Our results confirmed that participants understood the task and chose higher value items in the like frame and lower value items in the dislike frame (Figure 1B,C). This effect was modulated by confidence (Figure 1B) similarly to previous studies (De Martino et al., 2013; Folke et al., 2017; Boldt et al., 2019). For a direct comparison of the differences between the goal manipulations in the two tasks (Value and Perceptual) see Appendix 1 (Appendix 1—figure 1).

We then tested how attention interacts with choice by examining the eye-tracking variables. Our frame manipulation, which orthogonalised choice and valuation, allowed us to distinguish between two competing hypotheses. The first hypothesis, currently dominant in the field, is that visual attention is always attracted to high values items and that it facilitates their choice. The alternative hypothesis is that the attention is attracted to items whose value matches the goal of the task. These two hypotheses make starkly different experimental predictions in our task. According to the first, gaze will mostly be allocated to the more valuable item independently of the frame. The second hypothesis instead predicts that in the like frame participants will look more at the more valuable item, while this pattern would reverse in the dislike frame, with attention mostly allocated to the least valuable item. In other words, according to this second hypothesis, visual attention should predict choice (and the match between value and goal) and not value, independently of the frame manipulation.

Our data strongly supported the second hypothesis because we found participants preferentially gaze (Figure 2A) the higher value option during like (t(30) = 7.56, p<0.001) and the lower value option during dislike frame (t(30) = -4.99, p<0.001). From a hierarchical logistic regression analysis predicting choice (Figure 2B), the difference between the time participants spent observing the right over left item (ΔDT) was a positive predictor of choice both in like (z = 6.448, p<0.001) and dislike (z = 6.750, p<0.001) frames. This means that participants looked for longer at the item that better fits the frame and not at the item with the highest value. Notably, the magnitude of this effect was slightly lower in the dislike case (t(30) = 2.31, p<0.05). In Figure 2B are also plotted the predictors of the other variables on choice from the best fitting model.

Figure 2

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We then analysed the effect of attention on choice in the perceptual case to test the generality of our findings. As in the Value Experiment, our data confirmed that participants did not have issues in choosing the circle with more dots in the most frame and the one with least amount dots in the fewest frame (Figure 1D,F). Furthermore, as in the Value Experiment and many other previous findings (De Martino et al., 2013; Folke et al., 2017), confidence modulated the accuracy of their decisions (Figure 1E). Critically for our main hypothesis, we found that participants’ gaze was preferentially allocated to the relevant option in each frame (Figure 2C): they spent more time observing the circle with more dots during most frame (t(31)=13.85, p<0.001) and the one with less dots during fewest frame (t(31)=-10.88, p<0.001). ΔDT was a positive predictor of choice (Figure 2D) in most (z = 10.249, p<0.001), and fewest (z = 10.449, p<0.001) frames. Contrary to the results in the Value Experiment in which the effect of ΔDT on choice was slightly more marked in the like condition (Figure 2B), in the Perceptual Study the effect of ΔDT was the opposite: ΔDT had a higher effect in the fewest frame (ΔDT_Most-Few: t(31)=-2.17, p<0.05)(Figure 2D). However, and most importantly, in both studies ΔDT was a robust positive predictor of choice in both frame manipulations. To summarise, these results show that in the context of a simple perceptual task, visual attention also has a specific effect in modulating information processing in a goal-directed manner: subjects spend more time fixating the option they will select, not necessarily the option with the highest number of dots.

In both Value and Perceptual Experiments, the most parsimonious models were reported in the manuscript and in Figure 2B and D. For a full model comparison see Appendix 2—figure 1 and Appendix 2—table 1. More details on the choice models are reported in the Appendix 2.

In the Value Experiment in both frames, we replicated the last fixation effect and its modulation by value difference between the last fixated option and the other one (Figure 3A). In the like frame, the probability of choosing the last item fixated upon increases when the value of the last item is higher, as is shown by the positive sign of the slope of the logistic curve (mean β_Like = 0.922). Crucially, during the dislike frame the opposite effect was found: the probability of choosing the last seen option increases when the value of the non-chosen item is higher, seen from the negative slope of the curve (mean β_Dislike = −0.951; Δβ_Like-Dislike: t(30)=7.963, p<0.001).

