Directory UMM :Data Elmu:jurnal:J-a:Journal of Economic Psychology:Vol21.Issue6.2000:
Journal of Economic Psychology 21 (2000) 625±637
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On complexity and lotteries' evaluation ± three experimental
observations
Galit Mador, Doron Sonsino *, Uri Benzion
Technion, Israel Institute of Technology, 32000 Haifa, Israel
Received 16 July 1999; received in revised form 27 October 1999; accepted 27 October 1999
Abstract
We present experimental evidence suggesting that human subjects penalize lotteries for
complexity. Our results contradict the assumption that human agents follow the discounted
expected utility model in multi-period choice with uncertainty. In particular, we show that the
buying price oered for an inferior, simple multi-period lottery may sometimes signi®cantly
exceed the buying price oered for a better, yet more complicated, alternative, when the
lotteries are sold to a group of subjects in a ®rst-price auction. We discuss the possibility
to modify the existing models of choice to this ``complexity eect''. Ó 2000 Published by
Elsevier Science B.V. All rights reserved.
PsycINFO classi®cation: 2340
JEL classi®cation: C91; D81; D90
Keywords: Decision making; Complexity; Expected utility
*
Corresponding author. Fax: +972-4-823-5194.
E-mail address: sonsino@ie.technion.ac.il (D. Sonsino).
0167-4870/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 7 - 4 8 7 0 ( 0 0 ) 0 0 0 2 3 - 4
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
1. Introduction
The discounted expected utility (DEU) model serves as a basic building
block in modern economic theory. Thaler (1981), Benzion, Rapoport and
Yagil (1989), Loewenstein and Prelec (1991, 1992), Loewenstein and Elster
(1992) and others demonstrate, however, that human subjects may violate the
model's predictions in dierent circumstances. In particular, Thaler (1981),
Loewenstein and Prelec (1992) and others, present experimental evidence
suggesting that hyperbolic discounting suits subjects' behavior better than the
exponential discounting assumed by the DEU theory. Benzion et al. (1989)
demonstrate that the intensity of discounting may depend on the speci®c
characteristics of the lottery under evaluation. Loewenstein and Prelec (1991)
show that subjects may prefer an increasing stream of payos over a decreasing one (which suggests a negative discounting rate).
In this note, we argue that complexity aversion might be another source of
deviation from the DEU-model predictions. In particular, we demonstrate
that the buying price oered for an inferior, simple multi-period lottery may
sometimes exceed the buying price oered for a better, yet more complicated,
alternative, when the lotteries are sold to a group of subjects in a ®rst-price
auction. This suggests that decision makers penalize multi-period lotteries for
complexity: a negative complexity effect in intertemporal choice.
Previous experimental attempts to investigate the impact of complexity on
decision making (see, for example, Wilcox, 1993; Bruce & Johnson, 1996;
Johnson & Bruce, 1998, and the references therein) typically focused on the
complexity of a decision problem, compared to our focus on the complexity
of each alternative under consideration. These previous references investigated the eects of complexity in speci®c applications. 1 In a recent concurrent study, Hucks and Weizacher (1999) show (in the context of singleperiod lotteries) that the frequency of deviation from expected payo maximization in binary choice problems increases with the (maximal) number of
possible prizes. Moreover, the subjects reveal a tendency to choose the less
``complex'' alternative in such cases.
Intuitively, we suspect that complexity should play an especially strong
role in the evaluation of lotteries over streams of payoffs. The current study
1
Bruce and Johnson (1996) and Johnson and Bruce (1998) study the UK horse-race betting market;
Wilcox (1993) focuses on the case where ``simple'' lotteries are the reduced form of ``complicated'' multistage lotteries.
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
627
focuses on such multi-period lotteries. Our results provide strong evidence
for the existence of a negative complexity effect in lotteries' evaluation, as
described above.
2. DEU-theory and complexity
A multi-period lottery L is a tuple (p; X ) where p p1 ; p2 ; . . . ; pm is a
probability vector and X is an m T payoff matrix, so that with probability
pj the T-periods' payoff stream from the lottery will be the one given by the
jth row of the matrix X.
The DEU model assumes the existence of a utility function U : R 7! R and
a periodical discount factor d so that the value assigned to the lottery L is
given by
"
#
m
T
X
X
tÿ1
pi
d U xi;t ;
V L
i1
t1
where xi;t denotes the ith row, tth column element of the matrix X. The
certainty equivalent of lottery L; CE L, is de®ned by V CE L V L. The
discounted expected payoff (DEP) from L is the DEU from that lottery,
when U x x.
The DEU theory postulates that lottery L1 is (weakly) preferred to lottery
L2 if and only if V L1 P V L2 . In this study, we claim however that, in
reality, complexity might play a major role in subjects' evaluation of such
multi-period lotteries. Moreover, complexity aversion might lead to deviations from the DEU-theory predictions.
