Compound cue processing in linearly and nonlinearly separable environments

The Psychological Record, 2008, 58, 301–314

Compound Cue Processing in Linearly and Nonlinearly Separable Environments Ulrich Hoffrage Faculty of Business and Economics, University of Lausanne, Switzerland

Rocio Garcia-Retamero Max Planck Institute for Human Development, Berlin, and University of Granada, Spain

Uwe Czienskowski Max Planck Institute for Human Development, Berlin Take-the-best (TTB) is a fast and frugal heuristic for paired comparison that has been proposed as a model of bounded rationality. This heuristic has been criticized for not taking compound cues into account to predict a criterion, although such an approach is sometimes required to make accurate predictions. By means of computer simulations, it is shown that TTB-configural, an extension of TTB that processes compound cues, can outperform both TTB and a more demanding benchmark. Moreover, a review of experimental evidence suggests that people actually use TTB-configural when equipped with the corresponding knowledge about the causal structure of the environment.

In our everyday lives, we frequently have to make inferences. However, the information we need to do so is not always available, and because of our limitations in cognitive processing power, it might not be possible to consider all potentially available cues (i.e., pieces of information) that would allow us to improve the accuracy of our inferences. One recent approach, promoted by the Center for Adaptive Behavior and Cognition (Gigerenzer, Todd, & the ABC Research Group, 1999; Todd & Gigerenzer, 2000), suggests Ulrich Hoffrage, Faculty of Business and Economics (Ecole des Hautes Etudes Commerciales; HEC), University of Lausanne (Switzerland); Rocio Garcia-Retamero, Max Planck Institute (MPI) for Human Development, Berlin (Germany) and University of Granada, Spain; Uwe Czienskowski, MPI for Human Development, Berlin. Part of the present work has been presented as a poster at the Twenty-seventh Annual Conference of the Cognitive Science Society, Stresa, Italy (Hoffrage, Retamero & Czienskowski, 2005). The authors thank Anja Dieckmann, Jörg Rieskamp, Peter Todd, and Chris M. White for helpful discussions, and Anita Todd for editing the manuscript. Correspondence concerning this article should be addressed to Prof. Ulrich Hoffrage, Ecole des Hautes Etudes Commerciales (HEC), Université de Lausanne, Batiment Internef, CH-1015 Lausanne, Switzerland. E-mail: [email protected].

302

Hoffrage et Al.

that people use fast and frugal heuristics in these situations. That is, they use simple but nevertheless fairly accurate strategies to make probabilistic inferences quickly and with a minimum of information. These heuristics enable organisms to make smart decisions by exploiting the way information is structured in the environment1 (Gigerenzer & Goldstein, 1996; Martignon & Hoffrage, 2002). The ecological rationality view of decision making as promoted by Gigerenzer et al. (1999; see also Simon, 1990) thus brings the two elements — m ind and environment — together; it focuses on how boundedly rational minds with limited capacities are adapted to their environments and how the environments in which we make decisions shape our strategies. The fast and frugal heuristic that has received the most attention to date is take-the-best (TTB; Gigerenzer & Goldstein, 1996). This heuristic is designed for forced-choice paired comparisons, that is, it can be used to infer which of two objects, described on several dichotomous cues, has a higher value on a quantitative criterion, such as determining which of two university professors earns more money on the basis of cues such as gender and whether the professor is in the faculty of a state or a private university. TTB is constructed from building blocks, which are precise steps of information gathering and processing involved in generating a decision. Specifically, this heuristic has a search rule, which describes the order in which to search for information (TTB looks up cues in the order of their validity, i.e., the probability that a cue will suggest a right inference, given that it discriminates between the alternatives); a stopping rule, describing when the information search is to be stopped (TTB stops after the first discriminating cue); and a decision rule, which describes how to use the available information to make a decision (TTB chooses the alternative favored by the first discriminating cue). In sum, TTB is an example of one-reason decision making (it uses only the first discriminating cue). Moreover, this heuristic falls in the class of lexicographic strategies as it processes cues in a lexicographic ordering, specifically, one established by means of cue validity (for other variants, see Martignon & Hoffrage, 2002; Rakow, Newell, Fayers, & Hersby, 2005). When TTB was used to make predictions in real-world environments — for instance, to predict which of two cities has a higher homelessness rate, or which of two persons has a higher percentage of body fat — it turns out that it could compete well with more savvy strategies such as multiple regression, particularly in cross-validation (Czerlinski, Gigerenzer, & Goldstein, 1999). Cross-validation is a procedure in which a decision strategy is first fitted to one part of the environment (the training set) and subsequently tested on the other part (the test set). In other words, each strategy estimated its parameters from a subset of the objects before it was evaluated on the holdout cases. By being simple and focusing only on some, but relevant, information, TTB avoids being too closely matched to any particular environment and outperforms multiple regression in accuracy when generalizing to the test set. Multiple regression, in contrast, uses more information and is thus more likely to overfit the structure of information in the training set, that is, to adjust its parameters to noise in the training set that does not transfer to the test set. In sum, robustness can go hand in hand with simplicity and frugality. The fast and frugal research program has, however, been criticized (e.g., by Garcia-Retamero & Hoffrage, 2006; see also Hoffrage, Garcia-Retamero, & 1  When using the term environment in a more technical sense below, we refer to a set of objects that are described on a criterion value and on a set of attributes (i.e., cues).