Figure 3

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We observed the same pattern of results that in the Value Experiment (Figure 3B). In the most frame, it was more probable that the last fixation was on the chosen item when the fixated circle had a higher number of dots (mean β_Most = 1.581). In the fewest frame, the effect flipped: it was more likely that the last circle seen was chosen when it had fewer dots (mean β_Few = −0.944; Δβ_Most-Few: t(31)=3.727, p<0.001).

The previous set of analysis shows that the last fixation is modulated by the difference in evidence according to the goal that the participant is set to achieve. However, since the last fixation is in general followed by the participant response, one could suspect that the goal-dependent modulation of attention (i.e. ΔDT) we identified in our choice regression analysis (Figure 2) is entirely driven by the final fixation. This would be problematic since one would have similar results to the one presented in Figure 2 even if participants’ pattern of attention is not modulated by the goal (i.e. attention is directed in both frames to the most valuable item) or even if the pattern of fixation, before the last fixation, is random. To control for this possibility, we performed a series of further analyses:

First of all, we repeated the analysis presented in the previous section (hierarchical choice regression – Figure 2), removing the last two fixations when calculating the ΔDT. Note that we removed the last two fixations and not just the last one to avoid statistical artefacts (i.e. since the final fixation is mostly directed towards the chosen item there would be an increased probability that second to last fixation is on the unchosen item). In Appendix 2—figure 3, we show that once removed the last two fixations the pattern of results is unchanged.

Second, we specifically investigated the middle fixations. Previous studies (Krajbich et al., 2010; Krajbich and Rangel, 2011; Tavares et al., 2017) have reported that middle fixations duration increases when the difference in value ratings (or perceptual evidence) of the fixated minus unfixated item increases. We replicated this result for our like and most frames but critically the effect was reversed in dislike and fewest frames (i.e. middle fixations durations decreased when the relative value of the fixated item was higher). The results suggesting that the goal-relevant modulation of attention affects also the middle fixations are presented in the Appendix 3—figure 4.

Finally, we investigated in more detail how the relation between attentional allocation and difference in value or perceptual evidence changed over time in the context of the goal manipulation. We calculated the Pearson correlation between fixation position (0: left, 1:right) and the difference in evidence (i.e. ΔValue or ΔDots, in both cases right – left item) at different time points (Figure 3C). We observed that after an initial phase in which there was no clear gaze preference for any of the items (note that given the gaze-contingent design participants must explore both alternatives), fixations were correlated with the frame-relevant item: during like frame, fixations positions were positively correlated with ΔValue, that is the fixations were directed towards the item with higher value; during dislike frame the behaviour was the opposite: fixations were negatively correlated with ΔValue, indicating a preference for the option with lower value. Note that these results are in line with the ones reported by Kovach et al., 2014. We see a very similar pattern of results in the Perceptual Experiment too (Figure 3D).

To explore the effect that behavioural factors had over confidence, we fitted a hierarchical linear model (Figure 4A). As it was the case for the results presented above for the choice regression, the results for the confidence regression in the like frame replicated all the effects reported in a previous study from our lab (Folke et al., 2017). Again, we presented here the most parsimonious model (Appendix 4—figure 1 and Appendix 4—table 1 for model comparison). We found that the magnitude of ΔValue (|ΔValue|) had a positive influence on confidence in like (z = 5.465, p<0.001) and dislike (z = 6.300, p<0.001) frames, indicating that participants reported higher confidence when the items have a larger difference in value; this effect was larger in the dislike frame (t(30) = -4.72, p<0.01). Reaction time (RT) had a negative effect on confidence in like (z = −6.373, p<0.001) and dislike (z = −7.739, p<0.001) frames, that is, confidence was lower when the RTs were longer. Additionally, we found that, in both conditions, higher number of gaze switches (i.e. gaze shift frequency, GSF) predicted lower values of confidence in like (z = −2.365, p<0.05) and dislike (z = −2.589, p<0.05) frames, as reported in Folke et al., 2017.

