Probabilistic inferences are a common task for decision makers, given that to-be judged criteria are rarely known conclusively. Often enough, judgments must be made on the basis of some probabilistic cues (e.g., Gigerenzer, Hoffrage, & Kleinbölting, 1991), and it is thus a central issue to understand how people perform such tasks. In typical studies on probabilistic inferences, participants are shown pairs of objects (e.g., cities) and are asked to infer for each pair which of the two objects has the higher criterion value (e.g., population). Given that the answer is unknown, whereas both objects are familiar, participants may retrieve probabilistic cues about them (e.g., whether either has an international airport) to reach a decision (cf. Gigerenzer & Goldstein, 1996). If, however, only one or neither of the objects in a pair are known, other strategies are required. This is one of the main assumptions of the “adaptive toolbox” approach (Gigerenzer, Todd, & the ABC Research Group, 1999), which assumes that decision makers are equipped with a mental repertoire of specialized tools (mostly heuristics), selected ad hoc to solve a specific task.

As an alternative, Brown and Tan (2011) recently proposed that participants may engage in magnitude comparisons that are based on subjectively built orders of the objects involved. They tested a set of 16 cars (asking for the more expensive one in each pair) and found that participants’ decision times were better explained by assuming magnitude comparisons based on subjective linear orders than by assuming the use of specific heuristics, as predicted by the toolbox approach. However, Brown and Tan’s findings were limited to situations in which all of the objects are familiar. Thus, we here extend their approach to testing the role of linear orders with known and unknown objects (and all possible pairings of these). Also, we employ additional dependent measures, namely the deviation rate (measured in relation to one’s subjective order) and the rate of inconsistencies across repeated trials. Finally, we analyze the data from two experiments, thus assessing the robustness of the observed effects.

In the following sections, we will more fully describe the idea of linear orders as the basis for magnitude comparisons and what the toolbox approach predicts, assuming multiple specific strategies. We will then describe tests of these predictions in a reanalysis of a published experiment (Hilbig & Pohl, 2009, Exp. 3) and in a new experiment.

Linear orders

Brown and Tan (2011) suggested that decision makers rely on magnitude comparisons based on the subjective linear order of objects—thus avoiding the need to switch tools from trial to trial. Prior research had shown that in comparative tasks in which well-learned objects are ordered along a criterion, people possess or build subjective linear orders and subsequently base their decisions on this mental representation (see, e.g., Banks, 1977; Moyer & Bayer, 1976; Moyer & Dumais, 1978; Parkman, 1971). The central and most robust finding from these studies is the so-called “symbolic distance effect”: The larger the distance between two objects in their underlying order, the shorter the reaction time (and the fewer errors). This corresponds to what Brown and Tan found for pairs of recognized cars differing in price. In addition to this empirical argument, using such a linear order in repeated paired comparisons would arguably necessitate less effort than having to work through several potential decision strategies over and over (see Pohl, 2011).

Thus, it seems plausible that participants, if confronted with a new set of objects, quickly develop a subjective linear order of these objects and subsequently base comparisons on this order. Importantly, not only recognized objects could be so ordered, but unrecognized ones, too (cf. Marewski, Gaissmaier, Schooler, Goldstein, & Gigerenzer, 2010). For example, cues connected to the unrecognized object’s name might be used: Some city names simply sound big, and others small (see McCloy, Beaman, Frosch, & Goddard, 2010, for examples, and Marewski, Pohl, & Vitouch, 2011, for empirical findings on this topic). Or, participants might simply know more about an object than they admit, so that not all objects called “unrecognized” by participants would necessarily be truly novel (cf. Erdfelder, Küpper-Tetzel, & Mattern, 2011). Indeed, this could also explain why performance for pairs of unrecognized objects is often better than chance (see, e.g., Hilbig, Erdfelder, & Pohl, 2010).Footnote 1 In some cases, recognized objects might even score lower than unrecognized ones (cf. Smithson, 2010). For example, a recognized city could have negative cue values (e.g., no university, no airport, no industry) and therefore receive a relatively low position in the linear order. All of this information would be useful in ordering objects, both recognized and unrecognized.

