A simple two-choice single outcome valued decision under risk is presented ... Keywords: paradox, 'one-off', intransitivity, relative nature of decisions, structural ...
lAIN PATERSON AND ANDREAS DIEKMANN
A PARADOX
IN D E C I S I O N T H E O R Y
AND SOME EXPERIMENTAL
RESULTS:
T H E R E L A T I V E N A T U R E OF D E C I S I O N S
ABSTRACT. A simple two-choice single outcome valued decision under risk is presented which should show up the limitations in the classical approach of von Neumann, its extensions and its alternatives. An empirical testing of this hypothesis strongly supports this criticism. A rationale for explaining the apparent 'irrational decision' is put forward and the case is made for understanding the relative nature of decision choices especially when multi-criteria are involved.
Keywords: paradox, 'one-off', intransitivity, relative nature of decisions, structural dependence, relative dominance, multicriteria.
There is now a large and growing body of literature dealing with experimental refutations of the expected utility model (EU-model) and the strong Neumann-Morgenstern rationality criteria (e.g. Kahnemann and Tversky 1979, Holler 1983, Schoemaker 1982). Experimental results and various paradoxes like the 'Allais-Paradox' (Allais and Hagen 1979, Hagen 1986) gave rise to widely discussed concepts of bounded rationality (Klopstech and Selten 1984, Elster 1979) and alternative theories of rational behavior and decision making. Prominent variants of the EUmodel include Kahnemann and Tversky's (1979) 'prospect theory' and Hagen's (1979) 'positive theory of preferences under risk'. These alternative descriptive decision theories are successful in solving a great many paradoxes contradicting the classical EU-Hypothesis. However, in this paper we present a simple decision problem which is a challenge not only to the EU-model but also to the above mentioned refined theories of decision making.
THE DECISION PROBLEM
A simple two choice single outcome-valued decision under risk is presented in the following manner: t Theory and Decision 25 (1988) 107-116. 9 1988 by Kluwer Academic Publishers.
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lAIN PATERSON AND ANDREAS DIEKMANN
A
20
30
40
50
60
I0
B
10
20
30
40
50
60
DI
Subjects are asked to decide between alternatives A and B. Outcomes are equally likely with probability 1/6. After choosing the preferred alternative in problem D1 a 2 point additional bonus is offered to subjects willing to reverse their choice in problem D1. Thus, the compound decision problem for subjects who preferred A in D1 can be reconstructed as follows: A
20
30
40
50
60
I0
B
12
22
32
42
52
62
I)2
whereas subjects with a B-choice in DI are confronted with the analogous decision problem: A
22
32
42
52
62
12
B
I0
20
30
40
50
60
D3
All outcomes are equally likely with probability 1/6. PREDICTION OF CHOICE Clearly, in D1 not only the expected values but also the complete distributions of payoffs are identical for both alternatives A and B. Consequently, EU-Theory, most simple and more elaborate decision principles like Maximin, as well as prospect theory and Hagen's theory (which take into account expected value and higher moments of the distribution) predict indifference between A and B in D1, whereas another well-known hypothesis, Savage's (1954) principle of minimax loss/regret, would predict the choice of B: in the case of the outcome being 'six' alternative A would yield a maximal loss of 50 points. Also, Savage's principle, whereby probabilities of 1/6 are n o t assumed (i.e. a decision under
THE RELATIVE NATURE OF DECISIONS
109
uncertainty), implies that subjects after choosing B should reject the bonus of 2 points for reversing their choice mode in D1. All other theories predict acceptance of the bonus for changing the choice in D1. If subjects are indifferent in the original choice problem D1 an additional bonus for one alternative should make this alternative more attractive and therefore preferable. In contrast to the aforementioned choice principles we postulate a completely different behaviour. We expect that people consistently choose A in DI and even will resist the bonus offered for switching from A to B in D1. The postulated 'rationale' is that subjects compare payoffs for every outcome of the decision. All outcomes are equally likely whether alternative A or alternative B is chosen in DI: but, because in choosing A they are better off in five of the six possible outcomes, A will be preferred. The reasoning might be that you have to regret your choice with a high probability of 5/6 if B is chosen (and the fact that the long term average opportunity-loss of repeated identical decisions is equal for A and B is irrelevant because only one decision is to be made). EXPERIMENTAL RESULTS Eighty-two students of various scientific desciplines agreed to participate in an experiment on gaming and decision making. A questionnaire was formulated which included