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A,A,A,A,A or A,A,B,C,D? Over-Diversification in Repeated Decision Problems*
Ariel Rubinstein** School of Economics Tel Aviv University and Department of Economics Princeton University
*The work on this project has started in 1993 by Amos Tversky and myself. The death of Amos has caused the long delay in completing this work. I can just hope that Amos would not disapprove mentioning his contribution to the project were he reading the paper. My gratitude to Dana Heller and Michal Ilan, who were the research assistants in the project in its two phases, Dana in 1994 and Michal in 1999. Thanks to Tzachi Gilboa and George Loewenstein and Yossi Spiegel who commented on a draft of this paper.
**e-mail:
[email protected] url: http://www.princeton.edu/~ariel
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1.
Introduction
The starting point of this project was an observation made by Amos Tversky and myself during the analysis of experiments we had conducted for a very different project (see Rubinstein, Tversky and Heller (1996)). During the pilot stage of that research, we offered the subjects sequences of similar games, listed one after the other. In each game, a subject had to guess the site where an opponent had hid a treasure. The labels of the sites appeared in a row. The problems differ only in the labels given to the different sites (e.g., A,B,A,A; 1,2,3,4; or three smileys and one sad face). An action in each game was a choice of a site. It seemed that subjects employed rules to play the game (such as avoiding edges, choosing the salient label, and avoiding the salient label). Our impression was that subjects diversified the rules they used during the sequence of games they were asked to play. Namely, after employing a certain rule in one or two games, they switched to a different rule when playing the third game. This phenomenon introduced strong noise into our data. There was no clear rationale for this diversification. We concluded that it was an expression of a more general phenomenon: When people face a sequence of similar decision problems and are uncertain about the right action, they tend to diversify! This observation prompted us to investigate, in 1993, the issue of overdiversification by means of a sequence of class experiments. Amos Tversky died in 1995, and I was reluctant to continue our joint research for a long time. Only in 1998 was the work resumed. Though all the experiments reported here were conducted in 1998/9, the seeds of the ideas investigated are taken from my joint work with Amos.
The paper deals with repeated decision problems which are similar to the task of guessing the color outcomes of five independent spinnings of a roulette wheel, 60% of whose slots are covered in red and 40% in white, where each correct guess yields a prize of $1. The guess of 5 Reds clearly first order stochastic dominates any other strategy. (Note that if a decision maker chooses White for, let us say, the third outcome, he will just increase the probability that he will get the third dollar from 0.4 to 0.6 if he changes this choice to red.)
In each experiment the subject’s mission is to make five choices, each of which is a choice element of a fixed set of alternatives. The consequence of each of the five choices may be either "success" or "failure," according to the realization of a random factor. The rewards and the
Page 3 structure of the random factors imply that the best strategy for the decision maker is to choose the action which is most likely leading to success five times.
The results confirm the hypothesis that subjects over-diversify their choices when facing a repeated decision problem in situations where the choice is between lotteries with clear objective probabilities. The over-diversification is more intense when the subjects face uncertainty without objective probabilities, and weaker when the choice is stated as a choice problem which involves real-life actions.
The results are quite strong when considering the fact that all the subjects in this study were either undergraduates majoring in or graduate (MA) students of economics at Tel Aviv University. All the students had completed at least a basic course in statistics and the majority had taken a course in microeconomics or even game theory.
2.
Methods
As stated, the experimental work for this paper was conducted by surveying students in Tel Aviv University, all of whom were students of economics. Subjects were asked to imagine that they face a decision problem and were requested to write what they would do in such a situation. Some of the experiments were conducted via the Internet as part of the course assignments in an introductory course to game theory (see Rubinstein (1999) for a description and assessment of that method). The majority of surveys were conducted in class, where students were asked to fill out a form (the relevant form was actually sandwiched between forms for two totally different experiments). No monetary rewards were offered.
I am aware of the reluctance of some experimentalists to use an experimental method with no real payments made. I am also aware of the fact that classroom experiments suffer from the presence of subjects who do not take the experiment seriously. (By the way, the same can apply in real life.) I do not think that this method is perfect. However, I do not believe that asking students to enter into the world of a "short story" is less likely to make them think seriously about the situation than they would if paid a few dollars and play an artificial game. I do not believe that our research can, in any case, do more than demonstrate the existence of certain patterns of behavior. The methods cannot provide meaningful assessments of the frequencies of patterns of behavior in real-life.
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3.
Strong over-diversification in lottery-type decision problems
A very strong tendency to over-diversify was observed in repeated decision problems where the prospects of making the right decision were specified in objective probability terms and the choice was framed clearly as a choice between "lotteries".
