Dual process theories: A key for understanding the diversification bias?

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1 J Risk Uncertainty (2007) 34: DOI /s Dual process theories: A key for understanding the diversification bias? Christoph Kogler & Anton Kühberger Published online: 8 March 2007 # Springer Science + Business Media, LLC 2007 Abstract The diversification bias in repeated lotteries is the finding that a majority of participants fail to select the option offering the highest probability. This phenomenon is systematic and immune to classical manipulations (e.g. monetary rewards). We apply dual process theories and argue that the diversification bias is a consequence of System 1 (automatic, intuitive, associative) triggering a matching response, which fails to be corrected by System 2 (intentional, analytic, rational). Empirically, supporting the corrective functions of System 2 through appropriate contextual cues (describing the task as a statistical test rather than as a lottery) led to a decrease of diversification. Keywords Dual process theories. Diversification. Probability matching. Statistical independence JEL Classification D83. D81 Repetitions of similar risky choices are characteristic for everyday life, e.g. in betting, health decisions or when using seatbelts. Some of these repeated choices are practically identical, and assuming that preferences are relatively stable over time, it seems easy to formulate an optimal decision rule: find the best option when the choice comes up for the first time, and then stick to this option in all subsequent instances. Thus, if you believe that using a seatbelt is a good choice for your first trip, you should use it on any trip. As simple as this rule is, people do nevertheless violate it. For instance, Baron et al. (1993) reported that, although adolescent subjects believe that using seatbelts is beneficial, some failed to see that they then should use them on every single trip. Rather many respondents held the view that they should use seatbelts most, but not all of the time, to match the fact that there are C. Kogler: A. Kühberger (*) Department of Psychology, University of Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria anton.kuehberger@sbg.ac.at url:

2 146 J Risk Uncertainty (2007) 34: rare situations where this can do harm. Researchers named the tendency to choose differently in identical choice situations diversification. More specifically, when confronted with a choice between alternatives that have different rates of occurrence, experimental participants have been found to proportion their choices in accord with the relative expected rates. This phenomenon called probability matching has been demonstrated years ago in animals (see Herrnstein s 1961 classic demonstration of the matching law; and, for an overview, Gallistel 1990, chapter 11) and with humans (Fantino and Esfandiari 2002). In recent research, probability matching has been found in multiple decision problems with constant probabilities. Gal and Baron (1996) asked participants to predict the outcome of rolling a die with four red and two green faces repeatedly. Uniform prediction of red maximizes the chances of predicting correctly, but 37% of the participants failed to do so and the majority of them did not know that the best strategy was to constantly predict red, when explicitly asked about strategies. Similar results are reported by Loomes (1998), who identified probability matching as the dominant strategy in multiple choice problems with constant probabilities. That is, rather than predicting the most likely outcome for every single event of a series, the majority of people matched predicted outcomes to frequencies. For example, if the frequency of red is four out of six, and the frequency of green is two out of six, people tend to produce predictions of about 67% red and 33% green in the long run. A similar finding is reported by Rubinstein (2002): in his task five cards were chosen randomly from a deck of 100 cards (36 green, 25 blue, 22 yellow, and 17 brown). Then the cards were put into five separate envelopes and participants had to guess the color of the card in each envelope. Since p(green)=0.36 is the highest probability for any color, 1 green should be predicted for each envelope. However, the majority of participants failed to do so, but rather matched predictions to probabilities by predicting a mixture of colors that resembled their frequencies, for instance by predicting green, blue, yellow, brown, and again green. This probability matching strategy is very popular in multiple choice problems with constant probabilities (for an overview, see Vulkan 2000), and has been extensively documented in the experimental literature. Probability matching has been found with gains and losses (Myers et al. 1963; Suppes and Atkinson 1960), with asymmetric payoffs and different distributions of probabilities (Estes and Straughan 1954), and with monetary incentives (Healy and Kubovy 1981; Kogler and Kühberger 2006). Yet probability matching also occurs in more natural settings, representing more realistic everyday dilemmas (Fisher and Mazur 1997). Even if people correctly perceive the rate of items, they prefer to follow the probability matching strategy and thus fail to adopt the rule that maximizes their prospects of guessing correctly. Why do people prefer to adopt the probability matching strategy instead of the maximization rule? One can only speculate, but it seems that the explanation is partly grounded in the preference for a strategy that can guarantee full success (although with very little probability) over a strategy that would yield a relatively large number of correct guesses, but would inevitably lead to some errors (Arkes, 1 Even without replacement p(green) is maximal, since there are 11 green cards more than cards of any other color and only five draws. Although strict independence is not applicable to our task (which is without replacement), we will use this notion in its loose sense, of quasi-independence.