Compound Cue Processing

303

Czienskowski, 2005) for not taking compound cues (i.e., configurations of cues) into account to predict the criterion. Indeed, the use of configurations of cues may be considered to be against the “spirit” of one-reason decision making. However, it is sometimes required to make accurate predictions, and people can and do process such compound cues in certain environments (e.g., Edgell, 1993; Garcia-Retamero, 2007; Shanks, Charles, Darby, & Azmi, 1998; Williams & Braker, 1999; Young, Wasserman, Johnson, & Jones, 2000). The present article shows how compound cue processing can be integrated into the fast and frugal heuristic approach. It is structured as follows: We first describe two different structures of information in which it can be useful to process individual cues as compounds. Second, we discuss the importance of causal knowledge as a constraint on compound cue processing. Third, we introduce take-the-best-configural (TTB-configural), a fast and frugal heuristic that processes compound cues. Fourth, we present a computer simulation that demonstrates the superior performance of TTB-configural when compared with the original TTB and with another extension of TTB that is not constrained by causal knowledge. Finally, we review experimental evidence demonstrating that TTB-configural also performs well when used as a behavioral model.

Linearly and Nonlinearly Separable Environments Our language contains many concepts that are, in fact, compounds. A bachelor, for example, is someone who is both male and unmarried, or a mother is someone who is female and has at least one child. Seen the other way around, in these examples two individual attributes have been amalgamated by using the logical AND rule to form a compound. Compound cue processing can be important when making judgments or predictions. For example, some medications have side effects, such as nausea, if ingested together with alcohol, whereas neither the drug nor the alcohol would cause any problems if ingested alone (depending, of course, on the amount of alcohol consumed). Thus, to predict whether someone will show such side effects (y), we need to know about the ingestion of such a medication (x1), as well as the ingestion of alcohol (x2). Setting other causes aside, only the presence of both cues (x1 = 1 and x2 = 1) will lead to the prediction of nausea (y = 1). An environment that has this structure is called a linearly separable environment, because this pattern can easily be captured by a simple linear equation. In the present example, this equation would be s = x1 + x2, where the presence of the criterion is predicted, if and only if the sum (s) of both unweighted cue values surpasses a threshold that has to be greater than 1 but less than 2 (e.g., ˆy = 1 if and only if s > 1.5). Another type of linearly separable environment is constituted by a pattern in which the presence of one cue alone (x1 = 1 OR x2 = 1) would be sufficient to lead to the presence of a particular outcome. Here, the threshold for predicting the presence of the criterion would have to be set between 0 and 1 (e.g., ˆ y = 1 if and only if s > 0.5). One structure of information that has generated special interest in the literature on configural strategy use is found in an environment in which the XOR logical rule is applied to amalgamate two cues into one compound cue (Shepard, Hovland, & Jenkins, 1961; Waldmann, Holyoak, & Fratianne, 1995). In such an environment, the presence of one cue and the absence of the other hint at a higher criterion value than the absence or presence of both cues.

304

Hoffrage et Al.

