Experiment 1 (with n₁ = 31 participants) aimed at comparing different model predictions with respect to the novel set of trials we introduced (Figure 1B and C) and at establishing the core finding of Chau2014 by replicating it. In the decision-making task as introduced by Chau2014 (Figure 1A), two available options of different expected value, HV and LV, are presented in two randomly selected quadrants of the screen and participants are asked to choose the option with the higher expected value. In half of the trials, the remaining two quadrants are left empty, in the other half, a third distractor option, D, is shown in one quadrant but indicated as unavailable after 0.1 s. Then, participants have another 1.5 s to make their choice. The central behavioral analysis by Chau2014 was a logistic regression of relative choice accuracy (i.e., whether HV or LV was chosen while excluding all trials with other responses such as choosing D or the empty quadrant, or being too slow) on the value difference between the two available options, HV-LV, the sum of their values, HV +LV, the value difference of HV and D, HV-D, the interaction of value differences, (HV-LV)×(HV-D), and whether a distractor was present or not, D present. Most importantly, the authors reported a significantly negative regression coefficient of HV-D indicating that higher values of D increased choice accuracy.

To test whether our results replicate the findings of Chau2014, we analyzed decisions made in the (non-novel) trials that were identical to those used by Chau2014. Note that the choice sets in Chau2014 were generated by sampling magnitude and probabilities for HV, LV, and D until HV-LV and HV-D shared less than 25% variance. The options’ average expected values in the resulting trials were 5.13 for HV (SD = 3.08; min = 0.5; max = 10.5), 3.51 for LV (SD = 3.02; min = 0.25; max = 9), and 4.32 for D (SD = 2.11; min = 0.25; max = 9). Although the overall choice performance in our data was strikingly similar to the original data (Table S1 in Supplementary file 1), the negative effect of HV-D on relative choice accuracy could not be replicated. Instead, the average coefficient was positive but not significant (t(30) = 1.40, p = .171, Cohen’s d = 0.25; left panel of Figure 2A; complete results of all regression analyses for all experiments are reported in Tables S2-S5). Interestingly, analyzing absolute choice accuracy (i.e., including all trials) resulted in a significant positive regression coefficient of HV-D (t(30) = 5.14, p < .001, d = 0.92; right panel of Figure 2A). This suggests that a higher value of D lowered choice accuracy when all trials were taken into account.

Figure 2

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To better understand why D’s value has this negative impact on the probability of choosing HV, we analyzed the different possibilities of making errors (i.e., choosing LV, choosing D, choosing the empty quadrant, and being too slow) by testing whether they were predicted by D’s value. The value of D had a significant effect on the probability of choosing D (t(28) = 5.92, p < .001, d = 1.11) and also on being too slow (t(29) = 2.66, p = .013, d = 0.49), whereas the probability of choosing LV (t(30) = −0.47, p = .645, d = −0.08) or the empty quadrant (t(23) = 1.07, p = .294, d = 0.22) were unaffected (Figure 2B). Notably, higher values of D also slowed down response times (RT) for HV and LV choices (t(30) = 6.17, p < .001, d = 1.11; Figure 2C).

We then looked at choice accuracy in the novel trials that we added to differentiate between specific context effects and the models proposed by Louie2013 and Chau2014. Performance was higher in trials with Ds that were dominated by either HV or LV (Figure 2C), which is the opposite of what is predicted by the Chau2014 model (Figure 1D). This effect of Dominance was significant with respect to both relative choice accuracy (F(1,30) = 5.85; p = .022, ${\eta }_p^2$ = .16) and absolute choice accuracy (F(1,30) = 18.89; p < .001, ${\eta }_p^2$ = .39), the former being consistent with divisive normalization, the latter being consistent with value-based attentional capture (see Figure 1D; but note that–inconsistent with divisive normalization–we could not replicate the effect on relative choice accuracy in Experiments 2 and 3; Figure 5, Table S6-S7 for complete results). Overall, behavior in the novel trials supported the notion that higher-valued Ds impair multi-alternative decision making.

In summary, we could not replicate the results on relative choice accuracy and the IIA violations reported by Chau2014 but obtained opposite effects. At first glance, these results seem to support the divisive normalization account by Louie2013 as an appropriate mechanistic explanation. Note, however, that this account specifically predicts changes in relative choice accuracy, for which we found only weak evidence (i.e., no effect of HV-D in the regression, no effect of D’s value on LV-errors). On the other hand, the effects on absolute choice accuracy were stronger. Accordingly, we reasoned that the results are better explained by value-based attentional capture: This account states that attention is modulated by value such that value-laden stimuli attract attention and impair goal-directed actions, even if those stimuli are irrelevant to the task (Anderson, 2016; Anderson et al., 2011; Grueschow et al., 2015; Le Pelley et al., 2016; Peck et al., 2009; Yasuda et al., 2012). With respect to Chau2014’s paradigm, value-based attentional capture predicts that high-value Ds draw attention to a greater extent than low-value Ds, which leaves less cognitive resources to focus on the decision between HV and LV. Consequently, decisions slow down, leading to higher RT and to more too-slow errors, and choices of D increase, but the relative probability of choosing HV over LV is unaffected–implying no violation of IIA. In Experiment 2 and 3, we sought to establish value-based attentional capture as the underlying mechanisms by testing different predictions that this explanation (but not divisive normalization) makes.

If attentional capture is the driving force behind the performance decrease in the Chau2014 task, then increasing available attentional capacity should improve choice accuracy and reduce the detrimental effects of D. In contrast, divisive normalization effects do not seem to require the imposition of time pressure (see Louie2013). Based on this rationale, we conducted a second behavioral experiment in which we compared two groups: A high-time pressure (HP) Group (n_2,HP = 25) did exactly the same task as in Experiment 1, but a second, low-time pressure (LP) Group (n_2,LP = 24) was given more time to decide (6 s instead of 1.5 s). According to the attentional-capture account, we should replicate the results of Experiment one under high time pressure, but under low time pressure the negative influences of D on (absolute) choice accuracy should disappear. For the novel trial sets, we expected a choice accuracy pattern in line with multi-attribute context effects (see Figure 1D), given that these effects are known to become more prominent with longer deliberation time (Dhar et al., 2000; Pettibone, 2012; Trueblood et al., 2014).

Consistent with Experiment 1, we could not replicate the HV-D effect on relative choice accuracy reported in Chau2014. Again, our results showed a tendency in the opposite direction (t(24) = 0.50, p = .622, d = 0.10). With respect to absolute choice accuracy, there was a strong and significantly positive effect of HV-D in Group HP of Experiment 2 (t(24) = 5.05, p < .001, d = 1.01). In contrast and as predicted, this effect was much weaker and did not reach significance in Group LP (t(23) = 1.85, p = .079, d = 0.38; but note that the difference between groups was not significant: t(47) = 0.61, p = .544, d = 0.17; Figure 3A). Similarly, only participants of Group HP were less accurate in trials with D present compared to trials without D (Group HP: t(24) = -5.97, p < .001, d = -1.19; Group LP: t(23) = -0.06, p = .956, d = -0.01; group difference: t(47) = -4.07, p < .001, d = -1.16; Figure 3A). Choices of D were significantly more frequent in Group HP (5.8% vs. 0.9% of trials; t(47) = 4.15, p < .001, d = 1.20; Figure 3B). With respect to the novel trial set, we could not replicate the main effect of Dominance on relative choice accuracy in Group HP that we had found in Experiment 1 (F(1,24) = 0.25; p = .622, ${\eta }_p^2$= .01; left panel of Figure 3C). In Group LP, there was a significant main effect of Similarity (F(1,23) = 9.20; p = .011, ${\eta }_p^2$ = .25; right panel of Figure 3C) that was absent in any of our experiments conducted under high time pressure. This (IIA-violating) effect was due to a higher relative choice accuracy when D was more similar to HV than to LV in the two-dimensional attribute space and is consistent with a combination of an attraction effect (Huber et al., 1982) and a phantom-decoy effect (Pettibone and Wedell, 2007; Pratkanis and Farquhar, 1992; see Figure 1D and Materials and methods).

Figure 3

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Taken together, while Group HP replicated the results of Experiment one in most aspects, Group LP showed that the negative impact of D disappeared when time pressure was alleviated, lending further support for a value-based attentional capture account. In addition, only Group LP exhibited an IIA-violating choice pattern in the novel trials consistent with specific context effects, supporting the notion that such effects require longer deliberation times to emerge (Dhar et al., 2000; Pettibone, 2012; Trueblood et al., 2014).