3 J Risk Uncertainty (2007) 34: Dawes and Christensen 1986). Some scholars have argued that probability matching might actually be a rational strategy. For instance, in a competitive environment with multiple agents and constant payoff, opting for the less frequent event can be efficient. If most subjects choose the more frequent event, this payoff will be distributed among many. In contrast, choosing the less frequent event delivers the whole payoff, undivided, when it materializes (Gallistel 1990). Other researchers suggest that this type of diversification is a systematic mistake: Gal and Baron (1996) found that failure to rank different outcome patterns correctly in the die problem was strongly connected to a misunderstanding of the concept of statistical independence. Similarly Kogler and Kühberger (2006) found that correct ranking of different outcome patterns according to probability was strongly associated with application of the independence rule. The present research delves more deeply into the reasons for diversification and probability matching. We will use the card task developed by Rubinstein (2002) where participants have to predict the color of five cards randomly drawn from a deck of 100 cards (36 green, 25 blue, 22 yellow, and 17 red). We shall apply dual cognitive process theories to this task. Dual process theories propose the presence of two distinct thinking systems referring to collections of cognitive processes that differ in content, speed and control (see Epstein 1994; Sloman 1996; Stanovich and West 2000; Kahneman and Frederick 2002; Kahneman 2003). In general, System 1 works mainly on affective and concrete inputs and is rapid, automatic, and associative in nature, while System 2 mainly works on affectively neutral, abstract inputs in a slow, controlled, and deductive way. In typical decisions, the response is assumed to depend on both systems in a predetermined way: System 1 gives the preliminary, intuitive answer, which may or may not be corrected by System 2 (Gilbert 1989). We propose that the matching tendency is the result of System 1 processing. On presentation of the differently colored cards System 1 produces a prediction of the plausible outcome of five consecutive draws. This prediction is based on the similarity between sample and outcome. Since the drawing is from a sample of different colors, the outcome is similar to the degree that it also consists of different colors. The probability matching pattern is the pattern that is qualitatively most similar to the sample. Note that, according to Kahneman and Frederick (2002), similarity is an attribute that is assessed routinely and automatically, and forms the basis for a judgment by representativeness. In detail, prediction by similarity goes like this: There are cards of four different colors: green (36), blue (25), yellow (22), and red (17). Since the outcome needs to be similar to the sample, the first four cards will be predicted to be green, blue, yellow, and red. What will be the outcome of the fifth draw? Most probably, it will be the most likely color, that is green, because green is the most frequent color. Note, however, the last prediction cannot be based on similarity, but is based on a theoretical insight that frequency predicts probability. This chain of reasoning leads to a prediction of green, green, blue, yellow, and red (GGBYR), which we will term probability matching, since it is the pattern closest to probability matching, and exact probability matching is not possible in this task. Thus, a GGBYR pattern is, irrespective of the temporal order of predictions, indicative of a judgment by similarity, plus a judgment by probability. The former is, in all likelihood, an intuitive judgement, based on System 1 thinking, while the latter is based on System 2 thinking. We therefore have a mixture of cognitive processes,