Consider, for instance, an academic who is a father of a young boy and a young girl and who can decide whether to work at home or at the office. He prefers to stay at home as long as there is an undisturbed working atmosphere, so he wants to predict the level of distraction. If both his children are at home, they will play nicely with each other and do not need any attention, but if only one is there, he has to expect a lot of distraction. This structure cannot be handled with a simple linear strategy, that is, one that predicts the level of distraction on the basis of a linear function of the weighted cues without an interaction term (Zurada, 1992). Rather, the separation of all situations into those with high and low amounts of distraction would require a nonlinear (e.g., multiplicative) component,2 which is why such an environment is called a nonlinearly separable environment. In a linearly separable environment, a simple linear strategy could potentially solve paired comparisons among objects, that is, select the object with the higher criterion value. In contrast, in a nonlinearly separable environment such a linear strategy could not make full use of the structure of information in that environment. This deficit could potentially explain why learning is easier in linearly separable environments than in nonlinearly separable ones (see Kimmel & Lachnit, 1991; Lachnit & Kimmel, 1993; Smith, Murray, & Minda, 1997). Yet, this does not mean that people are unable to deal with nonlinearly separable environments (Waldmann et al., 1995). In fact, nonlinear structures can be handled by means of a configural strategy, that is, one that represents configurations of cues. Representing configurations of cues could, for instance, include the use of logical rules (such as the and rule or the xor rule) to amalgamate two cues into a new, compound cue. These compound cues could then, for instance, receive a relatively high weight in a linear strategy or a high rank in the cue ordering of a lexicographic strategy. Though using configural strategies can be adaptive, learning them can be hard. Because cues in the environment can be combined in many different ways, the acquisition of a configural strategy has to deal with the problem of a combinatorial explosion. Even if the construction of compound cues is constrained to pairs of cues (rather than triples or even higher orders) and to the application of the logical and, or, and xor rules, an environment with, say, 6 cues already yields 45 possible compound cues (15 pairs of cues times three logical rules). As a consequence, it may be extremely difficult to select from the set of all possible compounds those that are highly valid and stable. A strategy that, by default, processes all compounds that can be constructed from individual cues could thus be too computationally demanding and, therefore, also psychologically implausible (Kehoe & Graham, 1988). One possible solution to this dilemma would be that in their natural environment people do not process all possible compound cues and keep track of their validities but instead focus on only the subset of those compound cues that seem “plausible.” But what makes certain compounds more plausible than others?

The Adaptive Value of Causal Knowledge for Compound Cue Processing Some previous studies suggest that people’s prior knowledge about cues could facilitate compound cue processing. For example, Wattenmaker, Dewey, 2 If the presence of the son and the daughter is coded as x s = 1 and x d = 1, and their absence as x s = 0 and x d = 0, respectively, then one would predict a high level of distraction if and only if x s + x d − 2x s x d > 0.

Compound Cue Processing

305

Murphy, and Medin (1986) found that providing participants with a hint that encouraged the additive integration of features greatly facilitated learning of linearly separable categories in comparison with nonlinearly separable categories. When, in contrast, a hint induced encoding compatible with nonlinearly separable categories, then these categories were easier to learn than linearly separable ones (see also Waldmann et al., 1995). These results demonstrate that people are able to map their previously acquired knowledge to a learning task in a way that makes compound cues salient. People’s prior beliefs might also help them to focus on a manageable subset of compound cues in the environment. Specifically, people’s causal knowledge, that is, knowledge about causal relationships between events in the environment, may help in limiting selection to only some, but highly valid, compound cues and may, in turn, also help in their subsequent processing in decision making. When it is said that a cause brings about an effect, a stable causal link between the cause and the effect (Cheng, 1997) and an underlying causal mechanism that is an essential property of this link (Ahn & Kalish, 2000) are implied. Causal beliefs are not isolated but are tightly connected with other causal beliefs in a broad base of knowledge that represents the causal structure of the environment, henceforth referred to as a causal mental model (Garcia-Retamero, 2007; Garcia-Retamero & Hoffrage, 2006). In this article we are concerned not about how causal knowledge has been acquired (cf., Lagnado & Sloman, 2004; Novick & Cheng, 2004), but only about how, once present, it can be used to aid the selection of some highly valid compound cues from among the wide range of possibilities in the environment through a top-down process. We hypothesize that when people perceive that several cues in the environment act through a common causal mechanism in bringing about an effect, they will consider that these cues also interact with each other in bringing about that effect. This possibility may lead people to check whether the accuracy of their predictions would be increased through representing these cues as a compound cue. Such might be the case in the medicine-andalcohol example mentioned above. Specifically, the effect of a medical drug would be expected to be modified by the ingestion of alcohol, with both agents being taken up in the blood, and only the presence of both might cause nausea. Therefore, processing compound cues would be an emergent property of a causal mental model of the environment when the cues are perceived to act jointly through the same underlying causal mechanism. (In the present example, both alcohol and the medical drug act via the bloodstream.) As compound cue processing requires extra cognitive effort, we expect that when cues are perceived to act through different causal mechanisms (for instance, one substance acts via the bloodstream, the other via the hormonal system), they will be represented as individual elements.