Value-based attentional capture predicts an effect of the value of D on choice accuracy that is mediated by attention. In other words, a higher value of D leads to more attention to D, which in turn impairs absolute choice accuracy. This effect has been referred to as value-based oculomotor capture and is thought to underlie the (behavioral) effect of value-based attentional capture (Failing et al., 2015; Le Pelley et al., 2015; Pearson et al., 2016). To directly test whether value-based oculomotor capture drives the negative influence of D on decision making in the Chau2014 paradigm, we obtained eye-movement data as a measure of attention in Experiment three with n₃ = 23 participants using the same paradigm as in Experiments 1 and 2 (Group HP).

The behavioral results of this experiment were in line with our previous experiments (see Tables S2-S7 for statistical results): The negative HV-D effect on relative choice accuracy reported in Chau2014 could not be replicated, but its effect on absolute choice accuracy was significantly positive; the value of D led to more choices of D and slowed down choices of HV and LV; there was a main effect of Dominance on absolute (but not relative) choice accuracy in the novel trial set. With respect to the eye-movement results, we first tested whether options of higher value received more attention, defined as relative fixation duration (see Materials and methods). Thereto, we calculated within each participant the correlation between expected value and relative fixation duration separately for HV, LV, and D. These correlations were consistently higher than zero (HV: t(22) = 4.79, p < .001, d = 1.00; LV: t(22) = 3.84, p < .001, d = 0.80; D: t(22) = 6.25, p < .001, d = 1.30; Figure 4A). Second, we asked whether the dependency of attention to D on D’s value mediated the negative influence of the latter on choice accuracy. A path analysis confirmed this prediction (Figure 4B): In the path model, the value of D had a positive effect on attention to D (t(22) = 6.30, p < .001, d = 1.31), which in turn had a negative effect on choice accuracy (t(22) = −6.79, p < .001, d = −1.42). In fact, the influence of D’s value on choice accuracy was fully mediated by attention with the direct path being not significant (t(22) = −0.76, p = .455, d = −0.16). Finally, we tested whether participants whose attention was affected by D’s value to a greater extent also showed a stronger negative influence of D’s value on absolute choice accuracy. This was confirmed by a significantly negative correlation across participants between the coefficient quantifying the correlation between D-value and D-fixations and the coefficient quantifying the negative impact of D-value on absolute choice accuracy (r(21) = −0.61; p = .002; Figure 4C). Altogether, the eye-movement results strongly support value-based attentional and oculomotor capture as being the underlying mechanism of sub-optimality in the Chau2014 paradigm (see Materials and methods, Figure 4—figure supplement 1 and Figure 7—figure supplement 3 for further eye-tracking results).

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We conducted a fourth experiment, for which we used the exact trial sequences of Chau2014 (provided to us by the authors) and omitted the novel trials. Besides replicating the effects of value-based attentional capture, another goal of this Experiment 4 was to test whether we could find the HV-D effect on relative choice accuracy when making the experiment almost identical to Chau2014. The statistical results of this experiment with n₄ = 44 participants are provided in Tables S2-S5 and are in line with Experiment 1 to 3: The negative HV-D effect on relative choice accuracy could not be replicated, there was a significantly positive HV-D effect on absolute choice accuracy, and higher values of D led to significantly more choices of D and to longer RT when choosing HV or LV. Figure 5 summarizes the behavioral results collapsed over all experiments conducted under high time pressure. This summary provides clear evidence that there were no robust effects on relative choice accuracy (and thus no violations of IIA) but strong effects on absolute choice accuracy consistent with the value-based attentional capture account.

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Notably, most of the behavioral data analyses in Chau2014 and our study relied on the assumption that participants can integrate magnitude and probability to compute the expected value (EV = probability × magnitude) of each option and to choose the option with the larger EV. We tested this assumption by comparing a simple EV-based choice model with two other models that assumed that participants focused on only one attribute (i.e., either magnitude or probability) as well as an expected utility (EU) model (e.g., Von Neumann and Morgenstern, 1947) and prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992); see Materials and methods). In brief, we found that the EV model explained the choice data better than the single-attribute models, but that the EU model provided the best account of the data (Figure 5—figure supplement 1A). The additional complexity of prospect theory compared to EU was not justified by an increased model fit. For reasons of simplicity and comparability to Chau2014, we used the EV-based value estimates for most statistical and modeling analyses. As a robustness check, however, we reanalyzed our main behavioral tests and replaced EVs by EU-based subjective values. The results were largely unaffected by this adaptation (Figure 5—figure supplement 1B).

We also reanalyzed the data of Chau2014 to look for further evidence of value-based attentional capture. As in our own experiments, we found that HV-D was positively linked to absolute choice accuracy (t(20) = 4.53, p < .001, d = 0.99), and that higher values of D led to both more erroneous choices of D (t(19) = 4.67, p < .001, d = 1.05) and to higher RT when choosing HV or LV (t(20) = 2.94, p = .008, d = 0.64). Thus, the data of Chau2014 also supported value-based attentional capture. In addition, we show with multiple additional analyses that are detailed in Materials and methods and in Figure 5—figure supplement 2 to Figure 5—figure supplement 5 that the originally reported negative effect of HV-D on relative choice accuracy is a statistical artifact that is due to an incorrect implementation of the interaction term (HV-LV)×(HV-D) in the performed regression analysis. After correcting this error, the effect of HV-D disappears.

Although value-based attentional capture is a well-established empirical finding (Anderson, 2016; Le Pelley et al., 2016), to the best of our knowledge it has never been implemented into a decision-making model so far. In the following, we propose and test the Mutual Inhibition with Value-based Attentional Capture (MIVAC) model that accounts for the complex interplay of value and attention on choice (Figure 6; details are provided in Methods). MIVAC is an extended mutual inhibition (MI) model, a sequential sampling model that assumes a noisy race-to-bound mechanism of separate, leaky, and mutually inhibiting accumulators for each choice option (Bogacz et al., 2006; Usher and McClelland, 2001). Notably, the MI model (without the extensions we propose for MIVAC) is equivalent to a mean-field reduction of the cortical attractor model that was applied by Chau2014 (Bogacz et al., 2006; Wang, 2002; Wong and Wang, 2006), and it is indeed capable of predicting a positive effect of the value of D on relative choice accuracy (see Figure 6—figure supplement 1). MIVAC assumes an accumulator for each choice option, so four accumulators in the case of the Chau2014 paradigm (for HV, LV, D, and the empty quadrant). A choice is made as soon as an accumulator reaches an upper boundary that collapses in time (to account for the time limit of the task; for example Gluth et al., 2012; Gluth et al., 2013a; Gluth et al., 2013b; Hutcherson et al., 2015; Murphy et al., 2016). Based on our behavioral and eye-movement results, we propose three additional mechanisms. First and foremost, value-based attentional capture is implemented by assuming that the probabilities of fixating particular options depend on their expected values. In other words, more valuable options receive more attention. Second, the input to the accumulator of the currently fixated option is enhanced, consistent with the influence of attention on choice reported in previous work (Cavanagh et al., 2014; Krajbich and Rangel, 2011; Krajbich et al., 2010; Shimojo et al., 2003) and observed in our own data (see Figure 4—figure supplement 1 and Materials and methods). Finally, D can be identified as unavailable, in which case the expected value of D is assumed to be 0, and its accumulation rate and fixation probability are adjusted accordingly. This feature is a specific requirement for the Chau2014 paradigm in which participants are instructed not to choose D (this feature can be omitted for other experimental paradigms).

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We propose MIVAC to explain the central behavioral findings across all experiments. Furthermore, we illustrate that simpler models without the added mechanisms of MIVAC (in particular, without value-based attentional capture) cannot explain these findings. Thereto, we conducted rigorous quantitative model comparisons in which MIVAC was compared to three models, each of them missing one of the three novel components (i.e., without VAC = without value-based attentional capture; without AE = without attentional enhancement; without DD = without distractor detection) and a baseline MI model, without any of these components. When fitting the models to trials in which a distractor D was present, we found a clear advantage of MIVAC compared to the simplified versions in terms of both average model fit and best model fit per participant (Figure 7A). We then performed a generalization test (Busemeyer and Wang, 2000) by using the models’ estimated parameters from the trials with D to predict behavior in the trials without D. Again, MIVAC outperformed the simpler alternatives (Figure 7B). This generalization test provides strong support for a genuinely better description of the data by MIVAC and rules out an overfitting problem. Qualitatively, MIVAC predicts choice proportions of all potential actions very accurately, and it does so better than the simplified versions without value-based attentional capture with respect to both trials with D present and D absent (upper and middle panels of Figure 7C). The most important test for MIVAC was whether the model accurately predicts the observed increase in choices of D as a function of the value of D. As can be seen in the lower panel of Figure 7C, MIVAC predicts this pattern, and the alternative models either fail to do so or overpredict the frequency of choices of D. MIVAC also reproduces all RT effects reported in Figure 5, that is, negative effects of the difference and the sum of values of HV and LV, and positive effects of the value of D and the presence of D (Figure 7D; compare with Figure 5C). This is particularly remarkable because MIVAC was fitted only to the choice but not to the RT data. Finally, generalizing MIVAC to the novel trials shows that the model exhibits the observed qualitative patterns for both relative and absolute choice accuracy (Figure 7E; compare with Figure 5D). All these results hold when replacing the EV-based input to the accumulators by EU-based subjective values (Figure 7—figure supplement 2) or when simulating the model with fixations drawn from the empirical fixation duration distributions (Figure 7—figure supplement 3).

