Do people treat missing information adaptively when making inferences?

Abstract

When making inferences, people are often confronted with situations with incomplete information. Previous research has led to a mixed picture about how people react to missing information. Options include ignoring missing information, treating it as either positive or negative, using the average of past observations for replacement, or using the most frequent observation of the available information as a placeholder. The accuracy of these inference mechanisms depends on characteristics of the environment. When missing information is uniformly distributed, it is most accurate to treat it as the average, whereas when it is negatively correlated with the criterion to be judged, treating missing information as if it were negative is most accurate. Whether people treat missing information adaptively according to the environment was tested in two studies. The results show that participants were sensitive to how missing information was distributed in an environment and most frequently selected the mechanism that was most adaptive. From these results the authors conclude that reacting to missing information in different ways is an adaptive response to environmental characteristics.

Keywords

Adaptive decision making Incomplete information Missing information Probabilistic inferences Strategy selection

People often make inferences with incomplete information. For example, when a personnel manager tries to find a new employee for an open position, the candidates’ applications often are not complete. Frequently, important information, such as whether the candidates have social skills or are reliable, is not available. How should and how do people make inferences despite missing information? This is the central question of this article.

Although many models of cognition do not consider missing information (e.g., signal detection theory, Green & Swets, 1966; or causal attribution theory, Cheng, 1997; Kelley, 1973), there is increasing interest in how incomplete information influences people's judgements (see Ganzach & Krantz, 1990; Huber & McCann, 1982; Jaccard & Wood, 1988; Johnson & Levin, 1985; White & Koehler, 2004; Zhang & Markman, 1998, 2001). Yet there is no consensus on the answer. For instance, several authors have argued that people treat missing information as the average of the observed information (Ganzach & Krantz, 1990; Slovic & MacPhillamy, 1974; White & Koehler, 2004) or as the most frequently observed information (i.e., the mode; Jaccard & Wood, 1988). Alternatively, individuals sometimes treat missing information as negative (Huber & McCann, 1982; Jaccard & Wood, 1988; Johnson, 1987, 1989; Johnson & Levin, 1985; Lim & Kim, 1992; Lim, Olshavsky, & Kim, 1988; Meyer, 1981; Yamagishi & Hill, 1981, 1983; Yates, Jagacinski, & Faber, 1978), presumably assuming that missing information has been withheld because it would lead to a negative evaluation otherwise. Sometimes, individuals process unknown information as if it were positive, especially when positive information is more prevalent (e.g., Levin, Johnson, & Faraone, 1984; Levin, Johnson, Ruso, & Deldin, 1985; Levin, Mosell, Lamka, Savage, & Gray, 1977). For instance, casino managers initially assume that their guests can cover their bets unless a negative report has been made in the past. Likewise, road users expect good weather conditions as long as no warning is given on the radio. In these cases, missing information is positively correlated with the criterion that has to be predicted. Finally, people sometimes just use whatever information is available and ignore missing information (see Kardes, Posavac, Cronley, & Herr, 2008; Sanbonmatsu, Kardes, & Herr, 1992; Simmons & Lynch, 1991). In sum, the empirical evidence shows a large heterogeneity in how people treat missing information, leading to the question of why people treat missing information differently.

One way to tackle this question is to assume that human reasoning can often be understood as an adaptation to specific environmental situations (Anderson, 1991; Brunswik, 1943; Payne, Bettman, & Johnson, 1988, 1993). Along this line, Rieskamp and Otto (2006) argued that when people receive outcome feedback about the accuracy of their judgements, this initiates a learning process that most likely leads to the selection of a judgement strategy that performs well in the specific environment. Following this approach, we will assume that when people face an inference situation with missing information, they will, on the basis of learning, most likely apply the mechanism for treating missing information that leads to the best performance. We are aware that in this general form, such an adaptivity assumption is rather strong. Nevertheless, in the present article we examine to what extent it allows us to explain probabilistic inferences in situations with missing information.

The mechanisms for treating missing information are the building blocks of a complete inference strategy. Therefore, the study of people's responses to missing information requires analysing the inference strategies they use. To predict when people select which strategy, Rieskamp and Otto (2006) proposed the strategy selection learning (SSL) theory (see also Rieskamp, 2006a). In a nutshell, this theory assumes that people have a repertoire of strategies. When they receive feedback, people revise their initial strategies’ evaluations, so that the strategy that performs best becomes the one most likely to be selected. The SSL theory was supported in several experiments. Following this theoretical approach, it can be predicted that people will also select mechanisms for treating missing information adaptively. That is, the different mechanisms can be incorporated into inference strategies, and people will learn to select the strategy that incorporates the mechanism that leads to the overall best inferential accuracy, rather than, say, the same strategy incorporating alternative mechanisms.

Imagine the situation of choosing which of two job candidates will be more productive in the future, so that future productivity defines the inference criterion. A high criterion value implies that the candidate is very promising and will be very productive, whereas a low criterion value means that the candidate is not very promising. For this inference, the information of several cues could be used, such as whether the candidates have organizational skills or positive letters of recommendation. The cues could have either a positive or a negative value for each candidate. Each cue has a certain validity, defined as the conditional probability of making a correct inference under the condition that the cue discriminates (Gigerenzer & Goldstein, 1996; Martignon & Hoffrage, 2002). A cue discriminates when it provides positive evidence for one alternative and negative evidence for the other.

For such an inference problem, several inference strategies could be applied. Because in this article we focus on mechanisms for missing information, we will only consider two inference strategies. The first strategy is a simple lexicographic heuristic called “take the best” (TTB), which searches for the cue with the highest validity and selects the candidate with a positive value on that cue (see Gigerenzer & Goldstein, 1996). If the cue does not discriminate, then the second most valid cue is considered, and so on. The second strategy is a compensatory weighted additive strategy (WADD). WADD computes for each candidate the sum of all cue values multiplied by their cue validities and selects the alternative with the largest sum (see Bröder, 2000; Bröder & Schiffer, 2003; Rieskamp & Otto, 2006). WADD can also be implemented by determining a weighted difference; that is, the difference of the cue values is computed and multiplied by the cue's validity. The weighted differences of all cues are summed up, and a decision is made corresponding to the overall difference. For WADD, limited search is also possible. We assume that WADD searches for cues in the order of the validities and stops search when a preliminary decision on the basis of the cues acquired cannot be changed by the cues not acquired so far.