TTB-Configural: A Fast and Frugal Heuristic That Processes Compound Cues Building on both the literature about simple heuristics on the one hand and that about causal processing on the other, Garcia-Retamero, Hoffrage, Dieckmann, and Ramos (2007) recently proposed a configural version of TTB, namely the TTB-configural heuristic. This heuristic, like TTB, is precisely defined in terms of its building blocks of search, stopping search,

306

Hoffrage et Al.

and decision rules and, also like TTB, assumes that cues are ordered in a hierarchy according to their validities. The contrast to TTB, which only considers individual cues, is that TTB-configural also considers compound cues that result from the amalgamation of two individual cues and that are included as such in the cue hierarchy. When several cues are perceived to act through a common causal mechanism to determine the decision criterion, TTB-configural processes the combination of these cues as a compound. However, when cues are perceived to act through different causal mechanisms, TTB-configural processes these cues as individual elements. Consequently, TTB-configural includes both individual and compound cues in the cue hierarchy (the latter only for those individual cues that are perceived to act through the same mechanism). As with TTB, this cue hierarchy is ordered according to the cues’ validities and TTB-configural begins searching with the cue that has the highest validity — be it an individual or a compound cue. If the critical piece of information is a compound cue, TTBconfigural looks up the individual cues that jointly constitute that compound (see Baumeister & Seipel, 2002, for a similar treatment of compound cues in the context of multilevel set-covering models). In the simulation study reported next, we compared the performance of three heuristics that differed with respect to their treatment of compound cues. For simplicity, we only considered compounds of two cues. However, we see no reason why the beneficial effect of using prior knowledge to constrain which compounds are considered should not generalize to higher-order compound cues.

Computer Simulation Study In a series of computer simulations,3 we cross-validated the performance of three decision-making strategies in several environments. The strategies differ in whether they benefit from causal knowledge about the cues in the environment to process compound cues. Specifically, we focus on TTB, which just processes all individual cues as independent elements (i.e., it does not consider compound cues). In contrast, TTB-configural processes not only individual cues but also compound cues when the causal knowledge about the individual cues suggests that they act through the same causal mechanism. Thus, TTB-configural processes all of the individual cues in the environment and eventually also some of the possible compounds. Finally, take-the-best-all (TTB-all) processes all individual cues and all possible compound cues that can be constructed from these individual cues.

Procedure and Design The task was to infer which of two alternatives, described on several dichotomous cues, had a higher value on a quantitative criterion. The performance of the strategies was determined as the percentage of correct inferences in a complete paired comparison between all objects of a given environment. The environments we used were artificially created and consisted of 50 (or 100) objects described on six (or eight) individual cues. By using either the logical and rule or the logical xor rule, three (or four) pairs 3 The generation of environments and the evaluation of the strategies were programmed in Visual C# .NET (Microsoft .NET Framework 1.1, Version 1.1.4322 SP1).

Compound Cue Processing

307

of individual cues were each amalgamated into three (or four) compound cues. These are the compound cues, henceforth called the relevant compound cues, for which we assumed that the decision makers would have causal knowledge, suggesting that their components should be represented together as a compound cue. We created the cue values of the objects such that the relevant compound cues had high (or low) validities. As a control condition, we also created environments in which the amalgamation of the pairs of individual cues resulted in compound cues that had no predictive validity. In sum, in our simulation study, four factors were fully crossed — t he number of objects (50 vs. 100), the type of logical rule (and vs. xor) used to amalgamate individual cues into relevant compound cues, the number of relevant compound cues (three vs. four), and the validity of the relevant compound cues (high vs. low vs. chance) — y ielding 24 types of environments.

Generation of Environments When creating an environment, we first generated the values of the relevant compound cues. In environments with three relevant compound cues, the values were randomly generated with the constraint that the resulting validities equaled .95, .90, and .85 in the high validity condition and .90, .80, and .70 in the low validity condition. For environments with four relevant compound cues, their validities were .95, .90, .85, and .80 in the high validity condition and .90, .80, .70, and .60 in the low validity condition. In the chance condition, all validities were .50.