Figure 4

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We then looked at the effect of the summed value of both options, ΣValue, on confidence. As in Folke et al., 2017, we found a positive effect of ΣValue on confidence in the like frame (z = 3.206, p<0.01); that is, participants reported a higher confidence level when both options were high in value. Interestingly, this effect was inverted in the dislike frame (z = −4.492, p<0.001), with a significant difference between the two frames (t(30)=9.91, p<0.001) This means that, contrary to what happened in the like frame in which confidence was boosted when both items had high value, in the dislike frame confidence increased when both items had low value. This novel finding reveals that the change in context also generates a reassessment of the evidence used to generate the confidence reports; that is, confidence also tracks goal-relevant information.

We repeated the same regression analysis in the perceptual decision experiment, replacing value evidence input with perceptual evidence (i.e. absolute difference in the number of dots, |ΔDots|). We directly replicated all the results of the Value Experiment, generalising the effects we isolated to the perceptual realm (Figure 4B). Specifically, we found that |ΔDots| had a positive influence on confidence in most (z = 3.546, p<0.001) and fewest frames (z = 7.571, p<0.001), indicating that participants reported higher confidence when the evidence was stronger. The effect of absolute evidence |ΔDots| on confidence was bigger in the fewest frame (t(31)=-4.716, p<0.001). RT had a negative effect over confidence in most (z = −7.599, p<0.001) and fewest frames (z = −5.51, p<0.001), that is, faster trials were associated with higher confidence. We also found that GSF predicted lower values of confidence in most (z = −4.354, p<0.001) and fewest (z = −5.204, p<0.001) frames. Critically (like in the Value Experiment), the effect of the sum of evidence (ΣDots) on confidence also changes sign depending on the frame. While ΣDots had a positive effect over confidence in the most frame (z = 2.061, p<0.05), this effect is the opposite in the fewest frame (z = −7.135, p<0.001), with a significant difference between the parameters in both frames (t(31)=14.621, p<0.001). The magnitude of ΣDots effect was stronger in the fewest frame (t(31)=-10.438, p<0.001). For further details on the confidence models see the Appendix 4 (Appendix 4—table 2 and Appendix 4 —table 3).

The GLAM belongs to the family of race models in which evidence is independently accumulated for each option. Therefore, using the GLAM we were able to adapt the model to estimate a measure of confidence in the decision that is defined by the balance of evidence (Vickers, 1979; Vickers, 1970; Kepecs et al., 2008; De Martino et al., 2013) allowing us to characterise the pattern of the confidence measures. Balance of evidence is defined as the absolute difference between the accumulators for each option at the moment of choice, which is when one of them reaches the decision threshold (i.e., Δe = |E_right(t_final) - E_left(t_final)|) (Figure 6A). To estimate Δe, we performed a large number of computer simulations using the fitted parameters for each participant in both experiments.

Figure 6

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At the beginning of this experiment, participants were asked to report on a scale from £0 to £3 the maximum they would be willing to pay for each of 60 snack food items. They were informed that this bid will give them the opportunity to purchase a snack at the end of the experiment, using the BDM (Becker et al., 1964), which gives them incentives to report their true valuation. Participants were asked to fast for 4 hr previous to the experiment, expecting they would be hungry and willing to spend money to buy a snack.

After the bid process, participants completed the choice task: in each trial, they were asked to choose between two snack items, displayed on-screen in equidistant boxes to the left and right of the centre of the screen (Figure 1A). After each binary choice, participants also rated their subjective level of confidence in their choice. Pairs were selected using the value ratings given in the bidding task: using a median split, each item was categorised as high- or low-value for the agent; these were then combined to produce 15 high-value, 15 low-value, and 30 mixed pairs, for a total of 60 pairs tailored to the participant’s preferences. Each pair was presented twice, inverting the position to have a counterbalanced item presentation.

The key aspect of our experimental setting is that all participants executed the choice process under two framing conditions: (1) a like frame, in which participants were asked to select the item that they liked the most, that is, the snack that they would prefer to eat at the end of the experiment and (2) a dislike frame in which participants were asked to select the item that they liked the least, knowing that this is tantamount to choosing the other item for consumption at the end of the experiment. See Figure 1A for a diagram of the task.

After four practice trials, participants performed a total of 6 blocks of 40 trials (240 trials in total). Like and dislike frames were presented in alternate blocks and the order was counterbalanced across participants (120 trials per frame). An icon in the top-left corner of the screen (‘thumbs up’ for like and ‘stop sign’ for dislike) reminded participants of the choice they were required to make; this was also announced by the investigator at the beginning of every block. The last pair in a block would not be first in the subsequent block.