4 148 J Risk Uncertainty (2007) 34: leading to probability matching, where System 2 thinking only joins in at the penultimate stage. Of course, System 2 thinking can also inform predictions from the outset. If so, and if people understand that probability predicts frequency, they will realize that green is the most likely color in every singles draw and therefore they will predict GGGGG. Actually, Kogler and Kühberger (2006) have found that most people do understand that probability predicts frequency in this task. When people had to evaluate the likelihood of some predetermined patterns of drawing, they realized that GGGGB is more likely than GGGGR, because blue is more frequent than red. Note, however: at the same time they failed to see that GGGGG, i.e., four greens and one green, is more likely than GGGGB, because green is more frequent than blue. In sum, two processes seem to interact in this task. We construe these processes along the System 1 and System 2 distinction. What is the nature of this interaction? Gilbert (1999) distinguishes four architectures for the interaction of System 1 and System 2: (1) Selective architecture. The two different systems are activated by different stimuli. Thus, people do one thing at one occasion and do another thing at another occasion, because on all occasions two qualitatively distinct processes are possible, one of which is active and one of which is dormant. The (2) Competitive architecture is similar, with the exception that one process controls the output and the other does not. Thus, the noncontrolling process is active but ultimately ineffective. Both these architectures are not plausible in our case, since the data indicate both automatic as well as controlled processes having an effect on the output. (3) Consolidative architecture. Any stimulus activates both processes, and the result is a joint function of both processes. In our task such an architecture is involved, but presumably it is of the type of (4) Corrective architecture. Here any stimulus activates both processes, but the result is determined by only one process exclusively (e.g. winner takes all). Following Gilbert (1999) and Kahneman (2003), we assume that people will adhere to the maximization rule rather than do probability matching to the degree that System 2 succeeds in correcting the output of System 1, provided that System 1 produces an output. Thus, if we want people to overcome their tendency to rely on probability matching, we either have to arrange the task in a way that renders the similarity feature unavailable, or we have to strengthen the corrective functions of System 2 thinking. The following experiment is an attempt at doing the latter. 1 Experiment The following experiment is based on the card task of Rubinstein (2002) and on findings of Kogler and Kühberger (2006), where we showed that people have a strong tendency to emit the probability matching response in this task. Specifically, we found that only about 5% of participants predicted according to probability theory (i.e., they uniformly predicted the most frequent color). This pattern, which is termed probability maximizing, was clearly less common than the probability matching pattern, which was found in about 35% of participants (the rest being patterns resembling probability matching). In different versions of the task we investigated the robustness of probability matching and found that probability matching was robust across stimuli (symbols and colors) and incentives (real and hypothetical payoffs). In

5 J Risk Uncertainty (2007) 34: sum, participants withstood our aim to raise the percentage of probability maximizers at least up to the level of probability matchers (Kogler 2006). The following experiment is another try at accomplishing this. 1.1 Method Participants Ninety-seven undergraduates (78 females and 19 males; mean age=22.6 years) from six introductory statistics courses at the University of Salzburg participated voluntarily. Participants had basic knowledge about probabilities and statistical independence Design and procedure The experiment was designed to support the corrective functions of System 2. Kahneman (2003) suggests three procedures to strengthen System 2: (1) provide strong cues to the relevant rules; (2) increase the vigilance of the monitoring activities; and (3) extensive training in statistical reasoning. We implemented (1) and (2), keeping (3) constant. Participants were instructed on the card-task. They learned that five cards were to be randomly drawn from a deck of 100 colored cards and that each of the five cards had to be put into a separate envelope. The deck of cards consisted of 36 green, 25 blue, 22 yellow, and 17 red cards. Thus, each of the envelopes labelled 1 to 5 was to contain either a green, a blue, a yellow, or a red card. The participants task was to predict the color of the card in each particular envelope. Two different groups were formed and instructed according to Kahneman s(2003) advice to strengthen System 2. First, we provided cues to the relevant rules. This was done by referring to the task as a lottery task (group A), or as a statistical test (group B). In addition, participants in Group B were informed that the task was a test to find out about their level of statistical competence, and participants were encouraged to show the best of their ability in this test, presumably increasing the vigilance of the monitoring activities. Furthermore we advised them to take their time (low time pressure), and to carefully reconsider a second time their predictions (hint at correction). Generally speaking, participants in both groups received the same task, but the instruction for Group B represents an attempt to support the corrective functions of System 2. Thus, in accordance with dual process theories, less probability matching is predicted for Group B compared to Group A. 2 Results and discussion Table 1 presents the results in terms of groupings of response patterns: probability maximizing (GGGGG); probability matching (GGBYR); and other predictions. We found a higher rate of probability-maximizing predictions in the condition aiming at supporting the corrective functions of System 2. Whereas in the control group only 15% of participants repeatedly opted for the most frequent color, 43% of participants did so in the experimental condition. This difference is significant