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Thirty-one participants (21 female, age: 20 – 47, M = 27.71, SD = 6.59, 29 right-handed) completed Experiment 1. A total of 51 participants signed up for Experiment 2. Due to computer crashes, the data of two participants (one from Group HP and one from Group LP) were incomplete and excluded from the analyses, resulting in a final sample of 49 participants, 25 were in Group HP (13 female, age: 20–46, M = 26.88, SD = 6.62, 24 right-handed) and 24 were in Group LP (11 female, age: 19–35, M = 23.96, SD = 3.43, 20 right-handed). Participants were assigned randomly to the two groups. Thirty participants signed up for the eye-tracking Experiment 3. One participant was excluded for not passing the training-phase criterion, and additional six participants were excluded due to incompatibility with the eye-tracking device (for further details see section Eye-tracking procedures and pre-processing), resulting in a final sample of 23 participants (14 female, age: 18–54, M = 25.70, SD = 8.66, 19 right-handed). Forty-seven participants signed up for Experiment 4. Due to failing the training-phase criterion, three participants were excluded, resulting in a final sample of 44 participants (36 female, age: 18–46, M = 23.70, SD = 5.74, 40 right-handed). The sample size for Experiment 1 was based on the assumption that testing 1.5 times as many participants as in the original study (Chau2014) would suffice to replicate its main results. In Experiment 4, we tested more participants to ensure a statistical power of >0.90 for replicating the effect of HV-D on relative choice accuracy (note that the effect size in Chau2014 was d = −0.495). The sample size for Experiments 2 and 3 were based on observing strong effects (i.e., d ≥ 0.8) of value-based attentional capture in Experiment 1 (but it should be noted that no formal power analysis was conducted). All participants gave written informed consent, and the study was approved by the Institutional Review Board of the Department of Psychology at the University of Basel. All experiments were performed in accordance with the relevant guidelines and regulations. Data of all participants included in the final samples are made publicly available on the Open Science Framework at https://osf.io/8r4fh/.

The paradigm was very similar for all four experiments. Participants repeatedly chose between either two (binary trials) or three (distractor trials) two-outcome lotteries (gambles), each yielding an outcome of magnitude X in Swiss Francs (CHF) with probability p or 0 otherwise. The gambles were represented by rectangles, each shown in a random quadrant of the screen, whereby the rectangles’ colors represented outcomes X and the angles represented the probabilities p. Outcomes ranged from CHF 2 to CHF 12 in steps of CHF 2 and were represented by colors ranging from either green to blue or blue to green. Probabilities ranged from 1/8 to 7/8 in steps of 1/8 and were represented by orientation angles ranging from 0° to 90° or from 90° to 0° in steps of 15° (see Figure 1B for all colors and orientations). Associations between colors/outcomes and orientations/probabilities were counter-balanced between participants. In binary trials, participants saw the two options for 100 ms before orange frames appeared around each option (pre-decision phase). After the frames appeared, participants had up to 1.5 s to make a choice by pressing 7, 9, 1, or 3 on the numeric keypad for upper left, upper right, lower left, or lower right quadrant, respectively (participants belonging to Group LP of Experiment 2 had 6 s after appearance of the frames to decide). Distractor trials were similar to binary trials: All of the options were presented for 100 ms before frames appeared around them. Contrary to the binary trials, one of the options had a magenta frame (the distractor), signaling that it could not be chosen. Choosing the distractor resulted in a screen telling that the option was not available after which a new trial began. Similarly, choosing an empty quadrant resulted in a screen showing that the quadrant was empty after which a new trial began.

If a valid choice was registered, the trial continued with a choice-feedback phase for 1 to 3 s, in which the chosen option was highlighted in a dark red color. To ensure that participants paid attention to all available options, participants had to complete a ‘match’ trial before the choice-feedback phase in 15% of all trials. On match trials, one of the options from the decision phase (including the distractor, if distractor trial) was presented in the middle of the screen. Participants had up to 2 s (6 s in Experiment 2, Group LP) to press the key corresponding to the option’s quadrant. If correct, participants saw a screen saying ‘correct’ and an extra CHF 0.10 were added to the participant’s account, otherwise they saw a screen saying ‘wrong’. After every match trial, the trial continued with the choice-feedback phase.

After the choice-feedback phase, participants received feedback about the outcomes of the gambles for 1 to 3 s. The frames’ colors changed to grey if the option did not yield a reward (i.e., the outcome was CHF 0) or to golden yellow if the option yielded a reward. Participants also received feedback about the distractor’s outcome on distractor trials. After this outcome-feedback phase, a new trial began with an inter-trial interval of 1.5 to 3 s in which a fixation cross was shown.

After giving informed consent and filling out the demographic questionnaire, participants received detailed instructions about the task and were familiarized with the outcome and probability associations by making six judgments for each dimension in a paired comparison. In all experiments, participants completed a training phase and an experimental phase. In the training phase, participants encountered up to 210 trials, half of which were distractor trials. These trials were randomly generated with the boundary condition that 2/3 of the trials were not dominant (i.e., HV did not have a higher probability and a higher outcome than LV) and the rest were dominant. The training phase continued until participants encountered at least 10 dominant binary trials, and chose HV in at least 70% of the last 10 encountered dominant binary trials. The training phase ended when this criterion was reached and the experimental phase began. If the participants finished all 210 training trials without passing the criterion, the experiment ended. As reported above, participants who did not pass the criterion were excluded from the analysis.

The experimental phase consisted of either 412 (Experiment 1 to 3) or 300 (Experiment 4) trials. The 300 trials used in Experiment four were shared across all experiments and are those used by Chau2014. In Experiment 4, all trials were presented in exactly the same orders as in Chau2014’s experiment, whereas in the other experiments, the distractor trials were presented in the order provided to us by the authors of Chau2014 with the randomized binary trials interleaved. In addition to these trials, Experiments 1 to 3 included 56 novel distractor trials and, correspondingly, the 56 binary trials belonging to these novel trials (details are provided in the next two sections). Throughout the experiment, participants had the opportunity to make four breaks. After completing the experiment, participants received their show-up fee (CHF 5 for 15 min), the average reward of the chosen option (distractor and empty quadrant choices counted as no reward), and the accumulated match bonuses. If participants reached the experimental phase, the experiments took approximately 75 min.

Before conducting our experiments, we assumed that the positive relationship between the value of D and relative choice accuracy as reported by Chau2014 was a robust effect. To explain the apparent contradiction with the findings of Louie2013, we reasoned that the explicit presentation of two attributes (i.e., magnitude X and probability p of reward) in the Chau2014 task led people to compare the options on those attributes directly (i.e., a multi-attribute decision between attributes X and p instead of a decision between expected values, EV). Importantly, certain attribute-wise comparison processes are known to produce IIA violations, so-called ‘context effects of preferential choice’ (Berkowitsch et al., 2014; Gluth et al., 2017; Mohr et al., 2017; Pettibone and Wedell, 2007; Roe et al., 2001; Trueblood et al., 2014; Usher and McClelland, 2004). More specifically, our initial hypothesis was that individuals can recognize that D is either better or worse than LV and/or HV with respect to each attribute (e.g., D might dominate LV with respect to probability). Critically, since LV is per definition worse than HV, it is more likely that D dominates LV than that it dominates HV on some attribute. However, this is only true as long as D has not very low attribute values (leading to a low EV overall). Thus, the dominance relationship between D and LV/HV may help participants to identify the option with the highest EV. The positive relationship between D and relative choice accuracy would then be an epiphenomenon of this dominance relationship mechanism.