Previous research has shown that TTB was most suitable for predicting people's inferences when high strategy application costs existed (Bröder, 2000; Bröder & Schiffer, 2003; Dieckmann & Rieskamp, 2007; Garcia-Retamero, Hoffrage, & Dieckmann, 2007a; Garcia-Retamero, Hoffrage, Dieckmann, & Ramos, 2007b; Garcia-Retamero, Takezawa, & Gigerenzer, 2008; Rieskamp & Hoffrage, 1999, 2008), whereas WADD was most suitable for predicting participants’ inferences when it outperformed TTB (Garcia-Retamero & Dieckmann, 2006; Newell & Shanks, 2003; Newell, Weston, & Shanks, 2003; Rieskamp, 2006a; Rieskamp & Otto, 2006).

The impact of the environment on the accuracy of inference mechanisms for treating missing information

One crucial factor that influences the accuracy of the mechanisms for treating missing information is the way missing information is distributed in the environment. Many authors have assumed (e.g., White & Koehler, 2004) that the likelihood of being confronted with missing information is the same across alternatives (i.e., missing information is uniformly distributed). For instance, when predicting which of two job candidates will be more productive, one might assume that missing cue values will occur with the same probability across all cues for both candidates. Alternatively, missing cue values could be more likely to occur for alternatives with a low criterion value. Following our example, candidates who are less competent might more often have incomplete applications. Consequently, one could assume that unknown cue values are conditionally distributed depending on the criterion value. We argue that depending on the environment, different mechanisms for treating missing information are optimal for making inferences. If the performance of inference strategies and their mechanisms for treating missing information depend on the environment this could explain why people deal with incomplete information differently when aiming for high inference accuracy, and it would explain the mixed results in the literature on how people treat missing information.

In an earlier study (Garcia-Retamero & Rieskamp, 2008), we investigated the impact of the distribution of missing information on the accuracy of different mechanisms for treating missing information incorporated in the two inference strategies described above. By means of computer simulations, we explored whether it is more important to select the optimal inference strategy or the optimal mechanism for treating missing information when aiming for high inference accuracy. To avoid limitations that would result from focusing on a particular inference problem, we used the data from a large variety of inference problems that have been studied in different disciplines, including psychology, economics, and artificial intelligence. These problems differed in the number of objects considered (ranging from 24 to 181 objects) and the number of cues provided for making an inference (ranging from 3 to 12 cues).

We examined the impact of missing information that was either uniformly distributed or conditionally distributed depending on the criterion values. For the condition with uniformly distributed missing information, the cue values that were regarded as missing were selected with equal probability. For the condition with conditionally distributed missing information, the cue values that were regarded as missing were selected with a probability following an exponential function, f(z_n) = (1/b) exp(−z_n/b), with z as the z-transformed criterion value of each object and b as a scale parameter with a value of 4 for our simulations. Thus, in this condition, the smaller the criterion value the larger the probability of being selected to be regarded as missing. For each of our inference problems and two distributions of missing information, we simulated 1,000 instances of missing information. The inference accuracy of the following five mechanisms for treating missing information was tested: ignoring missing information, treating missing information as negative, treating it as positive, and treating it as the average¹ or the mode of the available information. Each mechanism was incorporated in the two inference strategies, TTB and WADD, described above.

When using binary cue values, a positive value can be coded as a value of 1.0 and a negative cue value (i.e., a nonpositive cue) can be coded as a value of 0, so that when positive and negative cue values occur equally often the average cue value is .50. When TTB compares an alternative that has a negative (positive) cue value with an alternative carrying a missing cue value that is replaced by the average, the alternative with the missing (positive) cue value will be selected. Thus, for TTB the exact worth of the average cue value is not crucial—using the average is identical with using any value between positive (1.0) and negative (0.0). In contrast, for WADD the exact average cue value was used to replace missing cue values, and the score that WADD determines is based on this average cue value.

The simulations showed that it was crucial for reaching high inference accuracy to select the most accurate mechanisms for treating missing information, regardless of whether the inference strategy TTB or WADD was used, but which mechanism was best depended on how missing information was distributed in an environment. Specifically, treating missing information as the average achieved the highest accuracy when unknown cue values were uniformly distributed in the environment. In contrast, processing missing information as negative outperformed the rest of the mechanisms in the environment with conditionally distributed missing information (see Figure 1). Interestingly, ignoring missing information did not lead to high inferential accuracy when compared with the other mechanisms. This result calls into question the assumption that ignoring missing information is adaptive (e.g., Gigerenzer, Hoffrage, & Kleinbölting, 1991; Kardes et al., 2008; Sanbonmatsu et al., 1992) and illustrates that it is better to form expectations about missing information and to use partial available information than simply to rely on information that is known while ignoring what is missing.

Surprisingly, the two strategies, TTB and WADD, had similar accuracies across all inference problems considered when they used the same mechanism for treating missing information. Relying only on the most valid cue, as practised by the strategy TTB, was not outperformed by WADD, which integrated the available information. Although integrating most of the available information appears to be an accurate inference strategy, one needs to consider that the additional information that WADD uses in comparison to TTB has low predictive accuracy. Thus, WADD's advantage of compensating mistakes of single cues by the integration of other cues has the cost of relying on less valid information. From this simulation result, an important conclusion can be drawn: To achieve high accuracy, it is more important to select a good mechanism for treating missing information than to select the best inference strategy.

In addition to evaluating the accuracies of the mechanisms for treating missing information, Garcia-Retamero and Rieskamp (2008) also examined the amount of information required by the different mechanisms when incorporated in the two strategies (i.e., the strategies’ frugality). We found that processing missing information as the average was by far the most frugal of the mechanisms, but only when it was implemented in TTB. Using the average to replace missing information makes discriminations between alternatives more likely, which stops TTB's information search. Specifically, a missing cue value that is replaced by the average cue values stops TTB's information search regardless of whether the other alternative has a positive or a negative cue value. In contrast, a missing cue value that is treated as positive (negative) only leads to discrimination when the second alternative has a negative (positive) value on this cue. Finally, when missing information is ignored the search only stops when a discriminating cue with complete information is found. This requires TTB to search for additional cues. In sum, the amount of required information was strongly influenced by the mechanisms used for treating missing information when incorporated in TTB. In contrast, the different mechanisms did not affect the amount of required information substantially when the mechanisms were incorporated in WADD, because this strategy always has to search for a large amount of information (even when assuming limited information search as described above).