If, as this reasoning implies, all objects are represented on the same continuum, a distance effect should be observed for all types of pairs—that is, with both objects known (labeled “knowledge case”), with only one object known (“recognition case”), or with no objects known (“guessing case”): The farther apart two objects in one’s subjective order, the faster the decision. By the same token, the probability of decisions deviating from one’s order (or showing inconsistencies) should decrease with the distance between objects, again for all types of pairs.

The toolbox approach

The toolbox approach assumes that decision makers possess a repertoire of specific strategies that are tailored to certain situations (Gigerenzer et al., 1999). When inferring which of two objects has the larger criterion value, several different cases and corresponding strategies would need to be differentiated (see Pohl, 2011, for a summary of these strategies). What would these strategies predict for the decision times for pairs of objects differing in their distances to each other on the underlying dimension?

In the case of two recognized objects (knowledge case), one potential strategy would be the fluency heuristic (FH; Hertwig, Herzog, Schooler, & Reimer, 2008; Schooler & Hertwig, 2005). This strategy exploits the difference in recognition times of the two objects and infers that if the difference is large enough, the object recognized more speedily would have the larger criterion value—ignoring any further cues or information (but see Hilbig, Erdfelder, & Pohl, 2011, for a critical evaluation of this aspect). Assuming that recognition times (fluency) decrease with objects’ positions on the continuum (Hertwig et al., 2008; Herzog & Hertwig, in press; Marewski & Schooler, 2011), one could deduce that recognition time differences should be larger for two distant than for two close objects. Thus, the FH could predict a distance effect in decision times. Another strategy for knowledge cases would be the take-the-best heuristic (TTB; Gigerenzer & Goldstein, 1996), which assumes that probabilistic cues are searched in the order of their validity until one is found that discriminates between the two objects. The inference is then based on this cue alone (but see Ayal & Hochman, 2009; Bröder, 2000). If none is found, participants resort to guessing. Arguably, two close objects would have more identical cue values than two distant objects, implying that the time needed to find a discriminating cue should be greater for two close than for two distant objects (see Bröder & Gaissmaier, 2007). Thus, TTB could also predict a distance effect for pairs of recognized objects. In contrast, Brown and Tan (2011) argued that in their study TTB did not predict the observed distance effect, because the pairs that they used could have been answered by considering the same highly valid cue, thus implying no decision-time differences.

When only one object is recognized, the toolbox proposes to apply the recognition heuristic (RH; Goldstein & Gigerenzer, 1999, 2002): The decision maker simply infers that the recognized object is the larger one—again ignoring all further information (see Hilbig, 2010, for an overview of critical findings). Marewski et al. (2010) recently proposed that the RH will be more often used when recognition times for the recognized object are short rather than long. In other words, recognition times (fluency) may act as a cue not only in the FH, but also in the RH.Footnote 2 Thus, again assuming that recognition times decrease with the objects’ relative positions, and further assuming that larger objects (with shorter recognition times) are, on average, more often included in pairs of two distant than in pairs of two close objects, a distance effect could be predicted: The fast RH would be more often applied in pairs of two distant than of two close objects.

Finally, if neither object is recognized, participants are typically assumed to know nothing about the two objects, implying that decision times should also not differ systematically between different distances.

A central prediction, independent of the distance between objects, concerns the decision times for different types of pairs (knowledge, recognition, and guessing). In the toolbox approach, the RH is assumed to be the first and foremost of all heuristics, and should thus lead to shorter decision times than would a strategy that must be selected if the RH cannot be applied (e.g., FH or TTB). So, on average, recognition cases should require less time than knowledge cases. Previous data have supported this prediction (e.g., Pachur & Hertwig, 2006), although only partially (Hilbig & Pohl, 2009). In any case, if all objects within a linear order were treated equally, one would not expect an effect of this nature. Main effects of the type of pair are thus difficult to reconcile with the linear-order perspective. We will return to this problem in the General Discussion.