Students participating in the experiments were asked to answer the following question (in Hebrew): Problem 1: "Guess the cards' colors" Imagine that you are participating in the following game: Five cards are chosen randomly from a stack of 100 cards; 36 of them Green, 25 Blue, 22 Yellow and 17 Brown. The five cards are then placed into five separate envelopes marked A,B,C,D,E. You have to guess (by marking an X in the matrix below) the color of the card in each envelope. Also, imagine that you will receive a prize for each correct guess (you can get up to five prizes). (Here the subject is presented with a 5x4 matrix, where each row corresponds to one guess).
In two separate undergraduate classes (of 50 and 74 students, respectively), only 42% and 38% of the students, respectively, chose five Greens. A noticeable alternative is the "probability matching" mode (a choice of two Greens and one from each other color). About 30% of the subjects (52% of the subjects who diversify in this experiment) chose the "probability matching" strategy, that is they diversify by choosing a mixture of actions in proportions similar to the probabilities of success (2 Greens, 1 Blue, 1 Yellow and 1 Brown).
I also conducted a modified version of Problem 1 experiment where it was added that one of the five envelopes had already been selected and that the subject would win a prize if and only if his guess regarding the envelope that had been picked is correct. I expected that this modification would increase significantly the proportion of "five Greens" answers. The reason for this choice, so I thought, is that many subjects go through the following reasoning stage: (i) construct a "representative event" and (ii) respond optimally to this typical event. While the representative event for Problem 1 is "the colors in the five envelopes are two greens, one blue, one yellow and one brown", the representative event here seems to be "the card in the envelope is Green".
In one MA economics class (using the Internet) a large proportion of students indeed chose "straight Gs" (61% of 49 participants; 33% of the subjects in that class used the "matching
Page 5 probabilities " strategy). But in another class of 46 undergraduates, 43% chose "5 Greens", only a bit more than the proportion of "5 Greens" answers for Problem 1. Thus, though I still maintain the above conjecture, much more research is needed to confirm the hypothesis.
4.
Extreme over-diversification in problems with subjective probabilities
When the subjects did not receive explicit information about the chances of success of the different alternatives, almost all of them over-diversified. Namely, they did not follow the rational rule of "assess the chances of each of the actions to succeed and take a straight choice of the most likely action".
The following problem was posed, via the Internet, to two undergraduate game theory classes in consecutive years: Problem 2: "Students majors" We have received a list of all the 3rd year students (approximately 250 students) taking a double major in economics and another subject. We have selected five of them completely randomly. Guess the second major of the five students. You will receive a lottery ticket for each "hit" (up to five tickets). One of the tickets will award the holder a coupon of 50 shekels to purchase books.
Near each of the guesses, a "window" was opened and the subject had to choose from among the eight most popular second majors, and an option "other" (see http://www.princeton.edu/~ariel/99/failures2_q.html ).
In the 1998 class, 28% of the total of 65 students in the class responded with five identical majors and in 1999, only 7% of the 41 student subjects made five identical selections. Thus, the vast majority of subjects over-diversified.
5. Weak over-diversification in task-type problems Once the problem was phrased to the subjects as a practical choice problem of a plan of a sequence of five actions, the over-diversification almost completely disappeared. Subjects were asked:
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Problem 3: "Caught the messenger " Imagine you are a detective at a shopping center. You know that every day, at noon, a messenger arrives with an envelope. The identity of the messenger is random; he is one of dozens of messengers who work for a delivery company. The shopping center has four gates although you have only one video camera, which you have to install each morning in one of the four gates. Your aim is to take photos of the maximum number of messengers as they enter the shopping center.
You have to choose a plan as to where to install the camera every morning. You have in hand the results of a reliable survey which provides statistics about the rates of messengers coming to each arrival gate: 36% of them enter through the Green gate, 25% use the Blue gate, 22% the Yellow and 17% come through the Brown gate.
Your plan: Day:
Sun
Mon
Tue
Wed
Thu
Gate:
___
___
___
___
___
In a class of 50 undergraduates, 70% selected a string of five Greens. When the problem was modified so that the messenger would arrive at the gate only once, the percentage of "five Greens" was retained: 72% (in a group numbering 46).
Comment: In Problem 2, the subjects faced subjective probability, a state which enhances overdiversification. In Problem 3, the problem was practical, with objective probabilities and the over-diversification almost disappeared. The question then raised is: What happens if the problem is "practical" and the uncertainty is not specified objectively? Hence, undergraduates in economics were asked the following question (stated originally in Hebrew): Problem 4: "Shirt color choice" Imagine that you are working for a company and that your manager asks you to purchase a present, an elegant shirt, for each of five men who provide services to your company. All five are Tel Aviv-type yuppies, 30-40 years old. The shirts are available in two colors: Burgundy and Black. Their tastes are unknown to you. Your task is to make them happy. Thus, you want to send each of them a shirt with a color that he would choose were he making the selection himself.