Garcia-Retamero and Rieskamp (2008) showed that the different mechanisms for treating missing information affect inferential accuracy, making it psychologically plausible that people use different mechanisms under different situations. However, Garcia-Retamero and Rieskamp (2008) did not provide evidence that people react adaptively to different distributions of missing information, which is the goal of the present article. On the basis of the simulation study's results, predictions about how people should deal adaptively with missing information can be derived. First, to obtain high inference accuracy, people should treat missing information as the average when it is uniformly distributed. In contrast, when missing information is negatively correlated with the criterion, missing cue values should be treated as negative. Second, when information search is costly, people should treat missing information as the average and use TTB for making inferences. The goal of Experiments 1 and 2, reported next, was to test whether people's behaviour follows these adaptivity predictions.

Experiment 1

Experiment 1 examined whether people select the mechanism for treating missing information adaptively. Participants had to infer which of two job candidates would be more productive on the basis of six cues and make inferences in one of three conditions. In the control condition, complete information was provided. In a second condition, missing information was uniformly distributed. In the third condition, missing information was conditionally distributed depending on the criterion value, yielding a negative correlation between the missing information and the criterion.

The simulation results of Garcia-Retamero and Rieskamp (2008) showed that the accuracy of TTB and WADD differed substantially when implementing different mechanisms for treating missing information depending on how missing information is distributed. To be most adaptive, people should treat missing information as the average when missing information is uniformly distributed. In contrast, people should treat missing information as negative when it is conditionally distributed. Garcia-Retamero and Rieskamp (2008) showed that the difference in accuracy between the two strategies, however, is less noticeable when the strategies use the same mechanism for treating missing information. Thus, it is most important to select the best mechanism for treating missing information to achieve high inferential accuracy, whereas the selection of the best inference strategy would be less important.

In contrast, the two strategies differ considerably in the amount of information they require to make an inference, which depends on the implemented mechanism for treating missing information. For simplicity, a strategy's performance can be defined in pure monetary terms—that is, the payoff the strategy produces when consistently applied. When the acquisition of information is costly, this definition implies that a strategy's performance depends on its accuracy in making correct inferences and on the amount of required information. Therefore, in a situation in which TTB and WADD have similar accuracies, because TTB requires less information, it outperforms WADD under costly information search. High information search costs should then prompt participants to select TTB combined with the most adaptive mechanism for treating missing information, rather than WADD. Furthermore, when selecting TTB, less information will be required when treating missing information as the average than when treating missing information as negative. Therefore, when selecting TTB in an environment with uniformly distributed missing information, participants should search for less information than when selecting TTB in an environment with conditionally distributed missing information.

Method

Participants

A total of 72 students (38 women and 34 men, average age 26 years) from the Free University of Berlin participated in the experiment. Participants were randomly assigned to one of three equally sized groups (n = 24) for each environment condition. The computerized task was conducted in individual sessions and lasted approximately 1 hour. Participants received one third of the total amount they accrued in the task, with an average payment of €12 (ranging from €10 to €35).

Stimuli and design

Participants had to infer which of two job candidates (displayed column-wise) would be more productive. To make these inferences, they could search for information provided by six cues describing the candidates (i.e., whether the candidates had organizational skills, social skills, positive letters of recommendation, computer skills, spoke foreign languages, or were reliable). These cues are common for assessing job candidates (see, e.g., Garcia-Retamero, Takezawa, & Gigerenzer, in press). The cues could have a positive, negative, or unknown value for each candidate, represented as a “ + ”, “ − ”, or “ ? ” sign, respectively (see Figure 2). The order in which the cues were presented on the screen was fixed for each candidate but varied randomly between participants. Likewise, the position of the two candidates (left or right on the screen) varied randomly.

We created 24 different item sets that allowed discrimination between the predictions of the five mechanisms for treating missing information when implemented in TTB and WADD (see Appendix). Each item set consisted of 30 pairs of objects described by six cues. In 8 item sets, complete information was provided. In another 8 sets, missing information was uniformly distributed. Finally, in 8 sets, the missing information was conditionally distributed depending on the criterion value. The item sets with missing information were generated on the basis of those with no missing information by replacing 25% of the actual cue values with unknown cue values. Specifically, when missing information was uniformly distributed, each cue value of each object was selected to be considered as missing with an equal probability of .25. Therefore, a cue with missing information was equally likely for the two candidates when missing information was uniformly distributed. In contrast, in the item sets with conditionally distributed missing information, each cue value of the object with the higher (lower) criterion value was selected with a probability of .15 (.35) to be considered as a missing cue value. Therefore, missing information was more likely for the alternative with the lower criterion value (see Figure 2). A total of 3 participants were assigned to each of the 24 item sets. The cue validities were on average: .77, .72, .67, .62, .57, and .52. To allow discrimination between the strategies’ predictions, the cue validities varied across the item sets, with deviations ranging between .002 and .08. Cue validities were not told to the participants. In sum, the experimental design had two factors: environment condition (no missing information, and uniformly or conditionally distributed missing information; between-subjects), and trial block, with seven repetitions of the 30 pair comparisons (within-subject).

Procedure

Participants first read the instructions for the experiments. The instructions explained that they had to imagine that they worked for a company that had grown substantially. It was their job to make recommendations for new personnel. Specifically, they had to choose between pairs of candidates and select the one who would be most productive on the basis of several properties that described those candidates.

After reading the instructions, participants made 210 inferences with no time constraints, in seven blocks of 30 trials each. For each participant, the same set of trials was presented within each block but in random order. To make decisions, participants could search for information about six cues describing each candidate by clicking little boxes on the computer screen. Once a box with information on one cue for one candidate was opened, the cue value remained visible until a decision was made. For each cue value looked up, 3 eurocents were deducted from participants’ overall payoffs. Participants made a decision by clicking on a button. Subsequently, outcome feedback was displayed. For a correct decision, participants earned 50 eurocents, and for an incorrect decision they lost 50 eurocents. The current balance of their account was always visible on the computer screen (see Figure 2).

Compared to the monetary gains of a correct inference, the search costs were relatively high (36 cents when acquiring all 12 cue values). Therefore, TTB performed better than WADD with respect to the monetary outcome.² Specifically, the consistent application of TTB led to a payoff of €32 in the condition with no missing information. When missing information was uniformly distributed, TTB led to a payoff of €32, €16, or €12 when missing information was treated as the average, as negative, or when it was ignored, respectively. Finally, when missing information was conditionally distributed, TTB led to a payoff of €19, €39, or − €10 when missing information was treated as the average, as negative, or when it was ignored, respectively.