Reanalyzed data

A previously published study (Hilbig & Pohl, 2009, Exp. 3) contained sufficient data to allow for a reanalysis that addresses the following questions: Is there evidence for the postulated distance effect in decision times? And, if so, does it hold for all types of pairs (i.e., recognition, knowledge, and guessing cases)? In that experiment, 68 participants made inferences for 91 pairs of cities (consisting of the 14 largest Swiss cities, excluding the largest, Zurich), judging for each pair which of the two cities is more populous.

To obtain individual rank orders, we assigned ordinal ranks to each city according to the relative choice frequency of that city, separately for each participant. The mean rank correlation (Kendall’s tau) between these subjective orders and the true order was τ = .47 (SD = .13), t(67) = 29.083, p < .001. When computed separately for recognized and unrecognized cities, the respective mean correlations were τ = .56 (SD = .34), t(60) = 12.881, and τ = .38 (SD = .26), t(60) = 11.517, both p < .001. Thus, participants’ subjective rank orders were typically valid; that is, they corresponded well with the true order of objects.

We initially analyzed the distance effect for the whole set (summing across types of pairs, leaving in the sample N = 45 who provided data for all distances).Footnote 3 The results show a clear distance effect (see Fig. 1); that is, decision times decreased substantially from Distance 1 (M = 2,329 ms, SD = 888 ms) to Distance 13 (M = 1,040 ms, SD = 449 ms), F(12, 528) = 38.937, p < .001, η 2p = .48. Next, we split the data conditional on the type of pair. Because not all participants provided data for each type of pair and each distance (especially not for the larger ones, which are inherently more rare), we included only those data cells to which at least ten participants contributed data. As is shown in Fig. 2, all three types of pairs showed a distance effect. Linear regressions yielded similar slopes of –70, –86, and –90 ms for the recognition, knowledge, and guessing cases, respectively. Finally, we also found a significant main effect of type of pair on decision times (see Fig. 2), F(2, 120) = 24.378, p < .001, η 2p = .09. Specifically, we observed shorter decision times for recognition cases (M = 1,518 ms, SD = 492 ms) than for knowledge (M = 1,852 ms, SD = 710 ms) or guessing (M = 2,000 ms, SD = 842 ms) cases, whereas the latter two did not differ significantly.

Fig 1
figure 1

Mean decision times (in milliseconds) as a function of the distance in each participant’s subjective rank order (data from Hilbig & Pohl, 2009, Exp. 3). Error bars represent one standard error of the mean

Full size image
Fig 2
figure 2

Mean decision times (in milliseconds) for recognition, knowledge, and guessing cases as a function of the distance in each participant’s subjective rank order (data from Hilbig & Pohl, 2009, Exp. 3)

Full size image

In sum, there was clear evidence for a distance effect in decision times that held for each type of pair, thus supporting the linear-order view. By contrast, the main effect of type of pair is inconsistent with a linear-order perspective. However, one potential caveat of these results is that individual rank orders were not explicitly stated by the participants, but only indirectly reconstructed from their choice data. Thus, we ran a new experiment in which we assessed subjective orders directly and, in addition, sought further dependent measures beyond decision times.