Page 7 You estimate that _____% of Tel Aviv’s yuppie population aged 30-40 would choose a Burgundy shirt and ___% would choose a Black shirt. Your choice of colors for the five people is: A)____ B)____ C)____ D____ E)___
Of the 48 students who responded to this question, only 8% chose the same color for all five. Since I asked the subjects to estimate the frequencies of the Burgundy and Black shirt-lovers, we can learn more precisely the alternative "strategies" used by the subjects. Apparently, 80% of the subjects who did diversify, did so in proportions approaching (+/- 10%) the estimated probabilities, an outcome which is similar to the "probability matching" strategy.
6. Over-Diversification and Over-Randomization I can think of two reasons for comparing the above results with the behavior of subjects in onestage decision problems. First, the results can provide a baseline to compare the frequencies of over-diversifiers found in the population. Second, it is interesting to compare the overdiversification phenomenon with its "twin", over-randomization in a one-stage decision problem.
The following question was given to a group of 56 third-year students in a course in game theory (via the Internet): Problem 5: "Meet your friend " The city mall has four gates. According to a statistical survey, visitors enter the mall through each gate in the following proportions:
Gate %
North East
South West
21
32
27
20
You want to meet your friend at the mall but you do not know through which gate he is going to enter. You can wait for him at only one gate. What are you going to do? (Choose one of the two options and fill in the appropriate details.) •
I will wait at Gate______
•
I will choose the gate randomly according to the following probabilities:
North: ___%; East: ___%; South: ___%; West: ___%
Page 8 About 69% of the subjects said that they would wait at the South gate. Almost all the others (28%) chose the gates randomly, according to the given probabilities. Thus, in spite of the fact that the framing of the question could pressure students to use a "mixed strategy", about 70% of the subjects made the "optimal choice". One can not expect a higher percentage of "correct answers" in a repeated choice problem.
The following similar question was given to a group of students in 1999. Problem 5*: "Arrest the suspect " You are a police officer wishing to arrest a suspect in a crime. The suspect is about to enter a mall through one of four gates. A statistical survey has revealed that the proportion of people who enters the mall through each of the four gates is as follows:
Gate
Blue
Green Red
Yellow
%
21
27
20
32
You have only one policeman to assign to one of the gates, according to the outcome of a roulette. You should assign the proportions of the roulette to each gate. What will you do?
I will assign the proportion of the outcomes of the roulette spins as follows: Blue: ___%; Green: ___%; Red: ___%; Yellow: ___%
In problem 5*, the pressure on the subjects to choose non-degenerate probabilities was stronger than in Problem 5 as subjects were forced to divide the 100% among the four colors. (Of course, they could still choose a "pure action" by assigning 100% to one option.) This time (i.e., Problem 5*, 1999), only 33% out of the 43 participants assigned the full 100% to the most likely gate. The rest randomized: 30% chose the matching probabilities and 14% attached equal probabilities to the four gates. It is quite plausible that the over-randomization in Problem 5* was triggered by the fact that the subjects perceived the situation as a zero-sum game (in which the subject wants to escape from the police officer) and not as a decision problem.
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7. Discussion The main conclusion of this paper is that individuals who plan a "repeated decision" very often over-diversify their choices relative to what would is suggested by any mode of behavior which is consistent with first order stochastic domination. This tendency is more extreme (about 90% of my subjects) when the uncertainty is framed without objective probabilities. The overdiversification is reduced (to about 60%) when the choice is between "lotteries" specified with objective probabilities. The proportion of over-diversifiers is further diminished to about 30% when the subjects have to make a plan of a sequence of actions. Most of the subjects who do not behave "optimally" follow the "probability matching" strategy.
As far as I can ascertain, the source of the over-diversification is a systematic mistake that appears with a frequency, which depends on the way it is phrased. It is my experience that after I explained the situation to the subjects, they did realize that they had made a mistake. The behavior reported in the paper is, in this respect, similar to nontransitivity, which is considered by subjects as a mistake once we point out a cyclical choice. However, the fact that a choice reveals a systematic mistake is not uninteresting, especially when its appearance relates to some characteristics of the problem.
Two literatures are related to the current paper. The closest paper to the current one is, to my knowledge, Read and Loewenstein (1995) (see also Simonson (1990)). Read and Loewenstein found that when subjects have to select, in advance, between products A and B for a string of days, they diversify their choices much more than if they had to do so daily, in a sequence of days, but just for one day at a time. The current study diverges from that literature in some basic ways. First, here, there is always a unique rational answer. This fact prevents the need to challenge the results with the simple conjecture that “individuals diversify because they prefer to diversify”. A second difference is that the current study allows us to compare the frequencies of the responses to the rational choice problem with another prominent strategy, "probability matching".