The payoff of a strategy is defined by the strategy's number of correct inferences multiplied by 50 eurocents, minus the strategy's number of incorrect inferences multiplied by 50 eurocents, and minus the strategy's number of cue values searched for multiplied by 3 eurocents.

In contrast, the application of WADD led to a payoff of − €3 when there was no missing information.³ When missing information was uniformly distributed, WADD led to a payoff of − €3, − €12, or − €9 when missing information was treated as the average, as negative, or when it was ignored, respectively. When missing information was conditionally distributed, WADD led to a payoff of − €19, €1.5, or − €28 when missing information was treated as the average, as negative, or when it was ignored, respectively. Thus, when missing information was distributed uniformly, treating missing information as the average led to the best payoff for both strategies. Likewise, when missing information was distributed conditionally treating missing information as negative led to the highest payoff for both strategies. However, in terms of payoff, TTB outperformed WADD due to its lower search costs. WADD required much more information to make an inference than TTB, with on average 8.2 cue values, compared to only 3.2 cue values for TTB. The strategies’ accuracies, however, were similar, with on average 73% correct decisions for both strategies.

To determine the search costs of WADD, we assumed limited information search. That is, we assumed that WADD would search for cues in the order of the cues’ validities and always consider the cue values for both alternatives. With this information, WADD computes the difference of the cue values multiplied by the cues’ validities. Information search was stopped whenever the preliminary decision based on the weighted difference could not be changed by the additional cues that had not been searched. For instance, when the three most valid cues spoke in favour of the first alternative, the remaining three cues could not change the preliminary decision, and the search would stop. Because it implements limited search instead of assuming exhaustive search, WADD provides a fair test against TTB, in that participants’ limited information search becomes consistent across the two strategies.

In each trial, we recorded four dependent variables: (a) how many cues were searched for, (b) which cues were searched for, (c) in which order the cues were searched for, and (d) what decision was made. Additionally, after the sixth trial block, participants had to estimate the cue validities. Specifically, for each cue they were asked: “How often, out of 100 decisions, does this cue make a correct prediction, given that one candidate has a positive cue value, and the other candidate has a negative cue value?” Participants had to provide numbers between 50 and 100. Afterwards, participants had to rank cues according to their subjective validity. Finally, in the seventh trial block, after making each decision, participants had to indicate how they treated the unknown cue values they encountered. For each cue value with missing information, participants had to select one of four possible options: (a) “I ignored the missing information”, (b) “I treated missing information as if it was negative”, (c) “I treated missing information as if it was positive”, and (d) “I treated missing information as if it was in between positive and negative information”. The last response corresponded to using the average for replacing missing cue values without requiring participants to compute or recall the exact average cue values. After the seventh trial block, participants were asked again to estimate the cue validities and to rank the cues accordingly.

Results

We first examined participants’ accuracy and their ability to estimate the cue validities correctly. Thereafter, we analysed how well the different mechanisms for treating missing information when implemented in the two strategies could predict participants’ inferences. We further analysed participants’ information search preceding their inferences.

Accuracy of decisions and of estimates of validities

We first conducted an analysis of variance (ANOVA)⁴ with participants’ accuracy as a dependent variable, the environment as a between-subjects factor, and the trial block as a within-subject factor. The main effect of trial block was significant, F(5.1, 348.7) = 6.06, p < .01. In all three conditions, the participants increased the accuracy of their inferences across the trial blocks, from 67% correct decisions in the first trial block to 72% in the last, F(1, 69) = 18.8, p < .01. In addition, an interaction between environment and trial block was also observed, so that only in the last trial block did participants make more correct decisions in the condition with no missing information than when missing information was conditionally distributed (76 vs. 69%), F(10.1, 348.7) = 3.04, p < .01.

Degrees of freedom for the analyses containing repeated measures factors were corrected by using the Greenhouse–Geisser technique (Greenhouse & Geisser, 1959). Note that this technique can only be applied when factors have more than two levels. We conducted Tukey's HSD (honest significant difference) test in post hoc analyses.

Second, we examined whether the participants were able to learn the objective cue validities. We determined the average correlation between the objective cue validities in the item set and the cue validities estimated by the participants, which was, on average, rather low (r = .35, SD = 0.48 when there was no missing information; r = .30, SD = 0.40 when missing information was uniformly distributed; and r = .21, SD = 0.31 when missing information was conditionally distributed). These correlations, however, differed significantly from zero, t(22) = 3.8, p < .01 when there was no missing information; t(22) = 3.9, p < .01 when missing information was uniformly distributed; and t(22) = 3.6, p < .01 when missing information was conditionally distributed. Nevertheless, the subjective cue validities differed substantially from the objective cue validities, and the participants did not learn the validities accurately.

We think there are two reasons for this: First the objective validities were very similar, making it hard to distinguish between them. Presumably more extensive learning opportunity needs to be provided to allow participants to identify validities. Second, due to the high search costs, participants limited their information search, so that they did not receive feedback about all cues in each trial. This might also have hindered the process of learning the objective cue validities. Given the discrepancies between the subjective and objective cue validities we determined the predictions of the strategies reported below on the basis of the subjective cue validities. For this purpose, the average estimated cue validities after the sixth and the seventh trial blocks were used for each individual to determine the strategies’ predictions.

How participants treated missing information

Did participants select the most adaptive mechanism for treating missing information? To answer this question, we computed for each trial block the percentage of inferences in which participants’ decisions were predicted by TTB or WADD when each of the five mechanisms for treating missing information was employed. The percentages of trials in which the strategies using different mechanisms made different predictions when using the subjective cue validities are reported in Table 1. This analysis was performed for the condition in which missing information was uniformly and conditionally distributed. We then compared the percentage of predicted inferences with the percentage of predicted inferences by TTB and WADD in the baseline condition with no missing information. Note that this analysis was performed exclusively on participants’ decisions and does not consider their search processes (for a similar procedure, see Bröder, 2000; Garcia-Retamero et al., 2007a, 2007b). How participants searched for information was analysed separately.

Table 1.

Percentage of trials in which the strategies using different mechanisms for treating missing information make different predictions for Experiments 1 and 2

Trial type	Prediction	No. of discriminating trials
Trial type	Prediction	Exp. 1	Exp. 2
Treating missing information as negative vs. as the average	(TTB_negative = WADD_negative) ≠ (TTB_average = WADD_average)	15	19
Treating missing information as negative vs. as positive	(TTB_negative = WADD_negative) ≠ (TTB_positive = WADD_positive)	41	39
Treating missing information as the average vs. as positive	(TTB_average = WADD_average) ≠ (TTB_positive = WADD_positive)	10	15
TTB vs. WADD	(TTB ≠ WADD)	54	55

Note: TTB: take the best heuristic. WADD: weighted additive strategy.