To give an example, let us assume that HV, LV, and D are specified as follows:

HV: p = 5/8; X = CHF 8 → EV = CHF 5

LV: p = 3/8; X = CHF 4 → EV = CHF 1.5

D: p = 6/8; X = CHF 6 → EV = CHF 4.5

In this case, D is superior to HV and LV with respect to probability. With respect to magnitude, however, D is superior to LV but inferior to HV. Hence, by counting the number of times D is superior to LV and HV on the attributes (i.e., two for LV vs. one for HV), a decision maker could correctly identify the option with the highest EV. Critically, this information is not helpful anymore when we replace D by a distractor D* of lower EV:

D*: p = 6/8; X = CHF 2 → EV = CHF 1.5

In this new case, the distractor is inferior to both HV and LV with respect to magnitude and (still) superior to both with respect to probability (i.e., the count is one for LV vs. one for HV). Thus, the option with the highest EV cannot be identified anymore based on the dominance relationship alone. This demonstrates that the high-value D might better support choosing between HV and LV than the low-value D*. Notably, the idea that the relative ranking of options influences decision making has also been suggested by others (Howes et al., 2016; Stewart et al., 2006; Tsetsos et al., 2016).

Besides this dominance relationship hypothesis, the paradigm of Chau2014 might also involve other context effects that influence behavior. We considered the attraction effect (Huber et al., 1982) and the phantom-decoy effect (Pettibone and Wedell, 2007; Pratkanis and Farquhar, 1992). According to the attraction effect, the preference between two options (in our case between HV and LV) can be changed by adding a third option (in our case D) that is similar but clearly inferior to only one of the two options (in our case, if D is similar to HV and worse than it, HV should be preferred; if D is similar to LV and worse than it, LV should be preferred). The phantom-decoy effect predicts that if an option is similar but worse than D (and D is unavailable, as in the Chau2014 task), it is more likely to be chosen. According to a combination of attraction and phantom-decoy effects, independent of whether D is worse or better than the similar option, the option being more similar to D should be more likely be chosen. In contrast to the context effects, the Chau2014 model, and the divisive normalization account, value-based attentional capture does not predict any influence of D on relative choice accuracy, but a negative effect on absolute choice accuracy (see Figure 1D). As described in the following section, we sought to distinguish between all these different context effects and the model proposed by Chau2014 by implementing a novel set of trials.

In the novel set of trials used to dissociate predictions from various models and context effects, the HV and LV options were arranged such that i.) HV was superior to LV with respect to one attribute (magnitude or probability) but ii.) inferior with respect to the other attribute, and iii.) D could be placed such that it either fully dominated HV/LV or it was fully dominated by HV/LV (for an example, see Figure 1C in the main text). In the Chau2014 task, there are 14 possible combinations of HV and LV that fulfil these criteria. For each of these 14 combinations, D was placed directly ‘above’ or ‘below’ HV or LV resulting in four trials per combination and 56 novel trials in total (we also added 56 binary trials without D, so that Experiment 1 to 3 had 112 trials more than the original study).

The (qualitative) predictions of the models and context effects with respect to these novel trials are outlined in Figure 1D. Chau2014’s biophysical cortical attractor model predicts a positive effect of the value of D on relative choice accuracy. Thus, the performance should be higher in trials with dominant distractors (i.e., D > HV and D > LV). The divisive normalization model by Louie2013 predicts higher accuracy for low-value Ds, or in other words, it predicts the opposite of Chau2014’s model. Our initial dominance relationship hypothesis predicts higher choice accuracy if HV dominates D (i.e., D < HV), or if LV is dominated by D (i.e., D > LV), because in these cases HV is better than D on more attributes than LV. The combination of attraction and phantom-decoy effects predicts more accurate choices when D is dominated by HV (i.e., D < HV) or dominates HV (i.e., D > HV), or in other words, when D is more similar to HV than to LV. Importantly, all these predictions refer to relative choice accuracy (for which we do not find any robust effects in the Chau2014 task with short deliberation time; see Figure 5D). On the contrary, value-based attentional capture predicts no effect on relative choice accuracy, but a reduction of absolute choice accuracy when D has a high value (i.e., D > HV and D > LV).

Note that the different predictions can also be formulated in terms of main effects and interactions of an ANOVA with the factors Dominance (D dominates or is dominated by HV/LV) and Similarity (D is more similar to HV or to LV). Within this ANOVA, the cortical attractor model predicts a main effect of Dominance. The divisive normalization model also predicts this main effect but in the opposite direction. The dominance relationship hypothesis predicts an interaction effect of Dominance and Similarity. The combined attraction/phantom-decoy effect predicts a main effect of Similarity. Value-based attentional capture predicts a main effect of Dominance in the same direction as the divisive normalization model but on absolute (not relative) choice accuracy.

In each trial, there was always a higher-valued option HV and a lower-valued option LV. We used two different dependent measures, relative and absolute choice accuracy. Relative choice accuracy refers to the proportion of HV choices among choices of HV and LV only, whereas absolute choice accuracy refers to the proportion of HV choices among all choices (including choices of D, choices of the empty quadrant, and missed responses due to the time limit). Importantly, an influence of the value of D on relative choice accuracy implies a violation of IIA, but an influence on absolute choice accuracy does not necessarily imply this. For each of the two dependent variables, we estimated intra-individual logistic regressions and tested the regression coefficients between subjects against 0 using a two-sided one-sample t-test with an α level of .05. We used the set of predictor variables reported in Chau2014, which consisted of the difference in EV of the two available options, HV-LV, the sum of their EVs, HV+LV, the EV difference between HV and D, HV-D, the interaction between HV-LV and HV-D, (HV-LV)×(HV-D), and whether it was a binary or distractor trial, D present. In the binary trials, the predictors HV-D and (HV-LV)×(HV-D) were kept constant (i.e., replaced by the mean values in the distractor trials in the regression analysis). The predictor variables HV-LV, HV+LV, and HV-D were standardized. Importantly, standardization was conducted before generating the interaction term (HV-LV)×(HV-D) in order to avoid nonessential multicollinearity (Aiken and West, 1991; Mahwah et al., 2003; Dunlap and Kemery, 1987; Marquardt, 1980). The interaction term itself and the binary predictor D present were not standardized (note that this would not have changed any statistical inferences but only the absolute values of coefficients). In addition, we analyzed the influence of the (standardized) value of D on the tendency to choose D (logistic regression) and on the RT of HV and LV choices (linear regression). The RT analysis included additional predictor variables with a significant influence on RT (i.e., HV-LV, HV+LV, D present). Note that only the 300 trials that were also used in the Chau2014 paradigm (but not our 112 novel trials) were included in these regression analyses. The novel trial sets used to dissociate different model predictions were analyzed by a 2 (Dominance) x 2 (Similarity) ANOVA (for details see above).

Because participants received feedback after each decision, we tested whether improvements over time affected the results of relative or absolute choice accuracy by re-analyzing the regressions with an additional predictor variable that coded for the (standardized) trial number. Although we found a small learning effect on absolute choice accuracy (t(122) = 2.12, p = .036, d = 0.19), this did not affect any other effects qualitatively. When adding the trial number predictor to the analysis of the influence of D’s value on the tendency to choose D, we found a strong learning effect (t(122) = −10.18, p < .001, d = −0.92), suggesting that participants improved in avoiding to choose D over the course of the experiment (but the effect of D’s value remained significant). Hence, future instantiations of our model could incorporate a dynamic component to accommodate this learning effect.

Experiment 3 was conducted while participants’ gaze positions were recorded using an SMI RED500 eye-tracking device. The experimental procedure was adapted to make it suitable for an eye-tracking experiment. Participants completed the experiment on a 47.38 × 29.61 cm screen (22" screen diagonal) with a resolution of 1680 × 1050 pixels. During the inter-trial interval, participants were instructed to look at the fixation cross and the random duration of the inter-trial interval was removed. Instead, there was a real-time circular area of interest (AOI) with a diameter of 200 pixels around the fixation cross. Participants’ gazes had to (continuously) stay within this AOI for 1 s for the trial to begin. This was done to make sure that participants were indeed looking at the fixation cross and to check the calibration at every trial. If this criterion was not reached within 12 s, the eye tracker was re-calibrated. This procedure was explained to the participants by the experimenter. In case these re-calibrations happened too frequently (e.g., three times in a row within the same trial, or at least three times in ten trials), the sampling frequency was reduced from the initial 500 Hz to 250 Hz. If the issues continued until the lowest possible frequency of 60 Hz was reached, the experiment was aborted and participants received their show-up fee and decision-based bonuses accumulated until then. The first calibration took place just before the training phase and the eye tracker was re-calibrated after each of the four breaks. This experiment took approximately 90 min to complete.

The raw gaze positions were re-coded into events (fixations, saccades, and blinks) in SMI’s BeGaze2 software package using the high-speed detection algorithm and default values. AOIs were defined around the positions where the frames of the options were, and all fixations inside the frame were counted towards the option within that quadrant. Fixations at empty quadrants as well as all fixations outside of the pre-defined AOIs were counted as empty gazes. Fixations within a trial were collapsed and summed to form the dependent variables relative fixation duration and number of fixations. We report results based on the relative fixation duration (i.e., the sum of the duration of all fixations on a specific quadrant divided by the sum of the duration of all fixations on any quadrant). Note that this measure is highly correlated with the number of fixations, which yielded similar results.