In Loomes (1998), subjects were asked to place 40 balls, 20 Greens and 20 Whites, into two bags, A and B, so that the number of balls in each bag will equal 20. They were told that one of the bags will be randomly selected with unequal probabilities (bag A will be
Page 10 chosen with probability 0.65) and that one of the 20 balls in that bag will be picked. If the chosen ball is green, the subject will get a prize. Apparently, less than 15% of the subjects put all 20 green balls in bag A. Instead, the vast majority of subjects put 12-14 green balls in bag A. If one thinks about their choice as a selection of a bag for each of the 20 green balls, then this problem is similar to the modification of problem 1, where one of the five envelopes is chosen and the subject gets the prize according to his selection of that envelope. A second related literature investigates decision makers’ strategies in repeated decision problems when the probabilities of "success" are fixed but unknown to the decision maker. This literature actually deals with learning. (For a survey of the literature, see Myers (1976) and Vulkan (1998); see also Shanks and McCarthy (1999)). That literature also identifies two strategies: (1)
Diversify in the first few stages and eventually "converge" to constantly choosing what
you perceive as the right decision; and (2)
The "probability matching" strategy, where subjects "learn" the probabilities and continue
to diversify over time.
Taking the sequential choice of "lotteries" (with objective probabilities) as a benchmark, we seek for two explanations for the situations examined: (1) Why do we observe a stronger tendency to diversify when the uncertainty is not quantified? One explanation follows Read and Loewenstein (1995). When the uncertainty is vague, people have a strong instinct to "seek information". Over-diversification here, although it does not provide information, may be a residual of this instinct to diversify for the sake of learning about the environment in which individuals and groups operate. (2) Why do we observe a weaker tendency to diversify in problems involving a choice of a string of actions? One explanation is that once we talk about plans of action, the instinct of preferring a simple course of action begins to work in favor of a constant strategy.
The emergence of the "probability matching" strategy as a "rival" to the optimal strategy may be related to the use of the following reasoning scheme. Facing uncertainty, people construct an event which represents a typical outcome of the random factors involved, after which they optimize given that event. As we know from the literature on the perception of randomness (see Kahneman and Tversky (1974)), "People expect that a sequence of events generated by a random
Page 11 process will represent the essential characteristics of that process even when the sequence is short". People find the typical random sequence of five realizations of uncertainty not to be the one which is the most likely string of events but one which matches the given probabilities (see also Bar-Hillel (1991)).
As always, the fact that we understand a bit better those circumstances where over-diversification appears is interesting but still only the first step in the construction of economic models that explain phenomena different from those explained by standard decision theories. This is indeed a challenge.
Page 12 References Bar-Hillel, M. (1991), "The Perception of Randomness", Advances in Applied Mathematics, 12, 428-454.
Kahaneman, D. and A.Tversky (1974) "Judgment under Uncertainty: Heuristics and Biases," Science, 185, 1124-1131.
Loomes, G. (1998), "Probabilities vs. Money: A Test of Some Fundamental Assumptions About Rational Decision Making", The Economic Journal, 108, 477-489.
Myers J.L. (1976) "Probability learning and sequence learning", in W.K.Estes (eds.) Handbook of learning and cognitive processes: Approaches to human learning and motivation 3, 171-205, Erlbaum.
Read, D. and G. Loewenstein (1995), "Diversification Bias: Explaining the Discrepancy in Variety seeking Between Combined and separated Choices", Journal of Experimental Psychology: Applied, 1, 34-49.
Rubinstein, A. (1999) "Experience from a Course in Game Theory: Pre and Post-class Problem Sets as a Didactic Device", Games and Economic Behavior, 28, 155-170
(see a "Second
Edition" on line at: http://www.princeton.edu/~ariel/99/gt100.html)
Rubinstein, A., A. Tversky and D. Heller (1996) "Naive Strategies in Zero-sum Games" in Understanding Strategic Interaction - Essays in Honor of Reinhard Selten, W.Guth et al. (editors), Springer-Verlag, 394-402.
Shanks, D. and J. McCarthy (1999) "A re-examination of Probability Matching and rational Choice", University College of London, mimeo.
Simonson, L. (1990) "The Effect of Purchase Quantity and Timing on Variety Seeking behavior", Journal of Marketing Research 32, 150-162.
Vulkan, N. (1998) "An Economist's Perspective on Probability Matching", mimeo.