Experiment

This experiment again used the typical city-size task. In addition, to test the robustness of the predicted distance effect across tasks, we manipulated (between subjects) whether participants were asked to indicate the more or the less populous city in each pair. Logically, both questions are equivalent and should not lead to different decision behavior. However, two previous studies that featured the same manipulation (Hilbig, Scholl, & Pohl, 2010; McCloy et al., 2010) both found that more choices were in line with the RH in the “larger” condition than in the “smaller” condition. Also, the first of these studies reported a main effect for decision times, which were shorter for the “larger” question than for the “smaller” question. The latter result corresponds to linear-order findings showing that the “larger” question yields faster answers for comparing large objects and the “smaller” question for comparing small objects—termed a “semantic congruity effect” (see, e.g., Banks, 1977; Banks, Fujii, & Kayra-Stuart, 1976; Moyer & Dumais, 1978). In most RH studies (including the present one), the materials have consisted exclusively of “large” objects from the upper end of the criterion scale. Consequently, we expected to find a main effect of the question format on decision times. More interestingly, we aimed to test whether the distance effect would be the same for both questions (as Banks et al., 1976, had found).

Finally, we repeated the whole set of paired comparisons for all participants, thus allowing us to measure the consistency of inferences as an alternative measure, which has rarely been considered in the probabilistic-inference literature but which—according to the reasoning above—should corroborate the predicted distance effect.

Method

Participants and materials

A total of 80 students (64 female, 16 male; age: M = 22.3 years, SD = 3.5 years) participated in return for a flat fee payment of €8 or course credit. They were randomly assigned to one of two groups, differing only in the instructions that they received in the paired-comparison task (see the Procedure section). Group 1 consisted of 42 participants, and Group 2 of 38. As materials, we exhaustively paired the 11 largest Belgian cities (excluding Brussels, the largest one, in order to avoid influences of criterion knowledge—that is, knowledge about a city’s rank position—that would make inferences obsolete; cf. Hilbig, Pohl, & Bröder, 2009; Pachur, Bröder, & Marewski, 2008), resulting in 55 pairs that were presented twice to each participant.

Procedure

First, participants were asked, for each single city (presented in random order), whether or not they recognized its name (indicated by pressing one of two keys on the keyboard). The reaction times of these answers (i.e., the recognition times) were recorded. Next followed the paired-comparison task. The 55 pairs of cities were presented one at a time and in random order. Participants were asked to indicate speedily and as accurately as possible which of the cities in a pair was the larger one (in Group 1) or the smaller one (in Group 2). Choices and reaction times were recorded. After completing the 55 pairs, the same set was presented again, but in a new random order and with altered positions (left/right) on the screen. This repetition was not announced or otherwise signified to the participants; thus, subjectively, the task consisted of 110 trials without any interruption. Finally, participants were asked to subjectively rank order the 11 cities by adjusting their relative positions on the screen (using a mouse) in an initially alphabetically ordered list. The experiment lasted about 30 min in total.

Results and discussion

Overall decision times

For all analyses, the median decision time of each person (per condition) served as the dependent measure. To test whether decision times differed between conditions, we ran a three-way ANOVA with Group (larger? vs. smaller?) as a between-subjects factor and Phase (1 vs. 2) and Type of Pair (recognition, knowledge, and guessing) as within-subjects factors.Footnote 4 The ANOVA revealed several effects: (1) Decision times were shorter in recognition cases (M = 1,107 ms, SD = 301 ms) than in knowledge cases (M = 1,468 ms, SD = 731 ms) or guessing cases (M = 1,397 ms, SD = 607 ms), F(2, 128) = 18.286, p < .001, η 2p = .18, while the latter two types did not differ significantly.Footnote 5 (2) The decision times of Group 1 (M = 1,235 ms, SD = 624 ms) trended toward being shorter than those of Group 2 (M = 1,418 ms, SD = 550 ms), F(1, 64) = 3.35, p = .07, η 2p = .05. In other words, the “larger” inferences were drawn faster than the logically equivalent “smaller” inferences. (3) Decision times were significantly shorter in Phase 2 (M = 1,186 ms, SD = 468 ms) than in Phase 1 (M = 1,461 ms, SD = 673 ms), F(1, 64) = 53.723, p < .001, η 2p = .46, which most likely resulted from learning or practice effects.