6 150 J Risk Uncertainty (2007) 34: Table 1 Frequencies (percentages in parenthesis) of basic response patterns Probability-maximizing Probability matching Other Control (N=46) 7 (15.2%) 28 (60.9%) 11 (23.9%) Corrective a (N=51) 22 (43.1%) 19 (37.3%) 10 (19.6%) Total (N=97) 29 (29.9%) 47 (48.5%) 21 (21.6%) a Instruction to support corrective functions of System 2. (χ 2 (1)=9.00; p<0.01). This goes to show that the rate of maximizing predictions increased via arranging the setting in a way to enhance System 2 thinking as proposed in the literature of dual process theories. As can be seen in Table 1, this increase was at the cost of matching choices, which decreased from 61% to 37% while other choices remained largely at their level (about 20%). Note that our experiment enables a very specific test of the dual processes idea. As we already discussed, the matching pattern involves both systems of thinking in a sequential way: first a judgment by similarity (System 1) leads to a prediction of different colors; second, an evaluation of frequency (System 2) leads to a prediction that green (the most frequent color) will appear two times, given that there are four colors and five cards. Thus, in this model System 2 joins in only at the second stage. Interestingly, our findings show that our attempt to strengthen System 2 did work, but not in stage 2, which actually contains the process that is in the realms of System 2. If we actually had strengthened the System 2 process, people would more frequently predict that the pattern contains two times green. We therefore would see a decrease of the patterns that are similar to matching (by containing four different colors, with only one green), and an increase in the matching pattern (four different colors, two greens). However, we found no such tendency. Rather, our finding of an increase of the maximizing-prediction indicates that participants did either rely more on System 2-thinking from the outset, or that they put more faith in the results of System 2- thinking such that the outcome of System 1 was overridden in some cases. Given that System 1 processes occur automatically and without intention (Stanovich and West 2002), we suggest that an intuitive answer was provided by System 1, but was suppressed. This is indicative of a consolidative or a corrective architecture. That means, while our stimuli seem to activate both processes, the result does not seem to be a joint function of both processes. Rather, we observe a the-winner-takes-all response: people do either stick to their intuitive answer (matching), or change it altogether according to reasoning based on frequency (maximizing). It does not seem to be the case that people apply the System 2 process more carefully. In addition, our data rule out the possibility of a selective or a competitive architecture, since these architectures would predict an increase in the frequency of other responses, which follows if matching and maximizing predictions are mixed. 3 General discussion Experimental research in simple repeated risky choices shows a striking violation of rational choice theory: the tendency to match probabilities by allocating the frequency of responses in proportion to their relative probabilities. We investigated

7 J Risk Uncertainty (2007) 34: this matching tendency in the card task developed by Rubinstein (2002). While our earlier research had shown that neither paying for correct prediction, nor making the independence of single outcomes salient, led to an increase of the rate of maximizing predictions (Kogler 2006), here we demonstrated that by strengthening the corrective functions of System 2, it was possible to significantly increase maximizing predictions. However, although our manipulation was successful, there was still a fair degree of matching (about 40%). What are the reasons for the robustness of the matching tendency? Of course, cognitive skills play an important role for coming up with an optimal prediction. For instance, people who do not know the principle of statistical independence will in all likelihood fail to behave accordingly. This is indicated by the finding that general cognitive ability is correlated with maximizing responses in probabilistic choice tasks: West and Stanovich (2003) showed that probability-matchers had lower SAT scores than optimal responders when predicting the outcome of rolling a die. Although lack of statistical knowledge may contribute to the predominance of matching, it is certainly not the only reason. In our task, people reported awareness of conflicting tendencies indicating that they had an idea about the maximizing answer, but they somehow failed to emit it. By appropriate contextual cues faith in the maximizing response obviously increased. Stanovich and West (2000) identify four alternative explanations for a gap between normative and empirical predictions: performance errors, computational limitations, a different construal of the task by the participant, and the wrong norm being applied by the experimenter. Performance errors as well as computational limitations are in all likelihood not the main reason for our findings because (1) we found a systematic matching tendency, rather than random predictions; (2) performance errors should decrease under conditions of incentive, which is not the case in our task (Kogler and Kühberger 2006); (3) the maximizing strategy is to always predict the same color, and this strategy is truly simple, being by far simpler than any prediction of a mixture of colors. We suggest that our appeal to cognitive ability led to a change in task construal to an understanding of the task as a test rather than a lottery. However, this does not by itself explain the higher rate of maximizing predictions. If people knew the correct answer, why should they withhold it in the lottery situation, but emit it in the test situation? The dual systems view of thinking is helpful here: the test situation is a strong cue for the reliance on System 2 thinking while the lottery situation fails to provide such a cue. This interpretation is also in accordance with other findings in the literature on dual processes (for an overview see Chaiken and Trope 1999), where the monitoring functions of System 2 are said to be quite lax, allowing many intuitive judgments to be expressed, including erroneous ones. Thus, people seem to be reluctant to think hard and are tempted to trust a plausible judgment that quickly comes to mind (Kahneman 2003, p. 699). There is also some neurophysiological evidence for dual processes in reasoning, in particular supporting the idea that System 2 can intervene or inhibit System 1 processes (Goel and Dolan 2003). Concerning the interaction between the two systems, these neurophysiological findings as well as our data favor the corrective type of architecture, where both processes (Systems 1+2) are said to be jointly activated, but finally one overrules the other. In our case we presume an automatic activation of the matching tendency