To test the hypothesis that the negative influence of D on absolute choice accuracy is driven by value-based attentional/oculomotor capture, we conducted the following three analyses: i.) We tested for the dependency of relative fixation duration of HV, LV, and D on their respective EVs (Figure 4A). Thereto, the correlations between the options’ relative fixation durations and EVs were calculated for each participant, and the individual Fisher z-transformed correlation coefficients were subjected to one-sample t-test against 0 on the group level (after checking for normality assumptions via the Kolmogorov-Smirnoff test at p < .1). ii.) We conducted a path analysis (within each participant) in which the influence of the value of D on absolute choice accuracy was hypothesized to be mediated by the relative fixation duration on D (Figure 4B). iii.) We run an (across-participant) correlation between the behavioral regression coefficients representing the influence of the value of D on absolute choice accuracy and the regression coefficients representing the influence of the value of D on relative fixation duration on D (Figure 4C).

To inform our computational model, we tested whether there is evidence for a direct effect of attention on choice (i.e., an option that receives relatively more attention is more likely to be chosen–independent of the option’s value) as reported by Cavanagh and colleagues (Cavanagh et al., 2014), or whether the effect of attention is modulated by value (i.e., only options with high value that receive more attention are more likely to be chosen) as proposed by Krajbich and colleagues (Krajbich and Rangel, 2011; Krajbich et al., 2010). Thereto, we conducted a logistic regression analysis with absolute choice accuracy as dependent variable and the following six predictor variables: the expected value of HV, the expected value of LV, the relative fixation duration of HV, the relative fixation duration of LV, the interaction of the value of HV and the relative fixation duration of HV, and the interaction of the value of LV and the relative fixation duration of LV. The first three predictors were standardized, and the interaction variables were generated based on standardized variables to avoid nonessential multicollinearity (Marquardt, 1980). A direct attention model predicts a positive effect of the relative fixation duration of HV and a negative effect of the relative fixation duration of LV. A value-dependent attention model predicts positive and negative effects of the interaction terms for HV and LV, respectively. The results are shown in Figure 4—figure supplement 1. Besides the positive effect of HV-LV, there were significantly positive and negative effects of the relative fixation duration of HV (t(22) = 9.40, p < .001, d = 1.96) and LV (t(22) = −8.20, p < .001, d = −1.71), respectively. Of the interaction terms, only LV was significantly negative (t(22) = −4.09, p < .001, d = 0.85), but HV was not significantly positive (t(22) = −1.21, p = .237, d = −0.25). Thus, we found unequivocal evidence for a direct effect of attention on choice (Cavanagh et al., 2014) but not for an interaction of attention and value (Krajbich and Rangel, 2011; Krajbich et al., 2010). Consequently, we implemented the influence of attention on choice in the computational model by a value-independent, additive increase of the input signal to the accumulator of the currently attended option (see Equation 9 below).

A central and often-tested prediction of the aDDM (Krajbich and Rangel, 2011; Krajbich et al., 2010) is that the direction of the initial fixation is independent of value (Konovalov and Krajbich, 2016; Krajbich and Rangel, 2011; Krajbich et al., 2010). However, the results of our eye-tracking experiment are inconsistent with this prediction. We found that the probability to fixate HV, LV, D first was positively linked to the respective option’s value (HV: t(22) = 6.03, p < .001, d = 1.26; LV: t(22) = 3.84, p < .001, d = 0.80; D: t(22) = 3.57, p = .002, d = 0.74). Importantly, the effects for HV and D remained significant when controlling for the eventual choice (HV: t(22) = 4.43, p < .001, d = 0.92; LV: t(22) = 0.62, p = .539, d = 0.13; D: t(21) = 2.70, p = .013, d = 0.58), indicating that the effect was not solely driven by an intention to choose a particular option. Instead, the effect can be explained by assuming that value-based oculomotor capture biases early competition on the saccade map (Pearson et al., 2016).

In addition, we tested whether the first fixation and its duration was predictive of choice (when controlling for HV-LV and D), which would be consistent with previous findings (Cavanagh et al., 2014; Krajbich and Rangel, 2011; Krajbich et al., 2010) and suggest a biasing influence of attention on evidence accumulation. Indeed, the probability to choose HV, LV or D was predicted by whether the first fixation was made to the respective option (HV: t(22) = 6.11, p < .001, d = 1.27; LV: t(22) = 4.98, p < .001, d = 1.04; D: t(20) = 7.02, p < .001, d = 1.53). Similarly, the duration of the first fixation predicted the choice of all three options (HV: t(22) = 7.26, p < .001, d = 1.51; LV: t(22) = 6.18, p < .001, d = 1.29; D: t(20) = 2.92, p = .008, d = 0.64). Yet, the value difference HV-LV remained being predictive of the choice of HV and LV (HV: t(22) = 8.09, p < .001, d = 1.69; LV: t(22) = −11.49, p < .001, d = −2.40), and the value of D remained being predictive of the choice of D (t(20) = 4.88, p = .003, d = 1.07). Thus, the first fixation and its duration contributed to the probability of choosing an option but was not fully predictive of it. In our view, these results are best explained by assuming that gaze time biases the evidence accumulation towards the fixated option as implemented in the MIVAC model. Note that the analysis of first fixations included trials from the training phase.

Many analyses in Chau2014 and our study, such as the regression analysis of relative choice accuracy, relied on the assumption that participants were able to integrate magnitude and probability information to calculate the expected value (EV) of each option and to choose the option with the largest EV. We tested this assumption by comparing five simple choice models with respect to how well they predicted whether participants chose HV or LV (note that in contrast to MIVAC, we do not propose any of these models to account for all behavioral findings but instead treat them only as auxiliary models to test the assumption that participants relied on EV when making decisions). The first two models assumed that participants relied only on magnitude information (‘OM’) or only on probability information (‘OP’). The third model assumed that participants relied on EV (‘EV’). The fourth model assumed that participants relied on subjective values given by expected utility theory (e.g. Von Neumann and Morgenstern, 1947) (‘EU’), specified as follows:

where α is a free parameter (α > 0) modulating the curvature of the utility function. The last model assumed that participants relied on subjective values given by prospect theory (Kahneman and Tversky, 1979; Tversky and Kahneman, 1992) (“PT”), specified as follows:

where α is again a free parameter (α >0) modulating the curvature of the utility function and where τ is an additional free parameter (τ >0) modulating the shape of the probability-weighting function. For each of these models, the probability of choosing HV was given by the logistic/soft-max choice function:

where δ is a free parameter (δ >0) modulating the sensitivity to value differences, and SV[HV]_Model and SV[LV]_Model refer to the model-specific values of options HV and LV, respectively. We used the same minimization algorithm and model-comparison approach as for MIVAC (see section Modeling procedures for MIVAC III: parameter estimation and nested model comparison). Models were compared on the basis of the choice data from the 300 trials that were part of the initial choice set of Chau2014 and from the 123 participants of our study who conducted the Chau2014 paradigm under time pressure. Because the EU model provided the most parsimonious account of the data (Figure 5—figure supplement 1), we reanalyzed the central behavioral tests of our study (shown in Figure 5A–5C) and replaced the options’ EVs by the EU-based subjective values estimates. Furthermore, we reran the MIVAC model to test whether using EU-based subjective values as input to the model’s accumulators would lead to qualitatively different predictions (Figure 7—figure supplement 2; details are provided in section Modeling procedures for MIVAC IV: MIVAC with subjective values as inputs). Note that the analyses and predictions for the novel trial sets (Figures 1D and 5D) do not depend on whether EV or EU is assumed as the underlying choice model.

The computational model we propose, MIVAC, is an extended version of the mutual inhibition (MI) model (Bogacz et al., 2006; Usher and McClelland, 2001). The MI model belongs to the sub-class of accumulator models within the framework of sequential sampling models (Ratcliff and Smith, 2004; Smith and Ratcliff, 2004). Sequential sampling models assume that evidence for choice options is accumulated over time until a decision threshold is reached after which a choice is made. Accumulator models use separate evidence accumulators for each choice option. MIs further assume mutual (or lateral) inhibition of these accumulators, that is, the more evidence for one choice option has been accumulated, the stronger this accumulator inhibits the competing accumulators (in contrast to feedforward inhibition models that assume inhibition between the inputs to accumulators). In addition, accumulators are leaky, that is, they tend to decay toward the starting point (i.e., 0). One reason why we chose this framework (and not, for instance, the aDDM) is that the MI is equivalent to a mean-field reduction of the biophysical attractor model that was applied by Chau2014 (Bogacz et al., 2006; Wang, 2002; Wong and Wang, 2006). Accordingly, without a value-based attentional capture mechanism, the MI model is able to predict a positive effect of the value of the distractor (D) on choice accuracy similar to the model proposed by Chau2014 (see Figure 6—figure supplement 1). In general, we begin with a baseline MI and then add additional assumptions (about the interplay of value, attention, and choice) that are necessary to explain the observed behavioral findings.