Subjective rank orders

The mean rank correlation (Kendall’s tau) between subjective orders and the true order was τ = .38 (SD = .19), t(79) = 18.534, p < .001. When computed separately for recognized versus unrecognized cities, the respective mean correlations were τ = .34 (SD = .41), t(53) = 5.940, and τ = .16 (SD = .33), t(79) = 4.269, both ps < .001. Thus, participants’ subjective orders generally had some objective validity. Moreover, recognized cities received predominantly higher subjective ranks than did unrecognized ones. Consequently, in only 6.4 % of all recognition cases was an unrecognized city subjectively considered larger than a recognized one.

Distance effect

To test for distance effects, we analyzed decision times, deviation rates, and inconsistencies conditional upon the distance in subjective rank-order positions of the cities in a pair. Due to the nature of linear orders, the number of cases declines with growing distance in the order. Analyses considering all experimental conditions thus led to many empty cells (or low sample sizes) for larger distances. Therefore, we initially collapsed the data across types of pairs and phases, and could thus keep all participants in the sample. We then ran two-way ANOVAs with Group (larger? vs. smaller?) as the between-subjects factor and Distance (1–10) as the within-subjects factor.

Decision times decreased significantly from Distance 1 (M = 1,451 ms, SD = 587 ms) to Distance 10 (M = 1,002 ms, SD = 318 ms), F(9, 702) = 31.604, p < .001, η 2p = .29, thus showing a distance effect (Fig. 3). As reported above, Group 1 had shorter decision times (M = 1,115 ms, SD = 420 ms) than did Group 2 (M = 1,260 ms, SD = 441 ms), F(1, 78) = 3.961, p = .05, η 2p = .05. There was no interaction between group and distance, F < 1; that is, the distance effect was similarly large for both question formats.

Fig 3
figure 3

Mean decision times (in milliseconds) for the two experimental groups as a function of the distance in subjective rank orders. Error bars represent one standard error

Full size image

Similarly, deviation rates (defined as the proportions of choices deviating from the choice implied by one’s subjective rank order) declined from Distance 1 (M = 36.4 %, SD = 11.6 %) to Distance 10 (M = 3.1 %, SD = 14.5 %), F(9, 702) = 136.726, p < .0001, η 2p = .64 (Table 1). Groups 1 and 2 (with Ms = 12.6 % and 14.7 %, respectively) did not differ significantly, and there was no interaction, both Fs < 1.

Table 1 Deviation rates (according to participants’ subjective rank orders) and inconsistency rates (between Phases 1 and 2) as a function of rank-order distance (1–10) and type of pair (recognition, knowledge, or guessing)
Full size table

In addition, we analyzed inconsistencies in choices between Phases 1 and 2, computing the probability of providing different judgments for the same pair. These inconsistency rates, too, declined significantly from Distance 1 (M = 8.8 %, SD = 0.4 %) to Distance 10 (M = 0.9 %, SD = 0.5 %), F(9, 702) = 55.719, p < .001, η 2p = .46 (Table 1). The two groups (with Ms = 3.9 % and 4.4 %, respectively) did not differ in their proportions of inconsistencies, F(1, 78) = 1.016, p = .32, and there was no interaction, F < 1.

Next, we split the decision time data (shown in Fig. 3) according to the type of pair, excluding cells to which fewer than ten participants contributed data. The resulting mean decision times showed distance effects for all three types of pairs (see Fig. 4). However, statistical analyses were feasible only for recognition cases, where a sufficient number of participants (viz. 72) entered the analysis. In a two-way ANOVA with Group and Distance as factors, the distance effect for recognition cases was significant, F(9, 630) = 14.729, p < .001, η 2p = .17. As before, the two groups also differed significantly, F(1, 70) = 10.058, p = .002, η 2p = .13. As can be seen from Fig. 4, the other two distance effects for knowledge and guessing cases were highly similar. Simple linear regression lines for the mean decision times found slopes of –44, –69, and –46 ms for recognition, knowledge, and guessing cases, respectively.