We conducted an ANOVA with the percentage of predicted inferences as the dependent variable, strategy, mechanism, and trial block as within-subjects factors, and environment as a between-subject factor. For the sake of simplicity, we only report the interactions that are relevant for our predictions. Note that the main effects are of no interest, because they cannot be meaningfully interpreted. Interactions of a higher order than those we tested in our analyses were not significant.

To test whether participants selected the mechanisms for treating missing information adaptively depending on the environment, the interaction effect between environment and mechanism is crucial, and this is significant in our data analyses, F(4.4, 150.6) = 20.53, p < .001. We examined which of the five mechanisms for treating missing information led to the highest fit of the strategies in each environment. Consistent with the adaptivity prediction, in the environment with uniformly distributed missing information the strategies reached the best fit with an average of 76% predicted inferences when missing information was treated as the average (p < .01 for all comparisons). In contrast and, again, consistent with the adaptivity prediction, in the environment with conditionally distributed missing information, treating missing information as negative led to the highest fit with an average of 77% predicted inferences (see Figure 3). Therefore, participants seemed to select the mechanism for treating missing information adaptively depending on how missing information was distributed in the environment.

To test which strategy participants in the different environments selected to make inferences, the interaction effect between strategy and environment is crucial. These two factors are also modulated by the trial block, F(12, 414) = 2.52, p = .01, which implies that the strategy that participants used to make inferences changed with experience. We then compared the fit of TTB and WADD across trial blocks and environment conditions. In the first trial block and across all three environments, WADD, with an average of 73% predicted inferences, predicted participants’ inferences better than TTB with an average of 68% (p < .01 for all comparisons). In line with previous results (e.g., Rieskamp & Otto, 2006), participants apparently had an initial preference to integrate the available information by selecting a compensatory strategy. However, TTB's fit increased over the seven trial blocks, leading to a higher fit for TTB than WADD in the last trial block in the environment with no missing information (80 vs. 74%, p = .01) and the environment with uniformly distributed missing information (77 vs. 69%, p < .01). In contrast, in the environment with conditionally distributed missing information, the fit of both WADD and TTB increased across trial blocks, so that the fit for WADD tended to be higher than that for TTB in the last trial block with 74 and 70% predicted inferences, respectively (p = .06).

Did participants’ reports of how they treated missing information coincide with the mechanisms they seemed to be using for making inferences? To answer this question, we analysed participants’ indications of how they treated unknown cue values in the last trial block. Specifically, participants were classified according to the mechanism that they reported using most frequently. Results showed that more participants in the environment with uniformly distributed missing information treated missing information as if it were in between positive and negative information, whereas more participants in the environment with conditionally distributed missing information treated missing information as negative (see Table 2). A χ² test for the contingency table with the five mechanisms and the environment as two dimensions showed that the frequency distribution differed for the two environments, χ²(3) = 17.00, p < .01.

Table 2.

Participants’ indications of how they treated missing information in the last trial block in Experiments 1 and 2

Experiment	Experimental condition	Treatment of missing information
Experiment	Experimental condition	Negative	Positive	Between positive and negative	Ignored
1	Uniform	2 (8)	6 (25)	15 (63)	1 (4)
1	Conditional	14 (58)	6 (25)	3 (13)	1 (4)
2	Uniform	4 (20)	4 (20)	10 (50)	2 (10)
2	Conditional	12 (60)	4 (20)	3 (15)	1 (5)

Note: The number of participants (with percentages in parentheses) classified according to the mechanism that they reported to be using most frequently is presented. Sample sizes in Experiments 1 and 2 differ. Experimental condition: Uniform or conditional refers to distribution of missing information.

Information search

In addition to participants’ inferences, we also recorded how many and which cues were searched for and in which order before making an inference. In this way, we analysed whether participants’ search behaviour was more alternative-wise (i.e., after searching for a cue value, they searched for another cue value for the same alternative) or more cue-wise (i.e., after searching for a cue value, they searched for the value of the same cue but for the other alternative). We determined the search index (SI) proposed by Payne (1976) to describe participants’ search processes, which is defined by SI = (n_a − n_c)/(n_a + n_c), where n_a is the number of alternative-wise search steps, and n_c is the number of cue-wise search steps. The index can take values between − 1 and + 1, where positive (negative) values indicate a more alternative-wise (cue-wise) search.

In the last trial block, participants’ information searches were more cue-wise than alternative-wise in all three environments. When there was no missing information or when missing information was uniformly distributed, the average value for the search index was − .76 (SD = .44) and − .71 (SD = .46) compared with an average of − .57 (SD = .65) when missing information was conditionally distributed. This result is consistent with a more frequent selection of TTB in the last trial block in the environment with no missing information or uniformly distributed missing information than in the environment with conditionally distributed missing information. Note that for asymmetric information boards with more cues than alternatives, as in our experiment, random search leads to positive values of the SI (Böckenholt & Hynan, 1994). Therefore, the negative values that we obtained for the search index can be interpreted as even stronger evidence for a cue-wise search.

How much information did the participants search for? Overall, participants searched for 50% of the available cue values (i.e., 6 of the 12 cue values). An ANOVA with the number of cue values that participants searched for as the dependent variable, trial block as a within-subject factor, and environment condition as a between-subjects factor showed a significant main effect of environment condition, F(2, 69) = 3.94, p = .02. Overall the results show that the number of cue values that participants searched for was lower in the environment with uniformly distributed missing information than in the environment with conditionally distributed missing information. The interaction between trial block and environment was also significant, F(12, 414) = 4.58, p < .01. In the environment with uniformly distributed missing information, the number of cue values that participants searched for decreased across trial blocks from 6.3 cue values in the first trial block to 4.9 in the last. In contrast, when there was no missing information, or when missing information was conditionally distributed, participants did not vary in the number of cue values that they searched for across trial blocks (6.6 and 6.4, respectively; see Figure 4), and they searched for much more information than participants in the environment with uniformly distributed missing information (p < .01). These results are consistent with the simulations of Garcia-Retamero and Rieskamp (2008), who found that processing missing information as the average was much more frugal than the other mechanisms when it was implemented in TTB.