Figure 1.  Scheme according to which relevant compound cues were split into two individual cues. The probabilities are conditioned on the value of the compound cue.

Next, we generated the individual cues by splitting up these relevant compound cues. This was achieved by taking the values of a given compound

308

Hoffrage et Al.

cue and by determining — according to the probabilities in Figure 1 — t he values of the two corresponding individual cues. Thereby, we used either the logical AND rule or the logical XOR rule. Seen the other way around, it was either the AND or the XOR rule that had to be used for the amalgamation of the two individual cues to result in the relevant compound cue. In a third step, we combined all the six (or eight) individual cues generated in the previous step to generate all possible compound cues that could be constructed from six (or eight) cues, using both the AND and the XOR logical rule. In this step the values of the individual cues were the inputs, and the values of all the possible compound cues were the outputs. Because six (eight) cues can be combined into 15 (28) pairs, the application of the AND rule and the XOR rule yielded 30 (56) compounds. The relevant three (four) compounds were included in this set. Together with the six (eight) individual cues, an environment thus had 36 (64) cues, either individual or compounded. The three steps explained above — generation of relevant compound cues, generation of individual cues from those compounds, and generation of all possible compounds from those individual cues — were repeated 100 times for each of the 24 types of environments. To be able to cross-validate the performance of strategies, we finally split each environment into two sets: the learning set and the test set, with 50% of the objects in each set. We repeated the assignment of the objects to the learning and test set 100 times.

Strategies and Predictions How do the three strategies process these cues? Recall that the strategies differed with respect to how they treated causal knowledge. In the present simulation, causal knowledge — or more precisely, knowledge about which cues acted through the same mechanism — w as considered by constraining TTB-configural to use only those compounds that were designed to have a high validity in the whole environment, whereas TTB used no compounds and TTB-all used all compounds. Specifically, TTB processes only the six (eight) individual cues. TTB-configural processes nine (twelve) cues, namely the six (eight) elemental cues and — d irected by causal knowledge — t he three (four) relevant compound cues. TTB-all includes all 36 (64) cues in its hierarchy. Evidently, the spirit of simple heuristics is no longer present with TTB-all, as this strategy does not try to avoid a computational explosion. In our simulations, all the strategies ordered the cues they used in a hierarchy according to the cues’ validities and searched for cue values in this order. What was our rationale for generating the environments according to the three steps explained above? The three (four) relevant compound cues were created so that they had some validity in the whole set of objects in the environment, and it can thus be assumed that this was also true for both the training and the test set. In contrast, the compound cues that were generated from the individual cues in the third step should only occasionally be valid in the whole set. For these cues, we expected that some would actually have a high validity in the training set, simply because of random fluctuations and because the set of all possible compounds is quite large. But these compound cues that have a high validity in the training set will, owing to statistical regression toward the mean, have much lower validities in the test set (as

Compound Cue Processing

309

those with low validities in the training set can be expected to have higher validities in the test set). As a consequence, TTB-configural should be more robust than TTB-all. Specifically, in the training set, TTB-all should outperform TTB-configural, because the set out of which valid cues — compound or individual — could be selected was larger for TTB-all. However, as most of the compounds that were highly valid in the training set would be useless in the test set, TTB-all should lose the advantage of selecting from more cues when evaluated in the test set.

Results Data were first aggregated across the 100 runs per environment (i.e., assignments of objects to training set and test set). Afterwards, the environment-specific means were computed across the 100 environments for a given type. The standard errors of the mean performance across the 100 environments for a given type ranged between 0.1 and 0.4 percentage points, and thus all factors that we manipulated turned out to be significant in a four-way analysis of variance. Still, the differences between the levels of the factor Number-of-Objects (50 vs. 100) and the factor Number-of-ElementalCues (six vs. eight) were relatively small and uninteresting from the present perspective; therefore, all analyses reported below were performed across these two factors.

Figure 2.  Proportion of correct inferences of TTB, TTB-configural, and TTB-all in the three validity conditions (high vs. low vs. chance) of the AND environments. The three leftmost circles within a validity condition depict performance of the three strategies in the training set; the three rightmost, that in the test set.