Fig 4
figure 4

Mean decision times (in milliseconds) for recognition, knowledge, and guessing cases as a function of the distance in participants’ subjective rank orders

Full size image

To test whether and how recognition times (fluency) relate to the distance effect in decision times, we ran additional analyses including only cells with data from at least ten participants. Unfortunately, tests of significance were not feasible, because the number of participants who contributed to all included cells was too low. Thus, we report only descriptive results and 95 % confidence intervals (95 % CIs).

For knowledge cases, the mean differences in retrieval times of the two recognized cities in a pair increased with increasing subjective distance of these cities, from Distance 1 (M = 321 ms, SD = 304 ms, 95 % CI [245, 397]) to Distance 6 (M = 621 ms, SD = 409 ms, 95 % CI [399, 843]). These increasing fluency differences might in turn have led to more frequent use of the fast FH, and thus to the observed distance effect in decision times. As such, the distance effect for two known cities may be compatible with the toolbox approach.

For recognition cases, we found that the mean fluency decreased with the recognized city’s subjective rank position, from Rank 1 (M = 1,104 ms, SD = 534 ms, 95 % CI [981, 1,227]) to Rank 6 (M = 1,733 ms, SD = 750 ms, 95 % CI [1,340, 2,126]). In other words, subjectively smaller cities were recognized more slowly than larger ones (Hertwig et al., 2008; Herzog & Hertwig, in press; Marewski & Schooler, 2011). However, when analyzed as a function of the distance in subjective rank orders, pairs of different distances showed nearly the same mean fluency for the recognized city, from Distance 1 (M = 1,136 ms, SD = 554 ms, 95 % CI [1,012, 1,259]) to Distance 10 (M = 1,083 ms, SD = 611 ms, 95 % CI [943, 1,223]). Thus, fluency cannot account for the observed distance effect in recognition cases, and thus, the effect is incompatible with the toolbox view.

Finally, we also split the deviation and inconsistency rates reported above according to type of pair. We found for both measures that the observed distance effect persisted for each type of pair (see Table 1).

In sum, we found that decision times, deviation rates (in relation to subjective orders), and inconsistencies (in repeated inferences for the same pair of cities) decreased with increasing rank difference of the cities in a pair. These effects were quite large. Most importantly, and replicating our previous findings reported above, the distance effect was not only observed for knowledge cases (with both cities known, as in the Brown & Tan, 2011, study), but also for recognition cases (with only one city known), and even for guessing cases (with neither city known). Assumptions derived from the toolbox heuristics (RH, FH, and TTB) could explain the distance effect in our data only for knowledge cases, but not for either recognition or guessing cases.

General discussion

In this article, we extended and tested Brown and Tan’s (2011) assumption that decision makers base their probabilistic inferences in paired comparisons on subjective linear orders of the tested objects. This contrasts to the toolbox approach (Gigerenzer et al., 1999), which assumes a repertoire of specialized heuristics for different types of comparisons.

We reanalyzed a previously published data set (Hilbig & Pohl, 2009, Exp. 3) and presented a new experiment to test the proposed role of linear orders: (1) The central result is the distance effect in decision times that was found in the reanalyzed data set, as well as in our new experiment: The larger the subjective distance between two cities, the faster the decision. This distance effect, moreover, was present for all three types of pairs (recognition, knowledge, and guessing). (2) The distance effect was further corroborated by alternative dependent measures, namely deviation rates and inconsistencies: Both declined with increasing subjective distance, and again for all types of pairs. In sum, these observations are well aligned with the predictions from a linear-order perspective. (3) As expected from the semantic congruity effect (Banks et al., 1976), the question format (larger? vs. smaller?) had a main effect on decision times, with shorter decision times for the “larger?” question (see also Hilbig, Scholl, & Pohl, 2010; McCloy et al., 2010). Nevertheless, the distance effects were comparable in magnitude for both types of questions (as Banks et al., 1976, had also found).