To explain why the participants were more frugal when missing information was uniformly distributed than when it was conditionally distributed, we examined in more detail how the mechanisms for treating missing information influenced the search process. When missing cue values are treated as the average, a cue with missing information for one of the alternatives discriminates between the alternatives regardless of whether the cue had a positive or a negative cue value for the second alternative. In contrast, when missing cue values are treated as negative a cue with missing information for one of the alternatives discriminates between the alternatives only if the cue had a positive value for the second alternative. Consequently, when participants apply the mechanism adaptively, it can be predicted that the search for cues should, on average, stop earlier in the environment with uniformly distributed missing information than in the environment with conditionally distributed information. We analysed how many cue values participants searched for in the last trial block after encountering a cue with missing information for one of the alternatives and with either a positive or a negative value for the second alternative. In line with the predictions, when participants encountered a cue with a negative value for one of the alternatives and with missing information for the second alternative, they looked up fewer cue values afterwards when missing information was uniformly distributed than when it was conditionally distributed (2.1 vs. 4.1), t(23) = 2.93, p < .01. In contrast, participants’ search did not differ in either environment after encountering a cue with missing information for one alternative and a positive value for the other alternative (3.8 vs. 3.7), t(23) = 0.45, p = .76. These results are in line with an adaptive use of the mechanisms for treating missing information for the two environments studied.

Discussion

The accuracy of the mechanisms for treating missing information depends crucially on how missing information is distributed in an environment (Garcia-Retamero & Rieskamp, 2008). From this result we derived several predictions about how people should treat missing information to behave adaptively. In line with these predictions, participants treated missing information as the average when it was uniformly distributed, whereas they treated it as negative when it was conditionally distributed. This result was found by analysing the fit of two inference strategies implementing the different mechanisms for treating missing information, and it was also found in the direct reports of the participants on how they treated missing information. Participants’ information search processes were consistent with these results.

However, the participants also showed maladaptive behaviour: Most participants initially selected WADD to make their inferences, which leads to a low payoff. Even more surprising, in the condition with conditionally distributed missing information this preference for WADD did not change over the course of the experiment. However, in the other two experimental conditions the participants seemingly learned to select TTB, as evidenced by the high fit of TTB in the last trial block.

Experiment 2

Experiment 1 showed that the participants selected the mechanisms for treating missing information adaptively. This finding, however, was observed under high search costs relative to the gains of correct inferences. The mechanisms for treating missing information influence the amount of information a strategy requires, because some of these mechanisms lead to more frequent discrimination between alternatives than others. For instance, treating missing information as negative or as the average requires less information than ignoring missing information. Thus, the adaptive selection of the mechanisms in Experiment 1 might be mainly due to the high search costs: Possibly, participants inferred the most likely value of the missing information instead of ignoring unknown cue values to avoid searching for much more costly information. It is therefore an open question whether participants’ adaptive behaviour can be generalized to situations with low information search costs. This generalization of Experiment 1's results appears necessary to provide a satisfying answer to the question of why people treat missing information differently as described in the literature reported above.

In a situation with low search costs, people might be less motivated to form expectations about missing cue values and simply ignore cues with partial information. Evidence from preferential choice research supports this hypothesis: People ignore missing information when they are not highly motivated to process information (e.g., when the consequences of the inferences are not important; Zhang & Markman, 2001). Ignoring missing cue values, however, would not be the most adaptive behaviour. Instead, treating missing information as the average or as negative still leads to higher inferential accuracy than ignoring missing information when missing information is uniformly or conditionally distributed (see Garcia-Retamero & Rieskamp, 2008). Thus, to behave adaptively, people should select these two mechanisms for treating missing information even in a situation with low search costs. This adaptivity prediction is tested in Experiment 2, in which the participants could search for information without any costs.

Method

Participants

A total of 60 students (34 men and 26 women), average age 25 (range 19–33) years, participated in the experiment. Participants were randomly assigned to one of three equally sized groups (n = 20) based on the environment condition. The computerized tasks were conducted in individual sessions and lasted approximately 1 hour. Participants received one fifth of the total amount they accrued in the task, leading to an average payment of €11 (ranging from €10 to €15).

Design and procedure

The instructions and the procedure of Experiment 2 were identical to those of Experiment 1, except that no search costs were imposed in Experiment 2. Participants earned 50 eurocents for a correct inference and lost 50 cents for a wrong inference. In the control condition with no missing information, the application of TTB and WADD led to a payoff of €53 and €55, respectively. In the condition with missing information uniformly distributed, treating missing information as the average, as a negative value, or ignoring it led to a payoff of €45, €35, or €30, respectively, when the mechanisms were implemented in TTB, whereas they led to a payoff of €50, €35, or €30, respectively, when implemented in WADD. In the third condition with missing information conditionally distributed, treating missing information as the average, as a negative value, or ignoring it led to a payoff of €36, €61, or €22, respectively, when the mechanisms were implemented in TTB, whereas they led to a payoff of €35, €65, or €25, respectively, when implemented in WADD. Thus, to receive the highest payoff it was most important to select the most adaptive mechanism for treating missing information, whereas whether TTB or WADD was selected was less important.

Results

We first examined participants’ accuracy. Thereafter, we analysed how well the different mechanisms for treating missing information could predict participants’ inferences. We further analysed participants’ information search preceding their inferences.

Accuracy in decisions and in validity estimations

The ANOVA with participants’ accuracy as a dependent variable, environment as a between-subjects factor, and trial block as a within-subject factor showed a main effect of environment, F(2, 57) = 6.59, p < .01, and trial block, F(4.4, 248.5) = 17.3, p < .01. Participants achieved higher inferential accuracy in the environment with no missing information (76%) than in the environment with uniformly distributed (71%) or conditionally distributed (73%) missing information. Furthermore, in all three environments, participants increased their inferential accuracy across trial blocks from 67% correct inferences in the first trial block to 75% correct inferences in the last trial block, F(1, 57) = 49.03, p < .01.

In line with the results of Experiment 1, on average the correlation between the subjective and the objective cue validities was rather low (r = .67, SD = 0.24, when there was no missing information; r = .18, SD = 0.40, and r = .14, SD = 0.48, when missing information was uniformly and conditionally distributed, respectively). Nevertheless, these correlations differed significantly from zero, t(18) = 4.8, p < .01 when there was no missing information; t(18) = 2.4, p = .03 when missing information was uniformly distributed; and t(18) = 2.3, p = .03 when missing information was conditionally distributed. Consequently, the strategies’ predictions were computed for each participant using the participant's subjective cue validities instead of the objective cue validities.