Formally, the MI assumes four accumulators representing the four choice options high-value option (HV), low-value option (LV), D, and the empty quadrant. The n by t matrix A representing all n (=4) accumulators is updated every time step Δt according to:

where S represents an n by n leakage and inhibition matrix with leakage parameter δ and inhibition parameter ϕ as on- and off-diagonal elements, respectively, I_t represents an n by t input matrix that–in the case of the baseline MI–contains the EV of each choice option (which is 0 for the empty quadrant), and E_t represents an n by t noise matrix with each element being drawn from a normal distribution with mean 0 and standard deviation σ. As soon as accumulator A_i,t reaches an upper threshold θ_t (i.e., A_i,t ≥ θ_t), option i is chosen (at time point t). Similar to previous work (Gluth et al., 2012; Gluth et al., 2013a; Gluth et al., 2013b; Hutcherson et al., 2015; Murphy et al., 2016), we assume that the decision threshold decreases (linearly) with time (note that we found the empirical response-time [RT] distribution in this task to lack the typical right-skewed shape, which is possibly due to the strict time limit and would be in line with a decreasing decision threshold; see Hawkins et al., 2015). In contrast to the leaky competing accumulator model (Usher and McClelland, 2001) but in line with suggestions of Bogacz et al., 2006, we allowed A to take on negative values.

Potentially, the baseline MI could have six free parameters: the leakage parameter δ, the inhibition parameter ϕ, the standard deviation σ of the noise vector, the height of the decision threshold at the start and end of the decision period θ₀ and θ_tmax, and a non-decision time parameter τ_non-dec. However, because we sought to extend this model by three additional components and because predictions of the MI cannot be derived analytically but must be simulated, we drastically constrained the model by fixing all but one of the aforementioned parameters (it should be noted at this point that we had to make several simplifying assumptions to simulate the model in a reasonable amount of time). Leakage and inhibition were set to δ = 0.96 and ϕ = −0.036, respectively, the start and end points of the decision threshold were set to θ₀ = 1’000 and θ_tmax = 200, respectively, and the non-decision time was set to τ_non-dec = 200 ms (so that tmax – τ_non-dec = 1’400 ms). Thus, only σ remained as a free parameter (this parameter controls the stochasticity of the decision process and allows to account for performance differences between participants). Importantly, fixed parameter values were chosen carefully so that the model made various sensible predictions (i.e., the relative choice ratio between HV and LV, the average RT, and the proportion of misses due to no threshold crossings until tmax were ensured to lie all within the range of the empirical observations).

For the MIVAC model, we propose three additional mechanisms that are required to account for the behavioral data and that describe the interplay of value, attention, and decision making as observed in the eye-tracking experiment. First, we implemented value-based attentional capture by modelling fixations to the choice options and by assuming that the probability of fixating a particular option depends on the option’s value (relative to the other options). Similar to a recent study on reinforcement learning and attention (Leong et al., 2017), we used a logistic function to describe the relationship of value and fixation probability:

 where F_i is the probability of fixating option i, γ is a free parameter, and nEV_i is the expected value of option i normalized with respect to all currently presented options (normalization was used to avoid that value-based attentional capture effects depend too much on the sum of all values). We assumed that every 200 ms, a new fixation is made, and with probability F_i it is made to option i (with the possibility to re-fixate the same option). This time window roughly corresponds to the average fixation duration of 230 ms in our data. We chose this implementation to maintain a high speed of model simulations and to make more conservative model predictions, as a more data-driven implementation of using empirical fixation patterns (e.g., Cohen et al., 2017; Krajbich and Rangel, 2011; Krajbich et al., 2010) may improve model fit but limit generalizability. In the section Modeling procedures for MIVAC V: MIVAC with fixation durations drawn from empirical distributions, we show that using empirical fixation durations (instead of a fixed duration of 200 ms) does not change the models’ predictions and the model comparison results qualitatively.

The second additional mechanism is the enhanced input to the currently attended accumulator. This assumption is based on previous work showing that options which receive more attention are more likely to be chosen (Cavanagh et al., 2014; Krajbich and Rangel, 2011; Krajbich et al., 2010; Shimojo et al., 2003), which we replicated in our eye-tracking experiment. Note, however, that we found evidence for a direct effect of attention on choice rather than for a modulatory effect that depends on the value of the attended option (see section Eye-tracking analysis II: direct vs. value-dependent influences of attention on choice and Figure 4—figure supplement 1). Accordingly, we assumed that the input I_a,t of the currently attended option a is enhanced additively rather than multiplicatively:

where EV(a) is the expected value of a, and β is a free parameter representing the attention-based enhancement of accumulation. Together with value-based attentional capture, this means that MIVAC assumes that options of higher value receive more attention, and that the evidence for an attended option is accumulated faster (but MIVAC does not assume that there is an interaction of value and attention, in the sense that the increase of accumulation due to attention is stronger for higher- compared to lower-valued options).

Finally, we assumed that participants can detect that the distractor is unavailable but occasionally fail to do so. This assumption was necessary because participants usually followed the instructions of the Chau2014 paradigm and avoided choosing D in most of the trials but still picked it more often than the empty quadrant: In trials with D present, there were 6.7% choices of D and only 0.1% choices of the empty quadrant (t(122) = 12.17; p<0.001); in trials without D, there were only 3.1% choices of both empty quadrants together. Therefore, we assumed that after every 100 ms, D can be identified as being unavailable with probability π, which is a free parameter (restricting the identification of D to every 100 ms was again necessary to maintain fast simulations of the model; note that 100 ms also corresponds to the point in time at which participants were told which of the options D is). If D is identified, the input to its accumulator is set to 0 (plus noise) just as for the empty quadrant. The probability to fixate D is also affected by its identification (i.e., nEV_D is set to 0 in Equation 8).

Taken together, MIVAC has four free parameters, the standard deviation σ of accumulation from the baseline MI model, the value-based attentional capture parameter γ, the attention-based enhancement β of accumulation, and the probability π to identify D as being unavailable (see Table S8 in Supplementary file 1 for estimated parameter values). In addition to the baseline MI and MIVAC, we tested three simplified versions of MIVAC that each lacked one of the additional features (i.e., either γ, β, π, or all of them were set to 0).

It should be noted that the proposed model does not predict violations of IIA, such as the attraction or the phantom-decoy effect (for which we found evidence in the experiment with long deliberation time). We refrained from implementing this because our central goal was to model the Chau2014 paradigm with short deliberation time, for which we did not find any IIA violations, and because we wanted to maintain a high speed of model simulations. Importantly, however, MIVAC can be extended to allow predicting IIA violations straightforwardly, for instance by implementing a stage of attribute-wise comparisons that modify the input I_t (Gluth et al., 2017; Hunt et al., 2014; Roe et al., 2001; Trueblood et al., 2014; Usher and McClelland, 2004). In the section Modeling procedures for MIVAC VI: Extending MIVAC to allow for violations of IIA, we provide one example of how such an extension could work, and show that it allows accounting for the IIA violation observed in the experiment with long deliberation time. Thus, in principle MIVAC is compatible with the rich literature on violations of IIA in value-based and perceptual decision making (Berkowitsch et al., 2014; Gluth et al., 2017; Huber et al., 1982; Mohr et al., 2017; Rieskamp et al., 2006; Spektor et al., 2018a; Spektor et al., 2018b; Trueblood et al., 2013; Tsetsos et al., 2012).

As mentioned, parameter estimation required time-consuming simulations of the models. Similar to our previous work (Gluth et al., 2013b), we chose a step-size Δt of 10 ms for fast simulations, and approximated the likelihood by simulating every trial 100 times. Trial-wise predictions were truncated to a minimum probability of .01 and a maximum probability of .99 (Gluth et al., 2013b; Nassar and Frank, 2016; Rieskamp, 2008). Note that the estimation of parameters was based on predicting choices but not RT, because our main goal was to explain the choice data, and because predicting RT would have required more simulations per trial and estimating the non-decision time (and possibly also the decision threshold), which would have prolonged computations to an infeasible amount of time. As shown in Figure 7D, however, the model reproduces all of the observed RT effects (i.e., higher value differences and sums of HV and LV decrease RT, higher values of D and the presence of D increase RT). Parameter estimation was realized by combining an initial grid search algorithm to obtain reasonable starting values, which were then passed on to a constrained simplex minimization algorithm (Nelder and Mead, 1965) as implemented MATLAB’s fminsearchcon to obtain the final estimates. For the grid search, 31 steps per parameter were used, equally spaced within the following bounds: σ between 5 and 50, γ between −1.12 and 3.08, β between −2.1 and 18.9, π between 0 and 1 (the same values were used as constrains during the simplex minimization). The simplex algorithm was run 16 times in total, 3 times with the parameter set obtained from the best grid search model fit, 9 to 13 times with a parameter set randomly selected from the best 1% of grid search model fits (20% in case of the baseline MI), and 0 to 4 times with the best parameter set obtained from the simplex search of nested models (e.g., all other models are nested in MIVAC, so the algorithm for MIVAC was started 4 times with the four other models’ best parameters as starting values). The final estimates of parameters were taken from the simplex search with the best model fit. Only trials of the Chau2014 set with D present were used to estimate parameters. This allowed testing a generalization (Busemeyer and Wang, 2000) of the models to the trials without D (Figure 7B) and to the novel trials that we added in Experiment 1 to 3 (Figure 7E).