To deal with complexity in multi-period lotteries' evaluation, one needs an
axiomatic framework or some formal complexity measure that might be used
to rank such lotteries according to their complexity levels. Yet, de®ning such
an axiomatic setup or an ``appropriate'' general complexity measure seems
like a delicate job. Intuitively, it seems ``right'' to assume that ``larger'' lotteries (i.e., lotteries with larger payo matrices) are more complicated to
evaluate than ``smaller'' lotteries. Yet, the size of the payo matrix might be
deceptive since, for example, the ``large'' payo matrix may contain only one
or two dierent payos while the smaller matrix will contain many more
dierent payos. Thus, it seems that the number of dierent elements in the
payo matrix is also relevant for determining the (relative) evaluation-complexity of the lottery. But if this indeed is the case, we should also consider
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
the number of different elements in the probability vector p. For instance, a
lottery that pays 100, 200, 300 and 400 with equal probabilities seems (intuitively) less complicated to evaluate than a lottery that pays the corresponding prizes with probabilities 0.24, 0.31, 0.29, and 0.16. All these
arguments together suggest the following:
Evaluation-complexity assumption *. If L1 p1 ; X1 and L2 p2 ; X2 are two
lotteries such that:
1. The number of rows in X1 is bigger than or equal to the number of rows
in X2 .
2. The number of columns in X1 is bigger than or equal to the number of columns in X2 .
3. The number of dierent elements in the matrix X1 is bigger than or equal
to the number of dierent elements in the matrix X2 ; i.e.,
Card fX1 g P Card fX2 g.
4. The number of dierent elements in the vector p1 is bigger than or equal
to the number of dierent elements in the vector p2 ; i.e.,
Card fp1 g P Card fp2 g,
then, L1 should be judged as more ``complicated to evaluate'' than L2 .
Note however that even this general assumption might seem ``wrong''
in certain circumstances. For instance, if lottery A pays 50 dollars with
probability 0.4, 40 with probability 0.1 and 0 with probability 0.5, while
lottery B pays 51.86 dollars with probability 765/1528 and 0.6565 with
probability 763/1528, the assumption above implies that B is less complicated
than A. Intuitively, this seems ``incorrect''. Thus, the number of digits in the
payos and the probabilities of the lottery (and the way these numbers
are presented) might also aect the perceived evaluation-complexity of the
lottery. This issue however seems irrelevant in the current study as all the
lotteries that we compare in the sequel have a similar structure: all the payos
are integers between 50 and 105 and all the probabilities are presented in
decimal numbers with only 1±3 digits left of the decimal point. We therefore
adopt assumption * for ranking the lotteries in this study according to their
evaluation-complexity levels. We use the expression COMP Li P COMP Lj
when referring to cases where the assumption implies that lottery i is more
complicated to evaluate than lottery j. We use a strict inequality COMP Li >
COMP Lj when one of the conditions in the assumption holds strictly.
In the sequel, we sometimes present a multi-period lottery as a combination of independent single-period lotteries, one lottery for each period (see,
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
629
for example, lottery 9 in Section 4). Clearly, every such multi-period lottery
can be presented in the canonical form described in the beginning of this
section. When lotteries i and j are presented in this ``independent'' form, we
say that lottery i is more complicated than lottery j when assumption * implies that the canonical form of lottery i is more complicated than the canonical form of lottery j.
3. Method
The experiment took place at the Faculty of Industrial Engineering and
Management at the Technion, Israel Institute of Technology. The subjects
were 73 undergraduate students who have recently passed the course ``Introduction to Economics''. The participants were randomly chosen from a
pool of respondents to an advertisement posted in the campus. 2
In the course of the experiment, each subject was asked to evaluate a sequence of lotteries. The lotteries were presented to each subject separately, in
a random order, one lottery at a time (i.e., the evaluation forms for the
``previous'' lotteries were collected from the subjects before submitting the
``next'' evaluation forms in the sequence). We have also used up to six different versions of each lottery with a dierent order of rows in each version. 3
All these measures were taken in order to avoid anchoring type of biases. The
experiment was divided into seven separate (similar) sessions, with 8±12
subjects participating in each session.
In the instructions, the subjects were asked to write down a price oer for
each lottery. Each subject was endowed with 60 New Israeli Shekel (NIS) 4
at the beginning of the experiment. The subjects were told that each lottery
will actually be sold to (and played for) the participant who has oered the
highest price for that lottery (in the corresponding session). The subjects were
also told that the winning buying-price will be deducted from the winner's
realized ``gains'' from the lottery, and that their ®nal check (for participating
in the experiment) will be increasing with the ``dierence between the amount
of payos that they have actually won, and the buying-prices they have oered
for the corresponding lotteries''. The instructions did not specify the exact
2
The average age of the subjects was about 24.5; 65% of the subjects were third-year undergraduate
students. A detailed description of the experiment and the results is provided in Mador (1998).
3
In the case of lottery 9 we could only produce four versions. For lotteries 7 and 8 we had two versions
for each lottery. In the case of lottery 5 we had only one version.
4
Approximately $17.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
formula according to which the ®nal checks will be determined. 5 Subjects
however were promised that their ®nal check will not be lower than 30 NIS.
4. Experimental observations
Observation 4.1 (The negative complexity effect). Consider the following
four lotteries:
Pi
Payment today
Payment in three months
Lottery 1
0.25
0.25
0.25
0.25
50
50
80
80
50
100
50
100
Lottery 2
0.17
0.33
0.17
0.33
55
50
76
80
50
100
50
100
Lottery 3
0.125
0.085
0.29
0.085
0.125
0.29
50
55
50
76
80
80
50
50
100
50
50
100
Lottery 4
0.13
0.14
0.21
0.17
0.17
0.18
50
52
51
80
82
81
51
53
52
101
102
103
5
In practice, we have used a 0.9 periodical discount factor to discount future payos. We have
calculated the present value of the realized payos for each subject, subtracted the corresponding buying
prices and added the initial endowment (60 NIS) to get a ``®nal payo score'' for every subject. The total
budget of the experiment ($1500) was divided among the subjects in proportion to these ®nal ``scores''.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
Lotteries 2, 3 and 4 were constructed out of lottery 1 so that:
1. The DEU of each lottery is higher than the DEU of lottery 1 for every concave utility function U and discount factor d.