Models’ fit and participants’ reports on how they treated missing information

Did the participants apply different mechanisms for treating missing information adaptively in the different environments as predicted? To test this prediction the interactions between strategy, trial block, mechanism, and environment in the ANOVA with the percentage of predicted inferences as the dependent variable are crucial. As in Experiment 1, main effects are not reported as they cannot be meaningfully interpreted. Interactions of a higher order than those we tested in our analyses were not significant. The interaction between the environment, the strategy, and the mechanism was significant, F(8, 228) = 2.13, p = .034. In line with the adaptivity prediction, in the environment with uniformly distributed missing information, with 82% predicted inferences WADD achieved the highest fit by treating missing information as the average. In the environment with conditionally distributed missing information, with 81% predicted inferences WADD achieved the highest fit when missing information was treated as negative (p < .01 for all comparisons; see Figure 5). In line with the results of Experiment 1, the participants in Experiment 2 in which no information search costs were imposed selected the mechanism for treating missing information adaptively depending on the environment.

The ANOVA also revealed a significant interaction between environment, strategy, and trial block, F(12, 342) = 2.32, p = .007. In the first trial block, WADD predicted more choices than did TTB in the three environments (76 vs. 64%). After the first trial block, this initial preference changed: When there was no missing information, the fit of TTB increased over trials, leading to an equal fit of TTB and WADD in the last trial block (78 vs. 75%). In contrast, when missing information was uniformly or conditionally distributed, the fit of both WADD and TTB increased over the trial blocks but overall WADD achieved the highest fit in all trial blocks (77 vs. 70% for WADD vs. TTB when missing information was uniformly distributed, and 74 vs. 67% for WADD vs. TTB when missing information was conditionally distributed). These results match the different performances of WADD and TTB: In the environment with no missing information WADD and TTB performed equally well, whereas in the environments with missing information WADD outperformed TTB.

Finally, the results of the analyses of participants’ reports of how they treated missing information coincided with the mechanisms they seemed to be using when they made their inferences. More participants in the environment with uniformly distributed missing information reported treating missing information as if it were in between positive and negative information, whereas in the environment with conditionally distributed missing information participants most frequently reported that they treated missing information as negative (see Table 2). A χ² test of the contingency table with the five possible reported mechanisms and the environment as the two dimensions showed that the frequency distribution differed for the two environments, χ²(3) = 8.1, p = .04.

Information search

Participants’ information search was generally cue-wise in all three environments but to a lesser extent than in Experiment 1. Specifically, in the environment with no missing information, the average value for the SI was − .51 (SD = .59) compared to an average of − .50 (SD = .53) and − .61 (SD = .52) when missing information was uniformly distributed and conditionally distributed, respectively. The predominantly cue-wise search is partly surprising, because for WADD, which predicted participants’ inferences well, at first glance an alternative-wise search would be expected. However, WADD can also be performed by determining a weighted difference for the two alternatives as described in the Introduction, for which a cue-wise search can be expected. Therefore, we conclude that many participants who applied WADD did not determine a score for each alternative separately.

Compared to Experiment 1, in which participants searched for 50% of the cues, participants in Experiment 2 looked up on average 90% of the available information. This result can be attributed to no search costs in Experiment 2. Also in contrast to the results of Experiment 1, when missing information was uniformly distributed, participants were not more frugal than when it was conditionally distributed. The ANOVA Environment × Trial Block on the number of cue values that participants searched for did not reveal a significant effect. On average, participants looked up 10.4 cue values per trial when there was no missing information. When missing information was uniformly distributed and conditionally distributed, they looked up 11.3 and 10.6 cue values per trial, respectively. This result is consistent with WADD's high fit in Experiment 2: When selecting WADD more information is required than when selecting TTB. In addition, treating missing information as the average when missing information is uniformly distributed only leads to a lower amount of required information than the alternative mechanisms when implemented in TTB (Garcia-Retamero & Rieskamp, 2008). In contrast, when selecting WADD the different mechanisms for treating missing information do not lead to different amounts of required information for the three environments, which is in line with the experimental results.

General Discussion

How do people make inferences with incomplete information? Previous research on the topic has led to a rather mixed picture, which can be explained by assuming that people select different mechanisms for treating missing information adaptively. The accuracy of different mechanisms for treating missing information crucially depends on the way missing information is distributed in the environment (Garcia-Retamero & Rieskamp, 2008). Specifically, treating missing information as the average of the available information is most adaptive in environments with uniformly distributed missing information. In contrast, treating missing information as if it was negative is most adaptive when missing information is conditionally distributed depending on the criterion value. Interestingly, ignoring missing information does not lead to a high inferential accuracy compared with the other mechanisms. In addition, the different mechanisms for treating missing information partly influence the amount of information a strategy requires. Processing missing information as the average is more frugal than the alternative mechanisms, but only when the mechanism is implemented within TTB. In contrast, for WADD the different mechanisms do not substantially affect the amount of required information (Garcia-Retamero & Rieskamp, 2008). These results enabled us to make predictions of how people should treat missing information adaptively, which were tested in two experimental studies.

Treating missing information adaptively

In Experiment 1 we imposed high information acquisition costs in comparison to Experiment 2 with no search costs. In line with our predictions, the results showed that participants selected the mechanism for treating missing information adaptively: They treated missing information as the average (negative) when unknown cue values were uniformly (conditionally) distributed in the environment. Furthermore, the selection of mechanisms for treating missing information was not influenced by the different search costs: We found these results both when there were high information acquisition costs and when there were none. Even in a situation of no information search costs, participant preferred to infer the most likely value of the missing information instead of ignoring unknown cue values. Participants’ reports of how they treated missing information coincided with the mechanisms they seemed to be using when they made decisions. Furthermore, the amount of information that participants searched for correlated with the mechanism they selected for treating missing information. In line with our predictions, participants searched for little information when missing information was uniformly distributed, and high information costs existed. Here the participants treated missing information as the average and selected TTB for making their inferences, which was the most frugal way of making inferences. These findings are important for how inference strategies should be defined. When considering that people treat missing information differently depending on the environment, this implies that defining only one single way of treating missing information for a strategy (see, for instance, Gigerenzer & Goldstein, 1996, or Gigerenzer et al., 1991) will not be very successful for predicting people's inferences.