The central goal of computational modeling was to compare MIVAC against simpler versions that lacked either one or all of the novel mechanisms. Thus, model comparison comprised the baseline MI, MIVAC, and three versions of MIVAC without either γ, β, or π (which were set to 0 for the respective model). Since all other models are nested within MIVAC, they could be compared using a likelihood ratio test (Lewandowsky and Farrell, 2011). However, we used the Bayesian Information Criterion (BIC; Schwarz, 1978) to compare the models, because it is more conservative when comparing the most complex model against simpler ones, and because the BIC allows to estimate participant-wise model evidence: For each participant, we classified model evidence as ‘weak’, ‘positive’, ‘strong’, or ‘very strong’, when the BIC difference x between the best and the second-best model was x < 2, 2 < x < 6, 6 < x < 10, or x > 10, respectively (Raftery, 1995). Following our previous work (Gluth et al., 2017), we tested the generalizability of the models by comparing their deviances for the trials without D and for the novel trials, which were not used for parameter estimation. We also compared the models on a qualitative level with respect to i.) how well they predict choices in D present trials, ii.) how well their predictions can be generalized to trials without D, and iii.) how well they account for the dependency of choosing D on the value of D (Figure 7C). Note, that all quantitative and qualitative model comparisons are based on out-of-sample predictions by running 10’000 new simulations per trial, participant, and model using the estimated parameters. A simple parameter recovery analysis (in which we generated synthetic data using MIVAC’s estimated parameters of each participant and the 150 trials from the Chau2014 paradigm and re-estimated parameters for these synthetic data) confirmed that the model is identifiable with the set of trials used in Chau2014 and our study (for all parameters, correlations between data-generating and re-estimated parameter values were significantly positive at p<0.001; see Figure 7—figure supplement 1).

Similar to our behavioral analyses, MIVAC assumes that people are able to integrate magnitude and probability information to estimate an option’s EV (since the EV serves as the basic input signal to each accumulator; see Equation 9). Because our analyses of the choice data suggested that an EU-based choice model provided a more accurate account of the data than an EV-based model (see Figure 5—figure supplement 1), we reran MIVAC and replaced the EVs as inputs to the options’ accumulators by the EU-based subjective value estimates. Because the range of EU-based values heavily depend on the power utility parameter α (if 0 < α < 1, the range is narrower than the range of EVs; if α > 1, the range is wider than the range of EVs), we ensured that subjective values were kept in the same range as the EVs by the following transformation (see Berkowitsch et al., 2015):

We simulated responses from all models 10’000 times per trial and participant with the previously estimated parameters. As can be seen from Figure 7—figure supplement 2, the predictions of the EU-based MIVAC model are very similar to the EV-based MIVAC model.

As stated in the main text, we did not draw fixations from empirical fixation distributions (as has been done, for instance, by Krajbich et al., 2010) when estimating parameters of MIVAC for two reasons. First, we sought to avoid overfitting MIVAC by equipping it with actual fixation patterns: If the main behavioral effects had been solely due to differences in fixation patterns across options, then using those patterns would have allowed MIVAC to make accurate predictions, even though the model’s proposed features (e.g., value-based attentional capture) would not have been critical. Second, without using empirical fixation distributions the model can be faster simulated and thereby fitted to the data in a feasible amount of time.

Here, we show that using empirical fixation patterns does not alter the predictions of MIVAC and its simplified competitor models qualitatively. Thereto, we first analyzed mean fixation durations by splitting fixations according to the factors Option Type (i.e., HV, LV, D, empty quadrant) and Fixation Type (i.e., first vs. middle fixations; see Krajbich et al., 2010; Figure 7—figure supplement 3A). A 4×2 repeated-measures ANOVA revealed significant main effects for both Option Type (F(3,18) = 29.51, p < .001, ${\eta }_p^2$ = .83) and Fixation Type (F(1,20) = 25.70, p < .001, ${\eta }_p^2$ = .56) and a significant interaction (F(3,18) = 18.24, p < .001, ${\eta }_p^2$ = .75). Accordingly, we approximated the empirical fixation-duration distributions by estimating the means and standard deviations of log-normal distributions (Krajbich et al., 2010) for each option and fixation type separately (Figure 7—figure supplement 3B). Finally, we simulated responses from all models 10’000 times per trial and participant with the previously estimated parameters. We sampled each fixation duration from the log-normal distribution with the respective means and standard deviations per option and fixation type (e.g., when the first fixation in a trial was on HV, then the fixation duration was drawn from a log-normal distribution with its mean and standard deviation taken from the fit to the HV/first distribution). As can be seen from Figure 7—figure supplement 3C, the predictions of MIVAC do not change qualitatively when empirical fixation durations are used. Moreover, using these fixation durations does not allow the simplified competitor models to catch up with MIVAC (Figure 7—figure supplement 3D).

As stated in the main text, MIVAC does not predict violations of IIA but can be extended straightforwardly to do so. Here, we provide one example of such an extension and show that it allows to predict the IIA violation we obtained in the experiment with long deliberation time (Experiment 2, Group LP). For the extension, we followed the assumption made by many multi-attribute sequential sampling models, including the Multialternative Decision Field Theory (Roe et al., 2001), the Leaky Competing Accumulator model (Usher and McClelland, 2004), and the Multi-attribute Linear Ballistic Accumulator model (MLBA; Trueblood et al., 2014), namely that the input to the accumulator of an option i is based on comparisons of this option with all other options. In our notation, the input of option i thus becomes

if i is currently fixated,

if i is currently not fixated,

 where I₀ is a free parameter of the model (I₀ ≥0), representing the baseline input to each accumulator, and V_i,j represents the output of a comparison process between option i and option j. In specifying V_i,j, we follow the MLBA, according to which this comparison process is the weighted sum of the attribute-wise differences:

where is w_i,j^M is the weight given to attribute M (for ‘magnitude’) in the comparison between options i and j, and u_i^P is the subjective representation of attribute P (for ‘probability’) for option i. In the full MLBA, the subjective representations can be subject to non-linear transformations. For the sake of simplicity, however, we used the objective magnitudes and probabilities as u^M and u^P, respectively. The weight given to an attribute X is a function of the similarity between the options with respect to that attribute X:

if u_i^X ≥u_j^X,

if u_i^X <u_j^X,

where λ₁ and λ₂ are two free parameters of the model that modulate the decay of attribute weights when the distance between the options’ attribute values increases. Note that this means that perceived similarity is not symmetrical: A can be perceived as more similar to B than B to A. As stated above, the outputs from this attribute-comparison layer are the inputs I_t to the accumulation process (Equation 9).

For our simulations, we set I₀ to 20, λ₁ to 3, and λ₂ to 0.05. While the value for I₀ was chosen simply to avoid negative inputs to accumulators, we chose to set λ₁ > λ₂ because then the weight w_i,j^X of attribute X in the comparison between option i against option j will be higher when i is better than j with respect to X compared to when it is worse, with this difference being largest at close distances (because of the decay function that links λ to w). As a consequence, option i will receive an enhanced input in the presence of a similar, inferior alternative, but it will not receive a reduced input in the presence of a similar, superior alternative. Such a disproportionate attention weight is a default assumption of MLBA and predicts both the attraction and the phantom-decoy effect, that is, an increased relative choice accuracy when (an inferior or superior) D is more similar to HV than to LV. All other parameter values were taken from the fit of the non-extended MIVAC model. The deliberation time was set to 6’000 ms. We ran 10’000 simulations per participant. As shown in Figure 7—figure supplement 4, the extended MIVAC is able to predict the pattern of IIA violation that we observed in Experiment 2 Group LP, namely that the probability of choosing HV is increased in the presence of a similar (inferior or superior) D option.