Participants’ eye movements were recorded throughout the choice task and the presentation of food items was gaze-contingent: participants could only see one item at a time depending on which box they looked at; following Folke et al., 2017, this was done to reduce the risk that participant, while gazing one item, would still look at the other item in their visual periphery.

Once all tasks were completed, one trial was randomly selected from the choice task. The BDM bid value of the preferred item (the chosen one in the like frame and the unchosen one in the dislike frame) was compared with a randomly generated number between £0 and £3. If the bid was higher than the BDM generated value, an amount equivalent to the BDM value was subtracted from their £20 payment and the participant received the food item. If the bid was lower than the generated value, participants were paid £20 for their time and did not receive any snack. In either case, participants were required to stay in the testing room for an extra hour and were unable to eat any food during this time other than food bought in the auction. Participants were made aware of the whole procedure before the experiment began.

Perceptual Experiment had a design similar to the one implemented in Value Experiment, except that alternatives were visual stimuli instead of food items. In this task, participants had to choose between two circles filled with dots (for a schematic diagram see Figure 1), again in two frames. In the most frame, they had to pick the one with more dots; and the one with fewer dots in the fewest frame. The total number of dots presented in the circles could have three numerosity levels (=50, 80 and 110 dots). For each pair in those three levels, the dot difference between the circles varied in 10 percentage levels (ranging from 2% to 20% with 2% steps). To increase the difficulty of the task, in addition to the target dots (blue-green coloured), distractor dots (orange coloured) were also shown. The number of distractor dots was 80% of that of target dots (40, 64, 88 for the three numerosity levels, respectively). Pairs were presented twice and counterbalanced for item presentation. After 40 practice trials (20 initial trials with feedback, last 20 without), participants completed 3 blocks of 40 trials in the most frame and the same number in the fewest frame; they faced blocks with alternating frames, with a presentation order counterbalanced across participants. On the top left side of the screen a message indicating Most or Fewest reminded participants of the current frame. Participants reported their confidence level in making the correct choice at the end of each trial. As in the previous experiment, the presentation of each circle was gaze contingent. Eye tracking information was recorded for each trial. Participants received £7.5 for 1 hr in this study.

Both tasks were programmed using Experiment Builder version 2.1.140 (SR Research).

We excluded individuals that met any of the following criteria:

1. Participants used less than 25% of the BDM value scale.

2. Participants gave exactly the same BDM value for more than 50% of the items.

3. Participants used less than 25% of the choice confidence scales.

4. Participants gave exactly the same confidence rating for more than 50% of their choices.

5. Participants did not comply with the requirements of the experiment (i.e., participants that consistently choose the preferred item in dislike frame or their average blink time is over 15% of the duration of the trials).

Since for Perceptual Experiment the assessment of the value scale is irrelevant, we excluded participants according to criteria 3, 4, and 5.

Forty volunteers gave their informed consent to take part in this research. Of these, 31 passed the exclusion criteria and were included in the analysis (16 females, 17 males, aged 20–54, mean age of 28.8). One participant was excluded for using less than 25% of the bidding scale (criteria 1). A second participant was excluded according to criteria 2 as they frequently gave the same bid value. A further four participants were excluded under criteria 4. Three participants were excluded due to criteria 5. In the latter case, one participant’s eye-tracking data showed the highest number of blink events and made choices without fixating any of the items; the other two did not comply with the frame manipulation. To ensure familiarity with the snack items, all the participants in the study had lived in the UK for 1 year or more (average 17 years).

Forty volunteers were recruited for the second experiment. Thirty-two participants (22 females, 10 males, aged 19–50, mean age of 26.03) were included in the behavioural and regression analyses. Three participants were excluded for repetition of the confidence rating (criteria 4). Five participants were removed for criteria 5: four of them had performance close to chance level or did not followed the frame modification, and one participant presented difficulties for eye-tracking. Due to instability in parameter estimation (problem of MCMC convergence), four additional participants were removed from the GLAM modelling analysis.

All participants signed a consent form, and both studies were done following the approval given by the University College London, Division of Psychology and Language Sciences ethics committee.