the decision problem D1. As mentioned, after answering problem DI subjects were asked if they were willing to change their choice in D1 for a 2-point bonus. It was stated that each point is worth 0.10 German marks, i.e. the maximal payoff for this decision problem amounts to DM 6.20 (Actually, after performing the experiment all subjects received the same amount for participating in the experiment). Experimental results are displayed in Table I. First, the experimental results positively indicate that choices were not made on a random, indifferent basis (two-way contingency test against equal cell values of 1/4 of 82 =20.5). Since there is a strong and highly significant group preference (84%) for alternative A in the original decision situation D1, the principle of minimum regret cannot explain the phenomenon. In addition it is unlikely (if possible) that the theories predicting indifference between A and B would produce an outcome of 69 to 13; however the results of the secondary question D2 strongly
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IAIN PATERSON
AND ANDREAS
DIEKMANN
TABLE I
Results of the decision experiment
Cho$ce
A
in DI
B Total
no c h a n g e of decision in DI f o r a 2-point-bonus 42
47
2-point bonus
27
35
Total
69
change cf d e c i s i o n in Dl
for
a
I3
82
contradicts not only the EU-model but also prospect theory, Hagen's extension, and several other decision principles. A significant majority (61%) of subjects choosing A do not alter their decision for the offer of the additional bonus. Also B-subjects are more inclined to revise their choice than subjects choosing A (although statistical significance is not demonstrated with as few as 13 such subjects).
THE RELATIVE NATURE OF DECISIONS
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However, the experimental results support our hypothesis: there is a strong preference among subjects for A over B in D1, strong enough for most of the A subjects to withstand the further inducement in D2. 2 There are also some possible objections to the results, in reference to alternative explanations of the observed decision patterns: (1) Subjects are prejudiced, e.g. in the belief that the probability of landing a 'six' is less than 1/6. (2) Perception error - Subjects think that A dominates B in all outcomes, not just in five out of six. (3) Subjects indifferent between A and B tend to choose the first named alternative, namely A. (This would account for the result of D1 but not for those of D2.) (4) 2 points bonus for switching decision in situation D2 is less than the psychological 'just noticable difference' (which objection renders also EU-theory irrelevant). Such objections can certainly be tested in further experiments designed to control against such effects, by appropriate permutations of the payoffs of the respective distributions.
CONSEQUENCES FOR DECISION THEORY The results are paradoxical to many descriptive and normative approaches to decisions. Some points of contention arising from the results are briefly listed here. (1) The choice of .4 in situation D1 is explainable 'rationally' (the rationale is provided above) in the sense that the reasons for preferring A to B are derived only from the facts stated implicitly in the decision problem and are independent of subjective factors (risk aversion, risk seeking, marginal utility of money) which might otherwise attempt to reconcile observed decision behaviour with EUtheor)/. (2) There is an essential difference between 'one-off' decisions (most situations as perceived by the decision maker) and repeated decisions:
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faced with the repetition of decision D1, say, 100 times one may well be indifferent between the choice of .4 or B - not so when the choice is made only once. Interestingly, frequentist arguments may be used to lend support to choosing .4 in the one-off decision D1: suppose many individuals, say 100, are each faced with the identical situations D1 but are preempted in their choice by being (randomly) allotted .4 or B; after the die is cast in each of the 100 independent trials, it is to be expected that 5/6 of those who were allotted A are satisfied, because they could not have chosen better themselves, whereas only 1/6 of those allotted B are satisfied and 5/6 are dissatisfied. The above argument is not put forward here as a general principle, but it enables a differentiation in this case between the alternatives .4 and B after first inspection reveals that the outcomes are identical, but asymmetric in relation to each other. (3) EU-Theory and its extensions are limited in their application: Consider the alternatives ` 4 - F (with A written twice for clarity) in Table II: TABLE II i
2
3
4
5
6
A
20
30
40
50
60
I0
B
10
20
30
40
50
60
C
60
I0
20
30
40
50
D
50
60
10
20
30
40
E
40
50
60
I0
20
30
•tcome
Alternat~ve~
where
each
F
30
40
50
60
10
20
A
20
30
40
50
60
10
outcome
has
a probability
of
I/6.