We also compared MIVAC to three different multinomial logit models as representatives of standard economic choice models (e.g., McFadden, 2001); see Figure 7—figure supplement 5). The first logit model (‘Exclude D’) assumes that participants are perfectly able to identify that D should not be chosen. Thus, irrespective of whether D is present or absent, the probability of choosing HV becomes a logistic function of the expected value (EV) difference between HV and LV:

where δ is a free parameter reflecting the participant’s sensitivity to EV differences. The second logit model (‘Include D’) assumes that participants are completely unable to identify that D should not be chosen. Thus, the probability of choosing an option i is:

with D being part of the choice set and where the sum of EVs is taken for all choice options. The third logit model (‘EV and Frame Value’) assumes that D is treated as a regular choice option (as in the ‘Include D’ model), and that choices are based on an additive combination of the options’ EVs with additional subjective values that are assigned to the differently colored frames (which indicate whether an option is a target or a distractor; see Figure 1A). Formally, the overall subjective value (SV) of an option i is:

where FV(i) is the ‘Frame Value’ of option i. We assumed that the Frame Value for distractors is 0 and the Frame Value for targets is a free parameter (restricted to positive values). The choice probability is then given by Equation 15 with EV replaced by SV.

Maximum likelihood estimates of the models’ parameter(s) were computed on the basis of trials with D present, using the same minimization algorithm as for MIVAC. Because the three models and MIVAC used different sets of data (i.e., the ‘Exclude D’ model used only trials in which either HV or LV were chosen, the other two logit models also included trials in which D was chosen, MIVAC further included trials in which the empty quadrant was chosen or no decision was made), the models could not be compared via quantitative criteria. Instead, we compared their ability to predict the observed choice proportions of HV, LV, and D in trials with D present, and of HV and LV in trials with D absent (Figure 7—figure supplement 5).

The authors of Chau2014 provided us with the behavioral data of their fMRI experiment that included 21 participants. We reanalyzed the data to check for evidence of effects related to value-based attentional capture: We tested for an effect of HV-D on absolute choice accuracy, for an effect of D’s value on the propensity to choose D, and for an effect of D’s value on RT. The results of all of these analyses were in line with predictions of a value-based attentional capture account (see Results).

When trying to reproduce the results of the original study, we also found that the HV-D effect on relative choice accuracy (i.e., the central behavioral result of Chau2014) was only significantly negative when the interaction term (HV-LV)×(HV-D) in the same regression analysis was generated on the basis of uncentered HV-LV and HV-D variables. In multiple regression analyses, centering (i.e., subtracting each entry by the mean) or standardizing (i.e., subtracting the mean and dividing by the standard deviation) predictors before generating their interaction term is critical to avoid nonessential multicollinearity (Aiken and West, 1991; Mahwah et al., 2003; Dunlap and Kemery, 1987; Marquardt, 1980). This is particularly important when one or both of the predictors do not have a meaningful 0 (i.e., are always positive or always negative). This is the case for HV-LV which is always positive by definition. Accordingly, the uncentered interaction term (HV-LV)×(HV-D) is correlated substantially with HV-LV (r = .287) and correlates particularly strong with HV-D (r = .862) (upper row of Figure 5—figure supplement 2). In contrast, the mean-centered interaction term is only weakly correlated with HV-LV (r = .004) and HV-D (r = −.064) (lower row of Figure 5—figure supplement 2).

To test how susceptible the original results of HV-D are to the generation and implementation of the interaction term, we conducted several control analyses. First of all, we repeated the regression analysis on relative choice accuracy, but used the correct interaction term (i.e., the term generated on the basis of standardized HV-LV and HV-D predictors). Strikingly (but in line with our own results), the effect of HV-D in the Chau2014 data was positive and not significant (t(20) = 1.56, p = .134; see also Figure 5—figure supplement 3, which shows the effect sizes of HV-D in all experiments when the correct vs. incorrect interaction term is used in the analysis). Second, we took the interaction term out of the regression analysis, which yielded similar results (t(20) = 0.67, p = .510). Third, we reanalyzed the original data in the following way: We separated the dataset by each level of HV-LV, which yielded 12 bins, and conducted the regression analysis on relative choice accuracy for each bin separately. This procedure allowed us to take out the interaction term (together with the predictor variable HV-LV itself), and to test for a ‘cleaner’ effect of HV-D in each bin. As shown in Figure 5—figure supplement 4 a significantly negative effect of HV-D on relative choice accuracy is seen in only 1 of the 12 bins, but there is also a significantly positive effect in another bin. Conducting this analysis with our own data yielded very similar results. Fourth, we simulated data from hypothetical decision makers who decide solely on the basis of HV-LV (via a logistic/soft-max choice function [see Equation 14 above] for which we set the choice-sensitivity parameter to 0.5 to roughly match the choice accuracy observed in the data). For these simulated participants, we know that their behavior is not influenced by HV-D. We conducted 1’000 simulations of 21 participants (the sample size of Chau2014) and performed four logistic regression analyses, either with or without the interaction term, and either with standardized or uncentered HV-LV and HV-D predictors. We found a spurious negative influence of HV-D on relative choice accuracy when including an interaction term with uncentered predictors (t(999) = −3.43, p < .001; upper row of Figure 5—figure supplement 2). This influence was absent when including the interaction term with standardized predictors (t(999) = 0.36, p = .717) or when excluding the interaction term with either standardized or uncentered predictors (both: t(999) = −0.89, p = .375; lower row of Figure 5—figure supplement 2). Taken together, the negative effect of HV-D on relative choice accuracy is a statistical artifact of the incorrect implementation of the interaction term (HV-LV)×(HV-D) into the logistic regression analysis.

The reanalysis of Chau2014 raises strong doubts on the reproducibility of the original results. Yet, theoretically there might still be a true negative HV-D effect that in principle could have been found in our own four experiments (while being absent in the original study due to its low statistical power). To test for the ‘replicability’ of this effect without reverting to the (equivocal) effect size of the original study, we conducted a recently proposed test of detectability that is independent of the reported effect size of the original study (Simonsohn, 2015). This approach uses only the sample size (and the statistical design) of the original study to specify the (hypothetical) effect size that would have given the study only 33% statistical power, which can be regarded as undoubtedly insufficient. If the effect size of the study that attempts to reproduce the effect of the original study is significantly below this d_33% threshold, then one can conclude that the studied effect is not large enough to have been detectable with the original sample size. The results of this test for the HV-D effect on relative choice accuracy show that the 95% confidence intervals of the effect sizes for each of our Experiments 1, 2 HP, 3, and four as well for all these experiments combined and for the original study itself (when using the correct implementation of the interaction term) were indeed closer to 0 than the d_33% threshold for the sample size of 21 participants used in Chau2014 (Figure 5—figure supplement 3). But even when using the incorrect implementation of the interaction term (that biases the results toward more negative HV-D effects), the confidence intervals of the effect size for our four experiments combined are closer to 0 than the d_33% threshold.

Finally, we conducted a Bayesian analysis of the HV-D effects on relative (and absolute) choice accuracy to be able to quantify the evidence in favor of the null hypothesis (Wagenmakers, 2007). Specifically, we applied the ttestBF function of the R package BayesFactor (with default settings) to obtain Bayes Factors (BF) in favor of the null (or alternative) hypothesis that the HV-D regression coefficients of relative and absolute choice accuracy are equal to (or deviate from) 0. We included the 123 participants from Experiments 1, 2 HP, 3, and 4 who performed the task under high time pressure. For the HV-D effect on relative choice accuracy we obtained a BF₀₁ in favor of the null hypothesis of 20.05 when testing against the alternative hypothesis of a negative HV-D effect (Figure 5—figure supplement 5, upper panel). This means the data are about twenty times more likely to have occurred under the null hypothesis, which is also seen as strong evidence for the null hypothesis (Kass and Raftery, 1995). When testing against the alternative hypothesis of a positive HV-D effect (which would be the opposite of what Chau2014 had reported and in line with the divisive normalization account of Louie et al., 2013), we obtained a BF₀₁ in favor of the null hypothesis of 3.14, which can be considered as positive evidence for the null hypothesis (Kass and Raftery, 1995). For the HV-D effect on absolute choice accuracy we obtained a BF₁₀ in favor of the alternative hypothesis of 4.07 × 10¹³, which is seen as decisive evidence (Figure 5—figure supplement 5, lower panel).

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

We thank Bolton Chau and Matthew Rushworth for providing us with the data from the original study and for helpful and constructive discussions. We thank Chris Donkin for pointing us towards literature on value-based oculomotor capture.

Human subjects: All participants gave written informed consent, and the study was approved by the Institutional Review Board of the Department of Psychology at the University of Basel. All experiments were performed in accordance with the relevant guidelines and regulations.

© 2018, Gluth et al.

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