In particular, note that U 50 U 80 < U 55 U 76 for every strictly concave monotone utility function U. This implies that the DEU of
lotteries 2 and 3 is higher than the DEU of lottery 1 whenever U is strictly
concave and monotonic. Note also that the distribution of payoffs in each
period in lottery 4 ®rst-order stochastically dominates the corresponding
distribution of payoffs in lottery 1. This immediately implies that the DEU of
lottery 4 is higher than the DEU of lottery 1 for any monotone utility
function U.
2. COMP L4 > COMP L3 > COMP L2 > COMP L1 , by assumption *.
3. The DEP from the four lotteries are very close (independently of the
discount factor d). The DEP at a periodical discount factor d 0:9 is
presented in the ®rst row of Table 1.
Fact 1 above implies that DEU maximizers should prefer each of the
lotteries 2, 3 and 4 to lottery 1. If this indeed is the case, we expect the price
oers for lotteries 2, 3 and 4 to be higher than the price oers for lottery 1. 6
Many of our subjects, however, have violated this prediction by oering a
signi®cantly lower buying price for some of the lotteries 2, 3 or 4. The third
row of Table 1 discloses the percentage of such violations for each of the
lotteries 2, 3 and 4. Note that the percentage of violations signi®cantly increases with the complexity of the DEU-superior alternative: only 28% of the
subjects oer a lower price for lottery 2 (than for lottery 1); 42% of the
subjects oer a lower price for lottery 3; 62% of the subjects oer a lower
price for lottery 4. Signed-tests suggest that the dierences are signi®cant at
Table 1
Lotteries comparison
Lottery
L1
L2
L3
L4
DEP at (d 0:9)
COMP-ranking
Violations of DEU (%)
Average buying price
132.5
Lowest
)
91.6
132.7
Second
28
110.6
136.2
Third
42
98.8
136.8
Highest
62
84.6
6
This argument implicitly assumes that the bidding function of the subjects (in the ®rst-price auction
conducted in the experiment) is monotonically increasing, so that the buying-price oers increase with the
discounted expected utility of the underlying lotteries.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
p 6 0:01. Clearly, this behavior is in line with our negative complexity eect
conjecture.
The last row in Table 1 gives the average buying price oered for each of
the four lotteries 1±4. The average price oered for lottery 2 is higher than
the average price oered for lottery 3 which in turn is higher than the average price oered for lottery 4. This provides additional support to our
negative complexity eect conjecture. In particular, note that the average
price oer for lottery 4 (84.6) is signi®cantly lower (t 2:6; p 6 0:02) than
the average price oer for lottery 1 (91.6), in spite of the fact that the
distribution of payos in each period in lottery 4 ®rst-order stochastically
dominates the corresponding distribution in lottery 1. The fact that the
average prices oered for lotteries 2 and 3 were higher than the average
prices oered for lottery 1 might be explained by the higher DEP of the
former lotteries (relatively to the DEP of lottery 1). These higher values
have pushed most subjects to oer a higher buying price for these lotteries
(compared to the buying price oered for lottery 1) in spite of their increased complexity (see Table 1). In the case of lottery 4, however, the
negative complexity eect was stronger (on average) than the ``positive''
DEP eect.
Finally note that since the payo distributions in lottery 4 ®rst-order
stochastically dominate the payo distributions in lottery 1 (for both periods
``today'' and ``in three months'') the observed contradictions to the DEU
theory cannot be resolved by any ``generalized'' model that satis®es the
dominance axiom. In particular, consider
P the case where the value of lottery
L; V 0 L, takes the form V 0 L Tt1 w tV Lt , where Lt denotes the
marginal distribution of payos for date t; w t the discount factor for period
t and V is an evaluation model for single-period lotteries that satis®es the
dominance axiom; e.g., the rank-dependent utility model. The contradictions
to the DEU theory observed in comparing the price oers for lottery 4 and
the price oers for lottery 1 cannot be resolved by any such model.
Observation 4.2 (Inconsistent evaluations). Lottery 4 was presented to all
subjects in the early stages of the experiment. 7 The individual price oers for
that lottery were then used to construct an individually tailored degenerated
``lottery'' as follows:
7
As mentioned above, the lotteries were presented to the subjects in a random order. However,
whenever required, we have changed the order to guarantee that lottery 4 will appear before lottery 5.
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
·
·
633
Lottery 5
100 dollars, today.
The price that was oered for lottery 4 (by that individual), in three
months.
We have also oered the subjects the next lottery that pays 105 dollars
today and replicates lottery 4 in three months.
·
·
Lottery 6
105 dollars, today.
The following lottery, in three months:
Pi
Payment in three months
Payment in six months
0.13
0.14
0.21
0.17
0.17
0.18
50
52
51
80
82
81
51
53
52
101
102
103
Assuming that the price oered by each subject for lottery 4 was lower
than the CE of that lottery for the subject, 8 the DEU theory implies that the
subjects should prefer lottery 6 to 5. In practice, 45% of our subjects have
oered a higher bid for lottery 5. 9 Again, we suggest that this follows from
the complexity eect. Lottery 6 is more complicated and the subjects penalize
it for its complexity. 10
Observation 4.3 (Sub-additivity of lotteries value). Along the experiment, the
subjects were asked to oer a buying price for each of the following simple
lotteries:
8
The experimental literature indeed shows that subjects discount their value when placing bids in ®rstprice auctions, see for example Kagel and Levin (1993).
9
The average bidding price for lottery 5 however was 149 NIS while the average bidding price for
lottery 6 was 164.8, a statistically signi®cant dierence at p 6 0:1.
10
Formally, one may present lottery 6 in the canonical form that was used in Section 2 by using a 6 3
payo matrix where the ®rst-period payo is always 105. Similarly, the degenerated lottery 5 may be
presented in the canonical form by using a 1 2 payo matrix. Assumption * then implies that lottery 6 is
more complicated than each of the lotteries 4 and 5.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
Lottery 7
Pi
Payment today
0.5
0.5
50
80
Lottery 8
Pi
Payment in three months
0.5
0.5
50
100
(Note that lottery 7 pays its prizes ``today'' while lottery 8 pays its prizes
``in three months''.)
The subjects were also asked to price another lottery (lottery 9) that was
composed of these two (independent) lotteries; i.e., lottery 7 that determines
the payo ``today'' and lottery 8 that describes the payo ``in 3 months''.
Lottery 9
Pi
Payment today
Pi
Payment in three
months
0.5
0.5
50
80
0.5
0.5
50
100
The DEU theory clearly predicts that the value assigned to lottery 9 should
equal the sum of the values assigned to lotteries 7 and 8; i.e., V L9
V L7 V L8 . Assuming that subjects are strictly risk averse, this implies
that CE L9 P CE L7 CE L8 . This in turn suggests that the bidding price
oered for lottery 9 should exceed the sum of bidding prices oered for
lotteries 7 and 8. 11 Yet, 70% of our subjects have violated this prediction. On
average, our subjects have oered 53.3 dollars for lottery 7, 55.4 dollars for
lottery 8 and only 91 dollars for lottery 9. We suggest that the increase in
complexity resulting from the composition of lotteries 7 and 8 into the larger
lottery 9 has caused this phenomenon.
11
This indeed is the case if the subjects discount the value of each lottery by some constant factor to
derive their bidding price.
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
635
Note also that the observed low average bidding price for lottery 7 (bidding price of 53.3 for a lottery with an expected payo of 65) indeed suggests
that our subjects are risk averse in their bidding behavior. The observed low
average bidding price for lottery 8 (an average bidding price of 55.4 for a
lottery with an expected payo of 75 in three months) suggests that in addition to risk aversion the subjects discount future payos quite heavily.
5. Discussion and future research
The results above suggest that complexity might have a negative eect on
multi-period lotteries' valuation. In particular, one may think of two basic
ways through which complexity can aect the evaluation of such lotteries.
Firstly, complexity per se may induce disutility; i.e., decrease the (average)
value of the lottery. Secondly, complexity might increase the noise in the
evaluation process. One may then come up with generalized evaluation
models that take these two eects into account. For instance, one may suggest a generalized discounted expected utility model, where the value assigned
to a given lottery L may be presented as follows:
V L n COMP L DEU L COMP L;
where n : R 7! 0; 1 is a monotonically decreasing function describing the
complexity eect, DEU L measures the discounted expected utility from the
lottery L and is some ``white noise'' (as, for example, in Carbone, 1997) with
a distribution that depends on the complexity of the underlying lottery. In
particular, it might be the case that the variance of COMP L takes the
form k 2 COMP L2 . It is easy to verify that this model may be used to explain (qualitatively) the experimental results described above. 12 Alternatively, one can take a dierent evaluation model (see, for example,
Loewenstein & Prelec, 1992), where the value assigned to a lottery L is V L
and modify it accordingly to claim that the actual value, V 0 L, is given by
V 0 L n COMP V L COMP L.
In our future work, we plan to extend the investigation to the case where
the subjects have to choose between dierent lotteries (choice-framing). It
12
More speci®cally, it is possible to adopt a numeric complexity measure e.g., (COMP L Nu of rows
in X Nu of columns in X Card fX g Card fpg, impose speci®c structure on the utility function U
(e.g., U x xa ) and the complexity function n (e.g., n y y b and derive the corresponding maximum
likelihood estimators (as, say, in Camerer & Ho, 1994). The number of problems investigated in this note
however is too small to apply these techniques.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
seems also interesting to examine the complexity eect more carefully (for
instance, try to check which one of the two explanations outlined above (if
any) plays a stronger role in inducing the complexity eect) and to think
more carefully about complexity measures in multi-period choice with uncertainty.
Finally note that the sample examined in this study consisted of relatively
sophisticated students that have previously attended at least one comprehensive introductory class in Economics. The fact that these subjects consistently violated the predictions of DEU theory makes us suspect that
similar violations would occur with other less sophisticated subjects as
well. 13 We therefore speculate that complexity aversion plays a signi®cant
role in everyday decision making in general. The conjecture that our qualitative results carry over to the case where the sample is less educated, however, has to be formally examined to establish this claim.
Acknowledgements
We thank David Budescu, Ido Erev, Uri Gneezy, Dov Monderer, Rosemarie Nagel, Mark Pingle, Abdulkarim Sadrieh, Lanna Volokh and Peter
Wakker for important comments and suggestions. We also thank the fund
for the promotion of research at the Technion for its ®nancial support.
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13
For a recent example of a comparative experimental study where sophisticated subjects' behavior is
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www.elsevier.com/locate/joep
On complexity and lotteries' evaluation ± three experimental
observations
Galit Mador, Doron Sonsino *, Uri Benzion
Technion, Israel Institute of Technology, 32000 Haifa, Israel
Received 16 July 1999; received in revised form 27 October 1999; accepted 27 October 1999
Abstract
We present experimental evidence suggesting that human subjects penalize lotteries for
complexity. Our results contradict the assumption that human agents follow the discounted
expected utility model in multi-period choice with uncertainty. In particular, we show that the
buying price oered for an inferior, simple multi-period lottery may sometimes signi®cantly
exceed the buying price oered for a better, yet more complicated, alternative, when the
lotteries are sold to a group of subjects in a ®rst-price auction. We discuss the possibility
to modify the existing models of choice to this ``complexity eect''. Ó 2000 Published by
Elsevier Science B.V. All rights reserved.
PsycINFO classi®cation: 2340
JEL classi®cation: C91; D81; D90
Keywords: Decision making; Complexity; Expected utility
*
Corresponding author. Fax: +972-4-823-5194.
E-mail address: sonsino@ie.technion.ac.il (D. Sonsino).
0167-4870/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. All rights reserved.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
1. Introduction
The discounted expected utility (DEU) model serves as a basic building
block in modern economic theory. Thaler (1981), Benzion, Rapoport and
Yagil (1989), Loewenstein and Prelec (1991, 1992), Loewenstein and Elster
(1992) and others demonstrate, however, that human subjects may violate the
model's predictions in dierent circumstances. In particular, Thaler (1981),
Loewenstein and Prelec (1992) and others, present experimental evidence
suggesting that hyperbolic discounting suits subjects' behavior better than the
exponential discounting assumed by the DEU theory. Benzion et al. (1989)
demonstrate that the intensity of discounting may depend on the speci®c
characteristics of the lottery under evaluation. Loewenstein and Prelec (1991)
show that subjects may prefer an increasing stream of payos over a decreasing one (which suggests a negative discounting rate).
In this note, we argue that complexity aversion might be another source of
deviation from the DEU-model predictions. In particular, we demonstrate
that the buying price oered for an inferior, simple multi-period lottery may
sometimes exceed the buying price oered for a better, yet more complicated,
alternative, when the lotteries are sold to a group of subjects in a ®rst-price
auction. This suggests that decision makers penalize multi-period lotteries for
complexity: a negative complexity effect in intertemporal choice.
Previous experimental attempts to investigate the impact of complexity on
decision making (see, for example, Wilcox, 1993; Bruce & Johnson, 1996;
Johnson & Bruce, 1998, and the references therein) typically focused on the
complexity of a decision problem, compared to our focus on the complexity
of each alternative under consideration. These previous references investigated the eects of complexity in speci®c applications. 1 In a recent concurrent study, Hucks and Weizacher (1999) show (in the context of singleperiod lotteries) that the frequency of deviation from expected payo maximization in binary choice problems increases with the (maximal) number of
possible prizes. Moreover, the subjects reveal a tendency to choose the less
``complex'' alternative in such cases.
Intuitively, we suspect that complexity should play an especially strong
role in the evaluation of lotteries over streams of payoffs. The current study
1
Bruce and Johnson (1996) and Johnson and Bruce (1998) study the UK horse-race betting market;
Wilcox (1993) focuses on the case where ``simple'' lotteries are the reduced form of ``complicated'' multistage lotteries.
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
627
focuses on such multi-period lotteries. Our results provide strong evidence
for the existence of a negative complexity effect in lotteries' evaluation, as
described above.
2. DEU-theory and complexity
A multi-period lottery L is a tuple (p; X ) where p p1 ; p2 ; . . . ; pm is a
probability vector and X is an m T payoff matrix, so that with probability
pj the T-periods' payoff stream from the lottery will be the one given by the
jth row of the matrix X.
The DEU model assumes the existence of a utility function U : R 7! R and
a periodical discount factor d so that the value assigned to the lottery L is
given by
"
#
m
T
X
X
tÿ1
pi
d U xi;t ;
V L
i1
t1
where xi;t denotes the ith row, tth column element of the matrix X. The
certainty equivalent of lottery L; CE L, is de®ned by V CE L V L. The
discounted expected payoff (DEP) from L is the DEU from that lottery,
when U x x.
The DEU theory postulates that lottery L1 is (weakly) preferred to lottery
L2 if and only if V L1 P V L2 . In this study, we claim however that, in
reality, complexity might play a major role in subjects' evaluation of such
multi-period lotteries. Moreover, complexity aversion might lead to deviations from the DEU-theory predictions.
To deal with complexity in multi-period lotteries' evaluation, one needs an
axiomatic framework or some formal complexity measure that might be used
to rank such lotteries according to their complexity levels. Yet, de®ning such
an axiomatic setup or an ``appropriate'' general complexity measure seems
like a delicate job. Intuitively, it seems ``right'' to assume that ``larger'' lotteries (i.e., lotteries with larger payo matrices) are more complicated to
evaluate than ``smaller'' lotteries. Yet, the size of the payo matrix might be
deceptive since, for example, the ``large'' payo matrix may contain only one
or two dierent payos while the smaller matrix will contain many more
dierent payos. Thus, it seems that the number of dierent elements in the
payo matrix is also relevant for determining the (relative) evaluation-complexity of the lottery. But if this indeed is the case, we should also consider
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
the number of different elements in the probability vector p. For instance, a
lottery that pays 100, 200, 300 and 400 with equal probabilities seems (intuitively) less complicated to evaluate than a lottery that pays the corresponding prizes with probabilities 0.24, 0.31, 0.29, and 0.16. All these
arguments together suggest the following:
Evaluation-complexity assumption *. If L1 p1 ; X1 and L2 p2 ; X2 are two
lotteries such that:
1. The number of rows in X1 is bigger than or equal to the number of rows
in X2 .
2. The number of columns in X1 is bigger than or equal to the number of columns in X2 .
3. The number of dierent elements in the matrix X1 is bigger than or equal
to the number of dierent elements in the matrix X2 ; i.e.,
Card fX1 g P Card fX2 g.
4. The number of dierent elements in the vector p1 is bigger than or equal
to the number of dierent elements in the vector p2 ; i.e.,
Card fp1 g P Card fp2 g,
then, L1 should be judged as more ``complicated to evaluate'' than L2 .
Note however that even this general assumption might seem ``wrong''
in certain circumstances. For instance, if lottery A pays 50 dollars with
probability 0.4, 40 with probability 0.1 and 0 with probability 0.5, while
lottery B pays 51.86 dollars with probability 765/1528 and 0.6565 with
probability 763/1528, the assumption above implies that B is less complicated
than A. Intuitively, this seems ``incorrect''. Thus, the number of digits in the
payos and the probabilities of the lottery (and the way these numbers
are presented) might also aect the perceived evaluation-complexity of the
lottery. This issue however seems irrelevant in the current study as all the
lotteries that we compare in the sequel have a similar structure: all the payos
are integers between 50 and 105 and all the probabilities are presented in
decimal numbers with only 1±3 digits left of the decimal point. We therefore
adopt assumption * for ranking the lotteries in this study according to their
evaluation-complexity levels. We use the expression COMP Li P COMP Lj
when referring to cases where the assumption implies that lottery i is more
complicated to evaluate than lottery j. We use a strict inequality COMP Li >
COMP Lj when one of the conditions in the assumption holds strictly.
In the sequel, we sometimes present a multi-period lottery as a combination of independent single-period lotteries, one lottery for each period (see,
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
629
for example, lottery 9 in Section 4). Clearly, every such multi-period lottery
can be presented in the canonical form described in the beginning of this
section. When lotteries i and j are presented in this ``independent'' form, we
say that lottery i is more complicated than lottery j when assumption * implies that the canonical form of lottery i is more complicated than the canonical form of lottery j.
3. Method
The experiment took place at the Faculty of Industrial Engineering and
Management at the Technion, Israel Institute of Technology. The subjects
were 73 undergraduate students who have recently passed the course ``Introduction to Economics''. The participants were randomly chosen from a
pool of respondents to an advertisement posted in the campus. 2
In the course of the experiment, each subject was asked to evaluate a sequence of lotteries. The lotteries were presented to each subject separately, in
a random order, one lottery at a time (i.e., the evaluation forms for the
``previous'' lotteries were collected from the subjects before submitting the
``next'' evaluation forms in the sequence). We have also used up to six different versions of each lottery with a dierent order of rows in each version. 3
All these measures were taken in order to avoid anchoring type of biases. The
experiment was divided into seven separate (similar) sessions, with 8±12
subjects participating in each session.
In the instructions, the subjects were asked to write down a price oer for
each lottery. Each subject was endowed with 60 New Israeli Shekel (NIS) 4
at the beginning of the experiment. The subjects were told that each lottery
will actually be sold to (and played for) the participant who has oered the
highest price for that lottery (in the corresponding session). The subjects were
also told that the winning buying-price will be deducted from the winner's
realized ``gains'' from the lottery, and that their ®nal check (for participating
in the experiment) will be increasing with the ``dierence between the amount
of payos that they have actually won, and the buying-prices they have oered
for the corresponding lotteries''. The instructions did not specify the exact
2
The average age of the subjects was about 24.5; 65% of the subjects were third-year undergraduate
students. A detailed description of the experiment and the results is provided in Mador (1998).
3
In the case of lottery 9 we could only produce four versions. For lotteries 7 and 8 we had two versions
for each lottery. In the case of lottery 5 we had only one version.
4
Approximately $17.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
formula according to which the ®nal checks will be determined. 5 Subjects
however were promised that their ®nal check will not be lower than 30 NIS.
4. Experimental observations
Observation 4.1 (The negative complexity effect). Consider the following
four lotteries:
Pi
Payment today
Payment in three months
Lottery 1
0.25
0.25
0.25
0.25
50
50
80
80
50
100
50
100
Lottery 2
0.17
0.33
0.17
0.33
55
50
76
80
50
100
50
100
Lottery 3
0.125
0.085
0.29
0.085
0.125
0.29
50
55
50
76
80
80
50
50
100
50
50
100
Lottery 4
0.13
0.14
0.21
0.17
0.17
0.18
50
52
51
80
82
81
51
53
52
101
102
103
5
In practice, we have used a 0.9 periodical discount factor to discount future payos. We have
calculated the present value of the realized payos for each subject, subtracted the corresponding buying
prices and added the initial endowment (60 NIS) to get a ``®nal payo score'' for every subject. The total
budget of the experiment ($1500) was divided among the subjects in proportion to these ®nal ``scores''.
631
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
Lotteries 2, 3 and 4 were constructed out of lottery 1 so that:
1. The DEU of each lottery is higher than the DEU of lottery 1 for every concave utility function U and discount factor d.
In particular, note that U 50 U 80 < U 55 U 76 for every strictly concave monotone utility function U. This implies that the DEU of
lotteries 2 and 3 is higher than the DEU of lottery 1 whenever U is strictly
concave and monotonic. Note also that the distribution of payoffs in each
period in lottery 4 ®rst-order stochastically dominates the corresponding
distribution of payoffs in lottery 1. This immediately implies that the DEU of
lottery 4 is higher than the DEU of lottery 1 for any monotone utility
function U.
2. COMP L4 > COMP L3 > COMP L2 > COMP L1 , by assumption *.
3. The DEP from the four lotteries are very close (independently of the
discount factor d). The DEP at a periodical discount factor d 0:9 is
presented in the ®rst row of Table 1.
Fact 1 above implies that DEU maximizers should prefer each of the
lotteries 2, 3 and 4 to lottery 1. If this indeed is the case, we expect the price
oers for lotteries 2, 3 and 4 to be higher than the price oers for lottery 1. 6
Many of our subjects, however, have violated this prediction by oering a
signi®cantly lower buying price for some of the lotteries 2, 3 or 4. The third
row of Table 1 discloses the percentage of such violations for each of the
lotteries 2, 3 and 4. Note that the percentage of violations signi®cantly increases with the complexity of the DEU-superior alternative: only 28% of the
subjects oer a lower price for lottery 2 (than for lottery 1); 42% of the
subjects oer a lower price for lottery 3; 62% of the subjects oer a lower
price for lottery 4. Signed-tests suggest that the dierences are signi®cant at
Table 1
Lotteries comparison
Lottery
L1
L2
L3
L4
DEP at (d 0:9)
COMP-ranking
Violations of DEU (%)
Average buying price
132.5
Lowest
)
91.6
132.7
Second
28
110.6
136.2
Third
42
98.8
136.8
Highest
62
84.6
6
This argument implicitly assumes that the bidding function of the subjects (in the ®rst-price auction
conducted in the experiment) is monotonically increasing, so that the buying-price oers increase with the
discounted expected utility of the underlying lotteries.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
p 6 0:01. Clearly, this behavior is in line with our negative complexity eect
conjecture.
The last row in Table 1 gives the average buying price oered for each of
the four lotteries 1±4. The average price oered for lottery 2 is higher than
the average price oered for lottery 3 which in turn is higher than the average price oered for lottery 4. This provides additional support to our
negative complexity eect conjecture. In particular, note that the average
price oer for lottery 4 (84.6) is signi®cantly lower (t 2:6; p 6 0:02) than
the average price oer for lottery 1 (91.6), in spite of the fact that the
distribution of payos in each period in lottery 4 ®rst-order stochastically
dominates the corresponding distribution in lottery 1. The fact that the
average prices oered for lotteries 2 and 3 were higher than the average
prices oered for lottery 1 might be explained by the higher DEP of the
former lotteries (relatively to the DEP of lottery 1). These higher values
have pushed most subjects to oer a higher buying price for these lotteries
(compared to the buying price oered for lottery 1) in spite of their increased complexity (see Table 1). In the case of lottery 4, however, the
negative complexity eect was stronger (on average) than the ``positive''
DEP eect.
Finally note that since the payo distributions in lottery 4 ®rst-order
stochastically dominate the payo distributions in lottery 1 (for both periods
``today'' and ``in three months'') the observed contradictions to the DEU
theory cannot be resolved by any ``generalized'' model that satis®es the
dominance axiom. In particular, consider
P the case where the value of lottery
L; V 0 L, takes the form V 0 L Tt1 w tV Lt , where Lt denotes the
marginal distribution of payos for date t; w t the discount factor for period
t and V is an evaluation model for single-period lotteries that satis®es the
dominance axiom; e.g., the rank-dependent utility model. The contradictions
to the DEU theory observed in comparing the price oers for lottery 4 and
the price oers for lottery 1 cannot be resolved by any such model.
Observation 4.2 (Inconsistent evaluations). Lottery 4 was presented to all
subjects in the early stages of the experiment. 7 The individual price oers for
that lottery were then used to construct an individually tailored degenerated
``lottery'' as follows:
7
As mentioned above, the lotteries were presented to the subjects in a random order. However,
whenever required, we have changed the order to guarantee that lottery 4 will appear before lottery 5.
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
·
·
633
Lottery 5
100 dollars, today.
The price that was oered for lottery 4 (by that individual), in three
months.
We have also oered the subjects the next lottery that pays 105 dollars
today and replicates lottery 4 in three months.
·
·
Lottery 6
105 dollars, today.
The following lottery, in three months:
Pi
Payment in three months
Payment in six months
0.13
0.14
0.21
0.17
0.17
0.18
50
52
51
80
82
81
51
53
52
101
102
103
Assuming that the price oered by each subject for lottery 4 was lower
than the CE of that lottery for the subject, 8 the DEU theory implies that the
subjects should prefer lottery 6 to 5. In practice, 45% of our subjects have
oered a higher bid for lottery 5. 9 Again, we suggest that this follows from
the complexity eect. Lottery 6 is more complicated and the subjects penalize
it for its complexity. 10
Observation 4.3 (Sub-additivity of lotteries value). Along the experiment, the
subjects were asked to oer a buying price for each of the following simple
lotteries:
8
The experimental literature indeed shows that subjects discount their value when placing bids in ®rstprice auctions, see for example Kagel and Levin (1993).
9
The average bidding price for lottery 5 however was 149 NIS while the average bidding price for
lottery 6 was 164.8, a statistically signi®cant dierence at p 6 0:1.
10
Formally, one may present lottery 6 in the canonical form that was used in Section 2 by using a 6 3
payo matrix where the ®rst-period payo is always 105. Similarly, the degenerated lottery 5 may be
presented in the canonical form by using a 1 2 payo matrix. Assumption * then implies that lottery 6 is
more complicated than each of the lotteries 4 and 5.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
Lottery 7
Pi
Payment today
0.5
0.5
50
80
Lottery 8
Pi
Payment in three months
0.5
0.5
50
100
(Note that lottery 7 pays its prizes ``today'' while lottery 8 pays its prizes
``in three months''.)
The subjects were also asked to price another lottery (lottery 9) that was
composed of these two (independent) lotteries; i.e., lottery 7 that determines
the payo ``today'' and lottery 8 that describes the payo ``in 3 months''.
Lottery 9
Pi
Payment today
Pi
Payment in three
months
0.5
0.5
50
80
0.5
0.5
50
100
The DEU theory clearly predicts that the value assigned to lottery 9 should
equal the sum of the values assigned to lotteries 7 and 8; i.e., V L9
V L7 V L8 . Assuming that subjects are strictly risk averse, this implies
that CE L9 P CE L7 CE L8 . This in turn suggests that the bidding price
oered for lottery 9 should exceed the sum of bidding prices oered for
lotteries 7 and 8. 11 Yet, 70% of our subjects have violated this prediction. On
average, our subjects have oered 53.3 dollars for lottery 7, 55.4 dollars for
lottery 8 and only 91 dollars for lottery 9. We suggest that the increase in
complexity resulting from the composition of lotteries 7 and 8 into the larger
lottery 9 has caused this phenomenon.
11
This indeed is the case if the subjects discount the value of each lottery by some constant factor to
derive their bidding price.
G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
635
Note also that the observed low average bidding price for lottery 7 (bidding price of 53.3 for a lottery with an expected payo of 65) indeed suggests
that our subjects are risk averse in their bidding behavior. The observed low
average bidding price for lottery 8 (an average bidding price of 55.4 for a
lottery with an expected payo of 75 in three months) suggests that in addition to risk aversion the subjects discount future payos quite heavily.
5. Discussion and future research
The results above suggest that complexity might have a negative eect on
multi-period lotteries' valuation. In particular, one may think of two basic
ways through which complexity can aect the evaluation of such lotteries.
Firstly, complexity per se may induce disutility; i.e., decrease the (average)
value of the lottery. Secondly, complexity might increase the noise in the
evaluation process. One may then come up with generalized evaluation
models that take these two eects into account. For instance, one may suggest a generalized discounted expected utility model, where the value assigned
to a given lottery L may be presented as follows:
V L n COMP L DEU L COMP L;
where n : R 7! 0; 1 is a monotonically decreasing function describing the
complexity eect, DEU L measures the discounted expected utility from the
lottery L and is some ``white noise'' (as, for example, in Carbone, 1997) with
a distribution that depends on the complexity of the underlying lottery. In
particular, it might be the case that the variance of COMP L takes the
form k 2 COMP L2 . It is easy to verify that this model may be used to explain (qualitatively) the experimental results described above. 12 Alternatively, one can take a dierent evaluation model (see, for example,
Loewenstein & Prelec, 1992), where the value assigned to a lottery L is V L
and modify it accordingly to claim that the actual value, V 0 L, is given by
V 0 L n COMP V L COMP L.
In our future work, we plan to extend the investigation to the case where
the subjects have to choose between dierent lotteries (choice-framing). It
12
More speci®cally, it is possible to adopt a numeric complexity measure e.g., (COMP L Nu of rows
in X Nu of columns in X Card fX g Card fpg, impose speci®c structure on the utility function U
(e.g., U x xa ) and the complexity function n (e.g., n y y b and derive the corresponding maximum
likelihood estimators (as, say, in Camerer & Ho, 1994). The number of problems investigated in this note
however is too small to apply these techniques.
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G. Mador et al. / Journal of Economic Psychology 21 (2000) 625±637
seems also interesting to examine the complexity eect more carefully (for
instance, try to check which one of the two explanations outlined above (if
any) plays a stronger role in inducing the complexity eect) and to think
more carefully about complexity measures in multi-period choice with uncertainty.
Finally note that the sample examined in this study consisted of relatively
sophisticated students that have previously attended at least one comprehensive introductory class in Economics. The fact that these subjects consistently violated the predictions of DEU theory makes us suspect that
similar violations would occur with other less sophisticated subjects as
well. 13 We therefore speculate that complexity aversion plays a signi®cant
role in everyday decision making in general. The conjecture that our qualitative results carry over to the case where the sample is less educated, however, has to be formally examined to establish this claim.
Acknowledgements
We thank David Budescu, Ido Erev, Uri Gneezy, Dov Monderer, Rosemarie Nagel, Mark Pingle, Abdulkarim Sadrieh, Lanna Volokh and Peter
Wakker for important comments and suggestions. We also thank the fund
for the promotion of research at the Technion for its ®nancial support.
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