How do people learn to treat missing information? The two experiments showed that people treat missing information differently depending on the environment. Because no prior knowledge about the environments was available, participants adapted to the environments on the basis of their own experience. How can such a learning process of treating missing information be described in detail? We assumed that people apply different strategies for solving inference problems they face, and based on the reinforcement of outcomes, the successful strategies become more likely to be selected (Rieskamp & Otto, 2006). In a situation with missing information, it can be assumed that the strategies’ successes using different mechanisms are explored. However, when considering five different mechanisms for treating missing information and two inference strategies, the implication is that 5 × 2 different strategies need to be explored. Due to this large number a lengthy learning process will be required. Because we observed a short learning process it appears reasonable that people do not only consider the strategies’ success as a whole, but also explore the success of the mechanisms for treating missing information separately, regardless of in which inference strategy they are incorporated. How this learning process can be computationally specified needs to be explored further.

How surprising is it that participants treated missing information adaptively? Recent research has shown that people also often show maladaptive behaviour when making inferences. For instance, people frequently search for too much information (Newell et al., 2003) or select maladaptive strategies (Bröder & Schiffer, 2003; Rieskamp, 2006a). In general, it is an open debate to what extent people are able to adapt to environments by showing optimal behaviour (Busemeyer & Myung, 1992; Rieskamp, 2006b; Rieskamp, Busemeyer, & Laine, 2003; Wallsten, 1968). Participants in our experiments did not behave completely adaptively, either. For instance, in Experiment 1, TTB led to a much higher payoff than WADD due to the high search costs. However, even at the end of the experiment after participants had gained substantial experience, in one experimental condition in which missing information was conditionally distributed in the environment WADD predicted slightly more inferences than TTB. Thus, when considering this maladaptive behaviour, the adaptive way participants treated missing information in both experiments was surprising and is an important finding of the present article.

Generalization to other inference situations

We examined a specific two-alternative inference situation with dichotomous cues. We had chosen this task because it has frequently been used by other researchers to study probabilistic inferences (e.g., Bröder, 2000; Gigerenzer & Goldstein, 1996; Newell & Shanks, 2003). These previous studies have neglected the issue of missing information, and our research shows that in situations with missing information people apparently modify their inference strategy by using different mechanisms for treating missing information depending on the environment.

It will be interesting to explore the generalizability of our findings to other inference situations. In particular it is worthwhile examining how people react to missing information in a situation with continuous cues. When replacing missing information by the average cue value it is less important to have an exact estimate of the average when dealing with dichotomous cues. That is, for TTB any value between a negative and positive cue value will lead to the identical inference. In contrast, in a situation with continuous values the exact estimates will be crucial, as they can lead to different inferences. Here one needs to explore whether the estimates depend on the cue values’ distribution. In cases in which missing information is correlated with the criterion value, this implies that it would be most adaptive to replace missing information with conditional means (average cue value for alternatives with low or high criterion values). In addition, it is an interesting question how people treat missing information when dealing with other tasks such as quantitative estimation. Instead of comparing alternatives as in our inferential choice task, in an estimation task people make inferences about the criterion value of one single alternative (e.g., von Helversen & Rieskamp, 2008). It is an interesting question, whether people will select the different mechanism for treating missing information also adaptively when encountering other inference problems such as quantitative estimations.

Our experiments used situations in which accurate feedback was given after each individual trial. Thus, a condition with good learning opportunities was used to test how people treat missing information. Whether adaptive behaviour is also observed when feedback is scarce and partly erroneous, as in many real-life situations, is an open question for future research. Possibly, in such situations individual learning will be much slower than that observed in our experiments.

Explaining previous mixed findings on how people treat missing information

Missing information is a common characteristic of everyday inference problems. Therefore it is surprising that previous work that tested the performance of different inference strategies often did not devote much attention to the issue (e.g., Gigerenzer & Goldstein, 1996, 1999; Hogarth & Karelaia, 2005, 2006, 2007). The research that has examined how people treat missing information has led to rather mixed results: People seem to be able to use various inferential mechanisms for treating missing information. However, it is unclear when people use a certain mechanism and how the selection of different mechanisms can be predicted.

Our results shed light on one of the factors that influence the accuracy of the different mechanisms for treating missing information and could possibly provide an explanation for the mixed results in the literature. In some of the previous research, the experiments were designed in such a way that the participants most likely believed that unknown cue values were negatively correlated with the criterion that had to be predicted (see Huber & McCann, 1982; Jagacinski, 1991; Johnson & Levin, 1985; Meyer, 1981, for examples). For instance, in an experiment by Jagacinski (1991) with a personnel selection task, participants were informed that the position required two crucial skills: management and computer programming skills. Participants were presented with candidates with missing information on the important skills. Jagacinski (1991) showed that candidates for whom the important pieces of information were missing were highly penalized. That is, participants were very suspicious of job candidates whose files lacked information and considered them as less competent than those who had complete applications. Thus, these participants apparently assumed that missing information was negatively correlated with the candidates’ capability and treated such missing information as negative. This belief might be adaptive as the cost of hiring an unqualified worker is generally very high (Cascio, 1987).

Alternatively, in other experiments, participants most likely assumed that missing information was uniformly distributed in the environment as there was no indication that missing information was conditionally distributed depending on the criterion that had to be predicted (see Ganzach & Krantz, 1990; Gluck & Bower, 1988; Nosofsky, Kruschke, & McKinley, 1992; Slovic & MacPhillamy, 1974; White & Koehler, 2004). In these cases, participants most frequently treated unknown cue values as the average. For instance, in a classic study by Slovic and MacPhillamy (1974), participants evaluated pairs of students described in terms of a common and a unique measure of ability. Therefore, all students had some missing information, and this missing information was completely uncorrelated with the common ability measure or the students’ capacity. Interestingly, participants often commented that they had assumed an average value for the measure that each member of the pair was missing. In sum, when making inferences with incomplete information, participants in various experiments often selected the mechanism for treating missing information depending on how they perceived missing information to be distributed in the environment.

We have argued that the way missing information is distributed in the environment has an important influence on the accuracy of the different mechanisms for treating missing information. Accordingly, when aiming for high accuracy, different mechanisms for treating missing information should be selected in different environments. This prediction was confirmed in two experimental studies. The adaptive response to characteristics of the environment is in line with recent research on probabilistic inferences (e.g., Bröder, 2003; Newell & Shanks, 2003; Newell et al., 2003; Rieskamp, 2006a; Rieskamp & Otto, 2006) and is consistent with the general view that people's reasoning can be understood as an adaptation to environmental situations (see also Anderson, 1991; Payne et al., 1988, 1993). We conclude that the different ways that people react to missing information is also an adaptive response to environmental characteristics.

Footnotes

Appendix

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