Figure 2 shows the performance of the three strategies, TTB, TTBconfigural, and TTB-all, in the AND environments. First, there was almost no difference in the performance of the strategies between the environments where the compounds had a high validity and those where the compounds had a low validity, but there was obviously a big difference between these

310

Hoffrage et Al.

environments and the chance condition, which did not contain any valid relevant compound cues. Second, TTB-all performed best in the training set, but, as expected, TTB-configural outperformed both TTB and TTB-all in the test set. Accordingly, TTB-configural was the most robust strategy: the differences between performance in the training set minus that in the test set for the AND environments were 6.6, 6.1, and 8.9 percentage points for TTB, TTB-configural, and TTB-all, respectively, averaged across all other factors, F(2, 1198) = 2351.9, p < .001. Figure 3 shows the performance of the three strategies in the XOR environments. Again, there was almost no difference between the validity conditions (except for the contrast between the high and low validity conditions on the one hand and the condition in which compound cues had no predictive validity on the other). However, unlike in the AND environments, the simple TTB, which did not process compound cues, was markedly outperformed by TTB-configural and TTB-all. This is hardly surprising, as the XOR environment is nonlinearly separable, and thus deciding based on one single, individual cue is not adaptive here. In the test set, TTB did not even perform better than chance, as would be expected. Furthermore, TTB-all outperformed TTB-configural in the training set, but just as for the AND environments, the pattern reversed in the test set. This was the case because, as we had predicted, TTB-configural was the most robust strategy. The differences between performance in the training set and test set were 13.6, 2.4, and 4.0 percentage points for TTB, TTB-configural, and TTB-all, respectively, F(2, 1198) = 2234.4, p < .001. Thus, TTB-configural was more robust than TTBall, even though TTB-all also included the relevant compounds in its cue hierarchy. However, TTB-all also processed other compound cues that could even be more valid in the training set but whose validities dropped in the test set. In other words, TTB-all relied on compound cues that were less robust to environmental changes (i.e., in cross-validation).

Figure 3.  Performance of the three strategies in the XOR environments. (For further explanation, see Figure 2.)

Compound Cue Processing

311

In sum, in linearly separable environments, where the relevant compound cues are constituted by amalgamating two individual cues into a compound cue with the use of the logical AND rule, TTB-all was best when fitting data; however, TTB-configural and TTB did not fall too far behind. Moreover, when generalizing to new data, TTB-configural was better than TTB and TTB-all. In nonlinearly separable environments, the same pattern could be observed with respect to the comparison between TTB-configural and TTB-all. However, TTB failed considerably in these environments, both when fitted to known data and in particular when generalized to new data.

Empirical Evidence for the Use of TTB-Configural In two series of experiments, we explored whether decision makers used TTB-configural if a highly valid compound cue existed in the environment, and their causal knowledge about the cues suggests that these cues may interact when determining the criterion (see GarciaRetamero et al., 2007; Garcia-Retamero, Hoffrage, & Dieckmann, 2007). In these experiments, both linearly separable and nonlinearly separable environments were used. The task was framed as a medical-diagnosis task. Participants were presented with information about two patients and had to infer which would show a higher body temperature. The information provided about the patients was whether they had ingested three different substances, namely, A, B, and C. In a first block of trials, participants were provided with information about all three cues in a given comparison at no cost and then had to make the decision. Outcome feedback about the correct option was given to enable participants to learn the cues’ validities. Subsequently, participants went through a decision-making phase in which the cue information was no longer available; instead they had to look it up sequentially. Again, outcome feedback was provided. We generated complex environments in which the amalgamation of two of the three cues, A and B, yielded a compound cue with a validity of 1.00. The validity of both cues A and B was 0.50, and the validity of cue C was 0.75. In a control environment, the only valid cue was cue C, which again had a validity of 0.75. The validity of the compound AB was 0.50, as was the validity of each of its component cues. We also manipulated participants’ causal knowledge about the cues by giving different instructions concerning the causal mechanism through which the substances acted. In particular, we differentiated among a configural, an elemental, and a neutral causal model condition. In the configural causal model condition, participants were told, “A crucial point is the mechanism of action of these substances. Please consider carefully how the substances act: They act via the same mechanism, that is, they act via X” (where X was either “the hormonal system,” “the bloodstream,” or “the nervous system”; it varied among participants). In the elemental causal model condition, participants were told, “A crucial point is the mechanism of action of these substances. Please consider carefully how the substances act: They act via different mechanisms, that is, they act via the hormonal system, the bloodstream, and the nervous system, respectively.” Finally, in the neutral causal model condition, participants were not given information about the cues’ causal mechanism. In short, having in mind a configural, an elemental,