An Eyelink 1000 eye-tracker (SR Research) was used to collect the visual data. Left eye movements were sampled at 500 Hz. Participants rested their heads over a head support in front of the screen. Display resolution was of 1024 × 768 pixels. To standardise the environmental setting and the level of detectability, the lighting was monitored in the room using a dimmer lamp and light intensity was maintained at 4 ± 0.5 lx at the position of the head-mount when the screen was black.

Eye-tracking data were analysed initially using Data Viewer (SR Research), from which reports were extracted containing details of eye movements. We defined two interest-areas (IA) for left and right alternatives: two squares of 350 × 350 pixels in Value Experiment and two circles of 170 pixels of radius for Perceptual Experiment. The data extracted from the eye-tracker were taken between the appearance of the elements on the screen (snack items or circle with dots in experiments 1 and 2, respectively) and the choice button press (confidence report period was not considered for eye data analysis).

The time participants spent fixating on each IA was defined the dwelling time (DT). From it, we derived a difference in dwelling time (ΔDT) for each trial by subtracting DT of the right IA minus the DT of the left IA. Starting and ending IA of each saccade were recorded. This information was used to determine the number of times participants alternated their gaze between IAs, that is, ‘gaze shifts’. The total number of gaze shifts between IAs was extracted for each trial, producing the gaze shift frequency (GSF) variable.

We fitted the model for each individual participant and for like and dislike frames, separately. To model participant’s behaviour in the like frame, we used as input for GLAM the RTs and choices, plus BDM bid values and relative gaze for left and right alternatives for each trial. The original GLAM formulation (as presented above) assumes that evidence is accumulated in line with the preference value of a particular item (i.e. ‘how much I like this item’). When information about visual attention is included in the model, the multiplicative model in GLAM assumes that attention will boost the evidence accumulation already defined by value. Our proposal is that evidence accumulation is a flexible process in which attention is attracted to items based on the match between their value and task-goal (accept or reject) and not based on value alone, as most of the previous studies have assumed. Since in the dislike frame the item with the lower value becomes relevant to fulfil the task, we considered the opposite value of the items (r_i,dislike = 3 - r_i,like, e.g. item with value 3, the maximum value, becomes value 0) as an input for GLAM fit. For both conditions, model fit was performed only on even-numbered trials using Markov-Chain-Monte-Carlo sampling, using implementation for No-U-Turn-Sampler (NUTS), four chains were sampled, 1000 tuning samples were used, 2000 posterior samples to estimate the model parameters. The convergence was diagnosed using the Gelman-Rubin statistic (|${\displaystyle \hat{R}}$ – 1|<0.05) and also corroborating that the effective sample size (ESS) was high (ESS >100) for the four parameters (ν, γ, σ, and τ). Considering all the individual models, we found divergences in less than 3% of the estimated parameters. Model comparison was performed using Watanabe-Akaike Information Criterion (WAIC) scores available in PyMC3, calculated for each individual participant fit.

Pointing to check if the model replicates the behavioural effects observed in the data (Palminteri et al., 2017), simulations for choice and response time (RT) were performed using participant’s odd trials, each one repeated 50 times. For each trial, value and relative gaze for left and right items were used together with the individual estimated parameters. Random choice and RT (within a range of the minimum and maximum RT observed for each particular participant) were set for 5% of the simulations, replicating the contaminating process included in the model as described by Thomas et al., 2019.

Additionally, we simulated the accumulation process in each trial to obtain a measure of balance of evidence (Vickers, 1970; Vickers, 1979) for each trial. The purpose of this analysis was to replicate the effect of ΣValue over confidence (check Results for details) and check if it arises from the accumulation process and its interaction with attention. Balance of evidence in accumulator models has been used previously as an approximation to the generation of confidence in perceptual and value-based decision experiments (Vickers, 1979; Smith and Vickers, 1988; De Martino et al., 2013). Consequently, using the value of the items and gaze ratio from odd-numbered trials, we simulated two accumulators (Equation 5), one for each alternative. Our simulations used the GLAM parameters obtained from participant’s fit. Once the boundary was reached by one of the stochastic accumulators (fixed boundary = 1), we extracted the simulated RT and choice. The absolute difference between the accumulators when the boundary was reached (Δe = |E_right(t_final) - E_left(t_final)|) delivered the balance of evidence for that trial. In total 37,200 trials were simulated (10 repetitions for each one of the trials done by the participants). A linear regression model to predict simulated Δe using |ΔValue|, simulated RT and ΣValue as predictors was calculated with the pooled participants’ data. This model was chosen since it was the most parsimonious model obtained to predict participant’s confidence in the Value Experiment (Appendix 4—figure 1). The best model includes GSF as predictor in the regression, but since GLAM does not consider the gaze dynamics we removed it from the model. Δe simulations using a GLAM without gaze influence (i.e. equal gaze time for each alternative) were also generated, to check if gaze difference was required to reproduce ΣValue effect over confidence. The parameters fitted for individual participants were also used in the no-gaze difference simulation. The same linear regression model (Δe ∼ |ΔValue| + simulated RT + ΣValue) was used with the data simulated with no-gaze difference.