Then we would expect that in subsequent decision situations where
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T H E R E L A T I V E N A T U R E OF D E C I S I O N S
only pairwise alternatives are available, analogous to our result A >-B (where '),-' represents 'is preferred to').
A~B,
B~-C,
C)'-D,
D~-E,
E~-F,
F~A
This intransitivity of choices is often regarded as 'irrational', whereas it is a corollary of our rationale in preferring A to B. EU-Theory and its extensions implicitly propose that a real function, e.g. f(A), can be used to evaluate each alternative in a decision under risk, and that the same alternative has the same value f(A) in different decision situations. This notion of an 'absolute' function is contradicted in those cases where intransitivity can be demonstrated: for real values cannot give rise to intransitivity. Such functions fail to evaluate an alternative relative to the other alternatives in a particular decision situation, and hence may be applicable in one, but not applicable in another, decision situation. At best EU-Theory and its extensions are a useful tool for many situations where they are a good approximation to explaining 'rationality', but there are situations where a rational choice is not covered adequately by them. Another situation which could be tested is the simultaneous choice among the six alternatives A-F. As there is now complete symmetry in this situation it is conjectured here that indifference between the choices will be shown. (4) Even the simple example of a decision under risk presented here disguises a multicriteria decision problem. As soon as more than one dimension (for example money-value) is revealed, intransitivities may occur (see Zeleny 1981). Two dimensions (at least) are considered relevant to situation DI. Just as EU-Theory and its extensions (Hagen 1979) rely on the distribution of the random variable 'outcome value' to make a one-dimensional evaluation, so analogously in our approach is the exact joint distribution in the 2-dimensional product space of ordered outcomes of A and B the object of our study. In the particular situation DI both the amount of the monetary values and the relative dominance of the outcomes are taken into account by the decision maker. 3 (5) A clear distinction should be made between interrelated and noninterrelated alternatives. The interrelations between one alternative and others are clearly demonstrated in an outcome/alternative array of payoffs, such as is commonly used to introduce decision theory in
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lAIN PATERSON AND ANDREAS DIEKMANN textbooks. However, many approaches,
in particular EU-theory and
its e x t e n s i o n s , t r e a t a l t e r n a t i v e s as i n d e p e n d e n t dent payoff-distributions,
l o t t e r i e s , ie i n d e p e n -
e v e n w h e n s t r u c t u r a l d e p e n d e n c e is g i v e n ,
a s is t h e c a s e i n s i t u a t i o n D 1 .