312

Hoffrage et Al.

or a neutral causal model, participants received an XOR environment, an AND environment, or a control environment.4 We expected that participants in the configural causal model condition would consider configurations of cues and thus predicted that TTB-configural would be better than TTB as a behavioral model. In contrast, participants both in the elemental and in the neutral causal model conditions should not consider configurations of cues, and we thus expected TTB to be the better behavioral model. To classify participants according to a particular strategy in our experiments, we used the Bayesian method for multiple-attribute decision making proposed by Bröder and Schiffer (2003). Since TTB and TTBconfigural showed the highest fit for most of our participants, we will focus on only these two strategies. Table 1 shows the percentage of participants for whom one of these strategies achieved the highest fit. (These percentages do not always add up to 100%, because some participants used other configural strategies, e.g., starting with unpredictive compounds or other simple lexicographic strategies like TTB but with a different cue order, e.g., starting with Cue A.) In accordance with our hypothesis, when participants had a configural causal mental model of the environment, a high percentage of them used the TTB-configural heuristic and decided according to the highly valid compound cue if it discriminated. Furthermore, when participants were told that these cues acted through different causal mechanisms (in the elemental causal model), or when no such information was given (in the neutral causal model), a high percentage of participants used the TTB heuristic even though there was a highly valid compound cue in the environment. Interestingly, these results were found regardless of whether the component cues were amalgamated into a compound by applying the XOR or the AND logical rule. Finally, when there was no highly valid compound cue in the environment, that is, in the control condition, the TTB was the most frequently used strategy. Table 1 Percentage of Participants for Whom TTB-configural and TTB Achieved the Highest Fit in a Given Causal Mental Model Condition Causal Mental Model

Environment AND XOR Control

Configural TTBconf. TTB 66.7 25 50 16.6 0 66.7

Elemental TTBconf. TTB 8.3 50 8.3 58.3 0 100

Neutral TTBconf. TTB 16.7 41.7 8.3 66.7 0 83.3

Note.  Participants have been classified by means of the Bayesian method for multiple-attribute decision making of Bröder and Schiffer (2003). TTB-conf. = TTBconfigural.

In sum, results in these experiments suggest that TTB-configural was used when the information structure in the environment and in the mind fitted together. That is, when the causal knowledge about the cues induced 4  For the sake of brevity and simplicity, here we will only summarize the results for those environments in which the compound was constituted by applying the logical AND or XOR rule, and omit results from environments in which the compound cue was constituted by applying the OR rule.

Compound Cue Processing

313

participants to search for highly valid compound cues in the environment and such a highly valid compound cue existed. When this was not the case, the classical TTB that considered only the individual but not the compound cues in its hierarchy was the best behavioral model in our experiments.

Conclusions The results from the simulation and from the experiments complement each other: When there are highly valid compound cues in the environment, TTB-configural, an extension of TTB that processes such compound cues as long as causal knowledge suggests that cues may interact with each other, outperformed, in cross-validation, both TTB and even another extension of TTB that processes all possible compounds, namely, TTB-all. Moreover, TTB-configural performed well as a behavioral model: If there was a valid compound cue in the environment and participants received a hint that cues may interact with each other, then the majority used this heuristic.

References Ahn, W., & Kalish, C. W. (2000). The role of mechanism beliefs in causal

reasoning. In F. C. Keil & R. A. Wilson (Eds.), Explanation and cognition (pp. 199–225). Cambridge, MA: MIT Press. Baumeister, J., & Seipel, D. (2002). Diagnostic reasoning with multilevel set-covering models. Proceedings of the 13th International Workshop on Principles of Diagnosis (DX–02). Semmering, Austria. Bröder, A., & Schiffer, S. (2003). Bayesian strategy assessment in multiattribute decision making. Journal of Behavioral Decision Making, 16, 193–213. Cheng, P. W. (1997). From covariation to causation: A causal power theory. Psychological Review, 104, 367–405. Czerlinski, J., Gigerenzer, G., & Goldstein, D. G. (1999). How good are simple heuristics? In G. Gigerenzer, P. M. Todd, & the ABC Research Group, Simple heuristics that make us smart (pp. 97–118). New York: Oxford University Press. Edgell, S. E. (1993). Using configural and dimensional information. In N. J. Castellan (Ed.), Individual and group decision making (pp. 43–64). Hillsdale, NJ: Erlbaum. Garcia-Retamero, R. (2007). The influence of knowledge about causal mechanisms on compound processing. The Psychological Record, 57, 295–306. Garcia-Retamero, R., & Hoffrage, U. (2006). How causal knowledge simplifies decision-making. Minds and Machines, 16, 365–380. Garcia-Retamero, R., Hoffrage, U., & Dieckmann, A. (2007). When one cue is not enough: Combining fast and frugal heuristics with compound cue processing. Quarterly Journal of Experimental Psychology, 60, 1197–1215. Garcia-Retamero, R., Hoffrage, U., Dieckmann, A., & Ramos, M. M.