The toolbox approach, with its assumption of multiple strategies that fit to different situations, can easily explain the distance effect in knowledge cases, either through the fluency heuristic (FH; Hertwig et al., 2008; Schooler & Hertwig, 2005) or the take-the-best heuristic (TTB; Gigerenzer & Goldstein, 1996). Brown and Tan (2011) interpreted their distance effect as incompatible with the TTB, because the same highly valid cue could have been used in pairs of different distances.

A potential explanation of the distance effect in recognition cases could be derived from a recent suggestion that decision makers use fluency as a cue not only in knowledge cases, but also in recognition cases (Marewski et al., 2010). However, in our experiment, the observed fluencies for the recognized city in pairs of different distances were highly similar and did not increase with distance. Thus, weighting the recognition cue by fluency cannot account for the observed distance effect in recognition cases. Consequently, both the original RH and recent extensions fail to account for this distance effect.

Finally, in guessing cases, with two unknown cities, participants should have nothing left but to guess. However, the data show that these cases were affected by subjective distance, too, suggesting that participants possess (or infer) some knowledge for subjectively ordering the “unknown” cities. That these subjective orders were not totally random, but at least partially built on valid knowledge, was shown by the positive rank-order correlations with the true order. To explain the distance effect from a toolbox viewpoint, one would need to argue that guessing cases were handled just like knowledge cases (see above), albeit with less knowledge.

The most problematic finding for the linear-order perspective, though it is well aligned with the toolbox assumptions, was that recognition cases were answered faster than either knowledge or guessing cases (Pachur & Hertwig, 2006). However, research has shown that the overall main effect of type of pair may actually be misleading, as it is not generally true that recognition cases yield faster decisions than knowledge cases (Hilbig & Pohl, 2009). Nonetheless, if linear-order representations include and treat all objects equally, irrespective of their recognition status, decision times should only depend on distance, not on type of pair. As such, our findings cannot be fully accounted for by the linear-order perspective.

Overall, neither the toolbox approach nor the linear-order perspective is fully compatible with all observations. A possible remedy to these theoretically inconsistent findings would be to consider an alternative approach, namely the difference in evidence (cf. Hilbig & Pohl, 2009; Newell, 2005). To derive predictions for different types of pairs, assume, merely for demonstration, that the amount of evidence is mapped on a scale from 0 to 4: Unrecognized city names, for example, will elicit no (0) or only very limited evidence (1) regarding their size. Thus, on average, the difference in evidence between two unrecognized cities would typically be small, making an inference rather difficult. Recognized city names, on the other hand, may elicit more diverse amounts of evidence, ranging from mere recognition without any further knowledge (2), over one or more probabilistic cues (3), up to full criterion knowledge (4; cf. Hilbig et al., 2009). Correspondingly, pairs of cities that differ more strongly in terms of the evidence that speaks to their criterion value will be answered more quickly than pairs with less differing evidence (see Hilbig & Pohl, 2009). Following this approach, the mean difference in evidence will typically be larger for recognition cases (with exactly one object recognized) than for guessing or knowledge cases (no matter how the evidence scale is defined), so that decision times should be fastest for recognition cases, thus explaining the main effect of type of pair. Further assuming that knowledge is typically correlated with the criterion (i.e., that people know more about larger cities than about smaller ones), it is clear that the mean difference in evidence increases with increasing distance of the two objects on the underlying continuum, thus also accounting for the distance effect.

As such, the difference in evidence (or, conversely, the degree of conflict) between two options determines decision times—as predicted by evidence accumulation (e.g., Busemeyer & Townsend, 1993; Newell, Collins, & Lee, 2007) and network models (e.g., Glöckner & Betsch, 2008a, b) of decision making: The larger the evidential difference, the faster the decision. This explanation should be tested by assessing (or manipulating) the subjective evidence for the objects in a set, and thereby predicting decision times. This method was not incorporated here, but appears to be a promising route for future research (see Pachur et al., 2008).