8 152 J Risk Uncertainty (2007) 34: (System 1) generated by the representativeness heuristic, and a subsequent possibility for System 2 to correct this intuitive judgment, or to accept it. Doubt is obviously a System 2 phenomenon and it seems that we enhanced it by arranging the setting in an appropriate way. Note however, that the rate of matching was still very impressive. Thus, it is surely possible that participants applied a norm other than statistical independence to the task. Probability matching is an instance of a more general tendency to opt for variety. Selecting variety in repeated choices is a risk-minimizing strategy (Read and Loewenstein 1995), and may be evolutionarily advantageous in many real-life situations (Gallistel 1990). The preference for variety is related to findings such as risk-sensitive foraging (e.g., Kacelnik and Bateson 1997), where it is argued that risky choices should reflect not only the expected outcome, but in addition the need of the organism, and the variance associated with each option (e.g., Rohde et al. 1999). Similar findings are reported in the field of consumer research (Simonson 1990; Read and Loewenstein 1995) where people often end up choosing more variety than they actually wanted when purchasing food. A way to reduce matching might thus be to use a more real-life task. Indeed, Rubinstein (2002) found that diversification was noticeably reduced when he used tasks like ours, however couched in a practical cover story (e.g., predicting the gate through which a messenger will enter a shopping mall on five consecutive days, given different probabilities for the gates). However, other research found matching even in practical tasks. For instance, in a medical task participants learned the correlation between certain symptoms and the likelihood of a disease. Then they were presented with the symptoms of several patients. To maximize the probability of correct diagnosis, participants should consistently choose the outcome that is more frequently associated with that particular symptom pattern. They failed to do so but rather their judgments were guided by the ratio of probabilities (Friedman et al. 1995). Admittedly, there are various differences between artificial and real-life diversification tasks. For instance, real-life diversification tasks are characterized by learning. Thus, repeated sampling (e.g. gaining experience by seeing the results of, say, 100 five-card-draws) might lead to a decrease in probability matching (Shanks, Tunney, and McCarthy 2002). For our task this is unlikely, however. What would be experienced by repeated sampling? People would probably just realize that patterns consisting of different colors are by far more frequent than the pattern of the single most frequent color. Thus experience would correctly inform people that mixed outcomes are more frequent than pure outcomes. They might easily over-generalize this to mean that matching is better than maximizing. Thus, experience might not always be the cure to probability matching. References Arkes, Hal R., Robyn M. Dawes, and Caryn Christensen. (1986). Factors Influencing the Use of a Decision Rule in a Probabilistic Task, Organizational Behavior and Human Decision Processes 37, Baron, Jonathan, Leonardo Granato, Mark Spranca, and Eva Teubal. (1993). Decision Making Biases in Children and Early Adolescents: Exploratory Studies, Merril Palmer Quarterly 39, Chaiken, Shelly, and Yaacov Trope (eds.). (1999). Dual-Process Theories in Social Psychology. New York: Guilford.

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