(2007). Compound cue processing within the fast and frugal heuristics approach in non-linearly separable environments. Learning and Motivation, 38, 16–34. Gigerenzer, G., & Goldstein, D. G. (1996). Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review, 103, 650–669.

314

Hoffrage et Al.

Gigerenzer, G., Todd, P. M., & the ABC Research Group. (1999). Simple

heuristics that make us smart. New York: Oxford University Press. Hoffrage, U., Garcia-Retamero, R., & Czienskowski, U. (2005). The

robustness of the take the best configural heuristic in linearly and nonlinearly separable environments. In B. G. Bara, L. Barsalou, & M. Bucciarelli (Eds.), Proceedings of the 27th Annual Conference of the Cognitive Science Society (pp. 971–976). Mahwah, NJ: Erlbaum. Kehoe, E. J., & Graham, P. (1988). Summation and configuration: Stimulus compounding and negative patterning in the rabbit. Journal of Experimental Psychology: Animal Behavior Processes, 14, 320–333. Kimmel, H. D., & Lachnit, H. (1991). Acquisition of a unique cue in positive and negative patterning? Integrative Physiological and Behavioral Science, 26, 32–38. Lachnit, H., & Kimmel, H. D. (1993). Positive and negative patterning in human classical skin conductance response conditioning. Animal Learning & Behavior, 21, 314–326. Lagnado, D. A., & Sloman, S. (2004). The advantage of timely intervention. Journal of Experimental Psychology: Learning, Memory, & Cognition, 30, 856–876. Martignon, L., & Hoffrage, U. (2002). Fast, frugal, and fit: Simple heuristics for paired comparison. Theory & Decision, 52, 29–71. Novick, L. R., & Cheng, P. W. (2004). Assessing interactive causal influence. Psychological Review, 111, 455–485. Rakow, T., Newell, B., Fayers, K., & Hersby, M. (2005). Evaluating three criteria for establishing cue-search hierarchies in inferential judgment. Journal of Experimental Psychology: Learning, Memory, & Cognition, 31, 1088–1104. Shanks, D. R., Charles, D., Darby, R. J., & Azmi, A. (1998). Configural processes in human associative learning. Journal of Experimental Psychology: Learning, Memory, & Cognition, 24, 1353–1378. Shepard, R. N., Hovland, C. L., & Jenkins, H. M. (1961). Learning and memorization of classifications. Psychological Monographs, 75, 1–42. Simon, H. A. (1990). Invariants of human behavior. Annual Review of Psychology, 41, 1–19. Smith, J. D., Murray, M. J., & Minda, J. P. (1997). Straight talk about linear separability. Journal of Experimental Psychology: Learning, Memory, & Cognition, 23, 659–680. Todd, P. M., & Gigerenzer, G. (2000). Précis of simple heuristics that make us smart. Behavioral & Brain Sciences, 23, 727–780. Waldmann, M. R., Holyoak, K. J., & Fratianne, A. (1995). Causal models and the acquisition of category structure. Journal of Experimental Psychology: General, 124, 181–206. Wattenmaker, W. D., Dewey, G. I., Murphy, T. D., & Medin, D. M. (1986). Linear separability and concept learning: Context, relational properties, and concept naturalness. Cognitive Psychology, 18, 158–194. Williams, D. A., & Braker, D. S. (1999). Influence of past experience on the coding of compound stimuli. Journal of Experimental Psychology: Animal Behavior Processes, 25, 461–474. Young, M. E., Wasserman, E. A., Johnson, J. L., & Jones, F. L. (2000). Positive and negative patterning in human causal learning. Quarterly Journal of Experimental Psychology, 53B, 121–138. ZURADA, J. M. (1992). Introduction to artificial neural systems. New York: West.