In the Perceptual Experiment, we repeated the same GLAM analysis done in Value Experiment. Due to instabilities in the parameters’ fit for some participants, we excluded four extra participants. Twenty-eight participants were included in this analysis. Additionally, the GLAM fit in this case was done removing outlier trials, that is, trials with RT higher than 3 standard deviations (within participant) or higher than 20 s. Overall less than 2% of the trials were removed. For most frame, relative gaze and perceptual evidence (number of dots) for each alternative were used to fit choice and RT. In a similar way to the consideration taken in the dislike case, we reassigned the perceptual evidence in the fewest frame (r_i,fewest = 133 - r_i,most+ 40 , considering that 133 is the higher number of dots presented and 40 dots the minimum) in a way that the options with higher perceptual evidence in the most frame have the lower evidence in the fewest frame. The same MCMC parameters used to fit the model for each participant in the Value Experiment were used in this case (again, only even-numbered trials were used to fit the model). As in the Value Experiment, model convergence was assessed using ${\displaystyle \hat{R}}$ and ESS. Overall, we observed divergences in less than 2% of parameter estimations across participants. Behavioural out-of-sample simulations (using the odd-numbered trials) and balance of evidence simulations (33,600 trials simulated in the Perceptual Experiment) were considered in this analysis. We tested the effect of ΣDots over confidence with a similar linear regression model than the one used in the Value Experiment. Pooled participants’ data for |ΔDots|, simulated RT and ΣDots was used to predict Δe. Δe simulations using a GLAM without gaze asymmetry were also calculated in this case. All the figures and analysis were done in Python using GLAM toolbox and custom scripts.

We examined how the frame manipulation impacted overall performance (Appendix 1—figure 1A). We defined ‘accuracy’ as the proportion of trials in which participant’s reported values (BDM bid) correctly predicted their binary decision, that is, they select the item with highest value in the like frame and the one with lowest value in the dislike frame. Overall accuracy was not significantly different in both frames (Mean_Like = 0.77; Mean_Dislike = 0.75, t(30) = 1.71; p=0.1). We also found that participants had slightly slower reaction times (RTs) in the dislike frame (Mean_Like = 2858.2 ms, Mean_Dislike = 3152.7 ms; t(30) = −2.52; p<0.05). Participants reported lower confidence in the dislike frame (Mean_{∆Confidence} = 0.19; t(30) = 4.49; p<0.001) and shifted their gaze (gaze shift frequency, GSF) between items more during dislike trials (Mean_∆|GSF| = −0.110; t(30) = −2.99; p<0.01). These results overall suggest that the subjects may have found the dislike condition slightly less intuitive. Although this did not affect their performance, it slightly reduced their confidence and increased RT and GSF.

As observed in previous studies (Folke et al., 2017; De Martino et al., 2013), we found that choice accuracy was modulated by confidence: decisions in which participants reported high-confidence were more accurately predicted by the value estimate collected before the experiment – the slope of the logistic curve is steeper in the case of high confidence (Figure 1B, Results section). In this study, this effect is replicated in both like (low confidence: β = 0.769; high confidence: β = 1.633) and dislike (low confidence: β = −0.642; high confidence: β = −1.363) frames. Note that the inversion of the sign of the slopes in like vs dislike frames indicate that participants were performing the task correctly (∆β_Like-Dislike: t(30) = 8.14, p<0.001), selecting the item with lower value during the dislike frame (Figure 1C, Results section). Choice accuracy (steepness of the slopes) was not significantly different between like and dislike frames (∆|β_Like-Dislike|: t(30) = 1.58, p=0.124).

