1953). Le comportement de l’homme rationnel devant le risque: critique des postulats etaxiomes 
de l’école Américaine. Econometrica 21, pp 503 546. 
Kahneman, D., Tversky, A., (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47, 
pp 313 327. 
Kahneman, D., Tversky, A., (1984). Choices, Values, and Frames. American Psychologist, Vol. 39, No. 4, pp 
341 350. 
Kontek,  K.,  (2009).  Absolute  vs  Relative  Notion  of  Wealth  Changes.  MPRA  Paper  http://mpra.ub.uni 
muenchen.de/17336/, Available at SSRN: http://ssrn.com/abstract=1474229 .  
Markowitz H., (1952A). The Utility of Wealth. Journal of Political Economy, Vol. 60, pp 151 158. 
von Neumann J., Morgenstern O., (1944). Theory of Games and Economic Behavior, Princeton University 
Press. 
Tversky A., Kahneman D., (1992). Advances in Prospect Theory: Cumulative Representation of Uncertainty. 
Journal of Risk and Uncertainty, vol. 5(4), October, pp 297 323. 

Absolute vs Relative Notion of Wealth Changes.

The Illusion of Irrationality

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics