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
www.elsevier.com/locate/joep
*Corresponding author. Fax: +972-4-823-5194.
E-mail address:sonsino@ie.technion.ac.il (D. Sonsino).
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 de-creasing 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: anegative 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 investi-gated the eects of complexity in speci®c applications.1 In a recent con-current study, Hucks and Weizacher (1999) show (in the context of single-period lotteries) that the frequency of deviation from expected payo max-imization 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 overstreams of payoffs. The current study
1Bruce and Johnson (1996) and Johnson and Bruce (1998) study the UK horse-race betting market;
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 andXis anmT payoff matrix, so that with probability pj theT-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 functionU :R7!Rand
a periodical discount factor d so that the value assigned to the lottery L is given by
where xi;t denotes the ith row, tth column element of the matrix X. The
certainty equivalent of lotteryL;CE L, is de®ned byV CE L V L. The discounted expected payoff (DEP) from L is the DEU from that lottery, whenU x x.
The DEU theory postulates that lotteryL1 is (weakly) preferred to lottery L2 if and only if V L1PV 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 devia-tions from the DEU-theory predicdevia-tions.
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'' lot-teries (i.e., lotlot-teries 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)
the number of different elements in the probability vectorp. For instance, a lottery that pays 100, 200, 300 and 400 with equal probabilities seems (in-tuitively) less complicated to evaluate than a lottery that pays the corre-sponding prizes with probabilities 0.24, 0.31, 0.29, and 0.16. All these arguments together suggest the following:
Evaluation-complexity assumption *.IfL1 p1;X1andL2 p2;X2are two lotteries such that:
1. The number of rows inX1 is bigger than or equal to the number of rows inX2.
2. The number of columns inX1is bigger than or equal to the number of col-umns inX2.
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 fX1gPCard fX2g.
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 fp1gPCard fp2g,
then, L1 should be judged as more ``complicated to evaluate'' thanL2.
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 LiPCOMP Lj
when referring to cases where the assumption implies that lotteryi is more complicated to evaluate than lotteryj. We use a strict inequality COMP Li>
COMP Ljwhen one of the conditions in the assumption holds strictly.
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 lotteriesi and jare presented in this ``independent'' form, we say that lottery i is more complicated than lottery j when assumption * im-plies that the canonical form of lottery i is more complicated than the ca-nonical form of lotteryj.
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 ``In-troduction 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 se-quence 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 dif-ferent 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
2The 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).
3In 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.
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 50 50
0.25 50 100
0.25 80 50
0.25 80 100
Lottery 2
0.17 55 50
0.33 50 100
0.17 76 50
0.33 80 100
Lottery 3
0.125 50 50
0.085 55 50
0.29 50 100
0.085 76 50
0.125 80 50
0.29 80 100
Lottery 4
0.13 50 51
0.14 52 53
0.21 51 52
0.17 80 101
0.17 82 102
0.18 81 103
5In practice, we have used a 0.9 periodical discount factor to discount future payos. We have
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 con-cave utility functionUand discount factor d.
In particular, note that U 50 U 80<U 55 U 76 for every strict-ly concave monotone utility function U. This implies that the DEU of lotteries 2 and 3 is higher than the DEU of lottery 1 whenever Uis 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 functionU.
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 d0:9 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 in-creases 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 (d0:9) 132.5 132.7 136.2 136.8
COMP-ranking Lowest Second Third Highest
Violations of DEU (%) ) 28 42 62
Average buying price 91.6 110.6 98.8 84.6
6This argument implicitly assumes that the bidding function of the subjects (in the ®rst-price auction
p60:01. 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 av-erage 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 (t2:6; p60:02) 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 in-creased 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 the case where the value of lottery L; V0 L, takes the form V0 L PT
t1 w tV Lt, where Lt denotes the
marginal distribution of payos for datet; w tthe discount factor for period tandV 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. 7The individual price oers for that lottery were then used to construct an individually tailored degenerated ``lottery'' as follows:
7As mentioned above, the lotteries were presented to the subjects in a random order. However,
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:
Assuming that the price oered by each subject for lottery 4 was lower than the CE of that lottery for the subject,8the 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:
Pi Payment in three months Payment in six months
0.13 50 51
0.14 52 53
0.21 51 52
0.17 80 101
0.17 82 102
0.18 81 103
8
The experimental literature indeed shows that subjects discount their value when placing bids in ®rst-price 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 atp60:1.
10Formally, one may present lottery 6 in the canonical form that was used in Section 2 by using a 6
3
Lottery 7
Lottery 8
(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
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 L9PCE 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 8.11Yet, 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.
Pi Payment today
0.5 50
0.5 80
Pi Payment in three months
0.5 50
0.5 100
Pi Payment today Pi Payment in three
months
0.5 50 0.5 50
0.5 80 0.5 100
11This indeed is the case if the subjects discount the value of each lottery by some constant factor to
Note also that the observed low average bidding price for lottery 7 (bid-ding 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 ad-dition 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 sug-gest a generalized discounted expected utility model, where the value assigned to a given lottery Lmay be presented as follows:
V L n COMP L DEU L COMP L;
where n:R7! 0;1 is a monotonically decreasing function describing the
complexity eect, DEU Lmeasures the discounted expected utility from the lotteryLandis 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 k2COMP L2. It is easy to verify that this model may be used to ex-plain (qualitatively) the experimental results described above.12 Alterna-tively, one can take a dierent evaluation model (see, for example, Loewenstein & Prelec, 1992), where the value assigned to a lotteryLisV L
and modify it accordingly to claim that the actual value, V0 L, is given by
V0 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
12More speci®cally, it is possible to adopt a numeric complexity measure e.g., (COMP
L Nu of rows inXNu of columns inXCard fXg Card fpg, impose speci®c structure on the utility functionU
(e.g.,U x xa) and the complexity functionn(e.g.,n y yband derive the corresponding maximum
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 un-certainty.
Finally note that the sample examined in this study consisted of relatively sophisticated students that have previously attended at least one compre-hensive introductory class in Economics. The fact that these subjects con-sistently 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 quali-tative results carry over to the case where the sample is less educated, how-ever, has to be formally examined to establish this claim.
Acknowledgements
We thank David Budescu, Ido Erev, Uri Gneezy, Dov Monderer, Ro-semarie 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.
References
Benzion, U., Rapoport, A., & Yagil, J. (1989). Discount rates inferred from decisions: An experimental study.Management Science,35, 270±284.
Bruce, A. C., & Johnson, J. E. (1996). Decision making under risk: Eect of complexity on performance.
Psychological Report,79, 67±76.
Camerer, C. F., & Ho, T. H. (1994). Violations of the betweenness axiom and nonlinearity in probability.
Journal of Risk and Uncertainty,8, 167±196.
Carbone, E. (1997). Investigation of stochastic preference theory using experimental data. Economics Letters,57, 305±311.
Duy, J., & Nagel, R. (1997). On the robustness of behavior in experimental beauty contest games.
Economic Journal,107, 1684±1700.
Johnson, J. E. V., & Bruce, A. C. (1998). Risk-strategy under task complexity: A multivariate analysis of behavior in a naturalistic setting.Journal of Behavioral Decision Making,11(1), 17.
13For a recent example of a comparative experimental study where sophisticated subjects' behavior is
Hucks, S., & Weizacher, G. (1999). Risk, complexity and deviations from EV maximization-results of a lottery choice experiment.Journal of Economic Psychology,20, 699±715.
Kagel, J. H., & Levin, D. (1993). Independent private value auctions: bidder behavior in ®rst-, second-, and third-price auctions with varying numbers of bidders.Economic Journal,103, 868±879. Loewenstein, G., & Elster, J. (1992).Choice over time. New York: Russel Sage Foundation Press. Loewenstein, G., & Prelec, D. (1991). Negative time preferences.American Economic Review,80, 347±352. Loewenstein, G., & Prelec, D. (1992). Anomalies in intertemporal choice: Evidence and an interpretation.
Quarterly Journal of Economics,107, 573±597.
Mador, G. (1998). Complexity and uncertainty in choice making. Thesis. Technion, Israel Institute of Technology, Haifa.
Thaler, D. (1981). Some empirical evidence on dynamic inconsistency.Economics Letters,8, 201±207. Wilcox, N. T. (1993). Lottery choice: Incentives, complexity, and decision time.Economic Journal,103,