We repeated the same analysis for the behavioural performance in most and fewest frames (Appendix 1—figure 1B). In contrast to the Value Experiment, we observed a slight reduction in accuracy in participant responses for the fewest frame (Mean_Most = 0.77, Mean_Few = 0.74, t(31) = 2.46; p<0.05); unlike the Value Experiment, however, we did not find differences in RTs (Mean_Most = 4029.57 ms, Mean_Few = 3975.59 ms; t(31) = 0.32; p=0.75). During the fewest frame participants reported lower confidence (Mean_{∆Confidence}=0.24; t(31) = 5.62; p<0.001) and shifted their gaze more between alternatives (Mean_∆|GSF| = -0.17; t(31) = -4.15; p<0.001), as observed in the Value Experiment.

Participants also reported higher confidence in trials that better discriminated the number of dots (Figure 1E, Results section). This effect was replicated in both most (low confidence: β = 1.142; high confidence: β = 2.164) and fewest (low confidence: β = −1.118; high confidence: β = −2.010) frames. The inversion of the sign of the slopes in most vs fewest frames also shows that participants were performing correctly (∆β_Most-Few: t(31) = 22.22, p<0.001); the magnitude of the slopes was not significantly different between the two frames (∆|β_Most-Few|: t(31) = 0.79, p=0.434; Figure 1F, Results section). This pattern of results mirrors the pattern seen in the Value Experiment.

Appendix 1—figure 1

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Appendix 1—figure 2

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Appendix 1—figure 3

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Appendix 1—figure 4

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Appendix 1—figure 5

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Using a logistic hierarchical regression model, we investigated which factors modulated choice-proportion, defined here as the probability of choosing the item on the right side of the screen. We report here the results of the most parsimonious model (i.e. the model with a lowest BIC; Appendix 2—figure 1) fitted to the like and dislike frames independently (Figure 2B, Results section). In Appendix 2—table 1, we present the parameters for each factor included in the model. In the like frame, the difference in the value of the right item minus left item (∆Value) had a positive influence on choice-proportion, that is, participants selected the items that had higher value. This is reversed in the dislike frame: ∆Value is now a negative predictor of choice, that is, participants selected the items that had lower value. In both conditions, confidence enhanced the effect of ∆Value, as shown by the interaction between ∆Value and confidence in the like and dislike frame. These results confirm the findings presented in Figure 1B (Results section) while controlling for other relevant variables. Unsurprisingly, confidence and summed value (ΣValue, the added value of both alternatives) were found to show no main effect on the choice-proportion. As discussed in the Results section, gaze allocation (difference in dwell time, ∆DT) is directed to the chosen item in both frames, that is, the parameters are positive for ∆DT in like and dislike frame (Appendix 2—table 2). 

As in the Value Experiment, we used a logistic hierarchical regression to determine the relevant factors modulating perceptual choice (choosing the circle with dots on the right side of the screen) (Figure 2D, Results section). We found that the most parsimonious model for choice was the same used in the Value Experiment, where like and dislike were replaced by most and fewest frames (Appendix 2—figure 1B). In the most frame, the difference in the number of dots of the right alternative minus the left one (∆Dots) had a positive influence over choice; that is, participants tended to select the circle with more dots. As expected, this pattern was reversed in the fewest frame: ∆Dots was a negative predictor of choice. As in the Value Experiment, confidence modulated the effect of ∆Dots in most and fewest frames. The sum of dots presented in both circles during a trial (ΣDots) was found not to have a significant effect on either frame, as expected. However, as discussed in the Results section, confidence was found to be a negative predictor of choice in most and fewest frames. This means participants had a bias to report higher confidence when they chose the left circle. In a similar way to the Value Experiment, participants spend more time fixating the chosen alternative in both frames, with ∆DT effect being positive in most and fewest frames (Appendix 2—table 3).