NOTES The decision problem was developed by lain Paterson after attending a series of lectures given by Ole Hagen at the Institute for Advanced Studies in Vienna. Andreas Diekmann conducted the experiment in the University of Munich as part of a series of decision situations concerning N-person mixed motive games. This research at the Institute for Sociology was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG). 2 In the questionnaire people were also asked to comment on their original decision. Although not all comments are clearly interpretable there seems to be two types of decision patterns. The relative majority (40 participants) let us know that they are indifferent (however, 9 out of the 40 are inconsistent with respect to actual choice). 20 people state that they prefer A because the payoffs for A are better in most cases than B-payoffs and/or t. the outcome '6' is unhkely tcompared to not being 6 ). A ty,~lcal v., statement is: " I f I choose A there are 5 out of 6 outcomes where i get a higher payoff". These people explicitly decide by the criterion of 'relative dominance'. With one exception these people choose A without switching to B for additional two points. Eight persons believe that outcomes are not equally distributed. There is the belief that extreme events like '1' or '6' are less likely than other numbers. These people, interestingly, also choose A without switching to B for the additional offer, i.e. all 8 behave like those perceiving relative dominance. The other 14 people gave sundry different or no comments, again half of them standing by their original choice of A. One person hints to a possible experimenter bias: "By your explanation of the problem 1 assume you want me to choose A ! " (see Appendix for explanation of the decision problem). 3 Bell (1982) put forward a two-dimensional utility function comprising an outcome-value and a 'regret' component. This function is used to evaluate an alternative relative to one other alternative, and may certainly lead to intransitivities. Whereas this approach is used to explain some paradoxes, it fails to explain the behaviour presented here, because, in a fashion analogous to that of Savage, large amounts of 'regret' lead to disproportionately less utility and hence alternative B would be preferred to A in situation DI. I
APPENDIX:
D
PRECISE TRANSLATION
OF T H E Q U E S T I O N N A I R E
In the following decision situation you have no other fellow players. The amount of points you obtain depends this time only on your decision and t h e r e s u l t o f t h e t h r o w o f a die. A c c o r d i n g t o w h e t h e r y o u d e c i d e o n A or B and according to the face-value of the die-throw,
w h i c h will b e
carried out by us after you have returned your questionnaire, receive the following number
of points:
y o u will
THE RELATIVE NATURE OF DECISIONS
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20
30
40
50
60
i0
10
20
30
40
50
60
If you choose for example, A and we afterwards throw a 'five', you will thus receive 60 points, if you select, e.g., B and we throw a 'three', you will thus get 30 points. Please denote your choice: I choose A .... I choose B .... Please explain your decision briefly:
If you have chosen A: Would you change your choice and select B, if you would in return be given 2 extra points whatever the outcome? Yes, I then choose B .... No, I stay with A .... If you have just chosen B: Would you change your choice and select A, if you would in return be given 2 extra points whatever the outcome? Yes, I then choose A .... No, I stay with B ....
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lAIN PATERSON AND ANDREAS DIEKMANN REFERENCES
Allais, M. and O. Hagen, Ed.: 1979, Expected Utility Hypotheses and the AIlais Paradox. Contemporary Discussions under Uncertainty with Allais Rejoinder. Dordrecht and Boston: Reidel. Bell, D. E.: 1982, 'Regret in decision making under uncertainty', Operations Research 30, 961-981. Elster, J.: 1979, Ulysses and the Sirens, Cambridge: Univ. Press. Hagen, O.: 1979, 'Towards a positive theory of preferences under risk', in: Allais, M. and Hagen, O., Eds., Expected Utility Hypotheses and the Allais Paradox. Contemporary Discussions under Uncertainty with Allais Rejoinder. Dordrecht and Boston: Reidel, 271-302. Hagen, O.: 1986, 'Some paradoxes in economics', in: Diekmann, A. and Mitter, P., Eds., Paradoxical Effects of Social Behaviour / Essays in Honor of Anatol Rapoport, Heidelberg and Vienna 1986: Physica. Holler, M.: 1983, 'Do economics students choose rationally? A research note', Social Science Information 22, 623-630. Kahnemann, D. and Tversky, A.: 1979, 'Prospect theory: an analysis of decision under risk', Econometrica 47, 263-291. Klopstech, A. and Selten, R.: 1984, 'Formale Konzepte eingeschr~,nkt rationalen Verhaltens', in: H. Todt, Ed., Normengeleitetes Verhalten in den Sozialwissenschaften, Berlin: Duncker und Humblot. Savage, L. J.: 1954, The Foundations of Statistics, New York, Wiley. Schoemaker, P. J. H.: 1982, 'The expected utility model: its variants, purposes, evidence and limitations', Journal of Economic Literature 20, 52%563. Zeleny, M.: 1982, Multiple Criteria Decision Making, New York, London: McGraw-Hill.
lain Paterson D e p a r t m e n t o f B u s i n e s s a n d Operational Research I n s t i t u t e o f A d v a n c e d Studies Vienna, A u s t r i a . Andreas Diekmann Zentrum fiir Umfragen, Methoden und Analysen (ZUMA) Mannheim, F.R.G.