In a study by Kovach et al., 2014 a design similar to our value-based experiment was implemented. Participants were required to indicate the item to keep and the one to discard. They found, similarly to our findings in the Value Experiment, that the overall pattern of attention was mostly allocated according the task goal. However, in the first few hundred milliseconds, these authors found that attention was directed more prominently to the most valuable item in both conditions. We did not replicate this last finding in our experiment, but one possible reason for this discrepancy is that the experiment by Kovach and colleagues presented both items on the screen at the beginning of the task -- unlike in our task, in which the item was presented in a gaze-contingent way (to avoid processing in the visual periphery). This setting might have triggered an initial and transitory bottom-up attention grab from the most valuable (and often most salient) item before the accumulation process started.

Appendix 2—figure 2

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Appendix 2—figure 3

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To explore variations in the process of accumulation of evidence characterised by GLAM, we compared the parameters obtained from the individual fit in like and dislike frames (Appendix 6—figure 1A). No significant variation between frames was found for the gaze bias (Mean γ _Like = -0.14, Mean γ _Dislike = 0.03, ∆γ _Like-Dislike = -0.17, t(30) = -1.66; p=0.11, ns), the scaling parameter (τ _Like = 2.81, τ_Dislike = 2.69, ∆τ_Like-Dislike = 0.115, t(30) = 0.313; p=0.75, ns), and the noise term (Mean σ_Like = 0.0075, Mean σ _Dislike = 0.0074, ∆σ_Like-Dislike = 0.00012, t(30) = 0.342; p=0.734, ns). We observed a significantly higher value of the drift term, ν, during the like frame (ν_Like = 5.60x10⁻⁵, ν_Dislike = 4.53x10⁻⁵, ∆ν_Like-Dislike = 1.06 x 10⁻⁵, t(30) = 3.44; p<0.01). This means that evidence is accumulated faster during the like frame, which gives us an insight into the differences in the evidence accumulation product of the change frame modification.

We also compared the parameters obtained from GLAM individual fit in the perceptual experiment (Appendix 6—figure 1B). No significant variation between frames was found for the scaling parameter (τ_Most = 0.34, τ_Few = 0.13, ∆τ _Most-Few = 0.212, t(27) = 1.43; p=0.16, ns) or the drift term (Mean ν_Most = 3.8x10⁻⁵, Mean ν _Few = 3.99x10⁻⁵, ∆ν_Most-Few = -1.92x10⁻⁶, t(27) = -0.465; p=0.645, ns). The gaze bias is larger during the fewest frame (γ_Most = 0.48, γ_Few = 0.26, ∆γ_Most-Few = 0.22, t(27) = 2.61; p<0.05). The σ parameter is also significantly different depending on the frame, with higher noise in the most frame (σ_Most = 0.0073, σ_Few = 0.0066, ∆σ _Most-Few = 0.0007, t(27) = 2.26; p<0.05). In summary, the accumulation process seems to be noisier and less affected by visual attention in the most frame. In both frames, the finding that γ < 1 indicates that gaze modulates the accumulation of evidence.

Appendix 6—figure 1

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Appendix 6—figure 2

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The models were fitted to choice and RT data independently for like and dislike frames in our Value Experiment and for most and fewest frames in the Perceptual Experiment. The odd trials of the pooled data from 31 participants in value-based data and 32 participants for perceptual case was used to fit the models. The model considers the available evidence (item value and number of dots) and the sequence of fixations for each trial. As in GLAM, we fitted the parameters in dislike and fewest frames considering a version of the input values/evidence that accounted for the change in the objective of the task (i.e. reporting item not preferred or the alternatives with fewer dots, respectively). To compare, we also fitted another model using the evidence ‘by default’ (i.e. BDM bid values or number of dots in the circles). To account for the different ranges of item valuation used by the participants we normalised the value reports by binning at a participant level. In the Value Experiment, the data were separated in six bins using quantiles-based discretisation. In the Perceptual Experiment, given the distribution of the evidence (i.e three numerosity levels and smaller dots differences between two alternatives), we separated the dots data in eight bins. The maximum likelihood estimation (MLE) procedure was carried in iterative steps searching over a grid with the three model parameters. Initial grid was set to [0.001, 0.005, 0.01] for d, [0.01, 0.05, 0.1] for σ and [0.01, 0.5, 1] for θ. The likelihood for choice and RT in odd-trials, conditional to the pattern of fixations, was calculated for each combination of parameters in the grid (check Tavares et al., 2017 for the details of the algorithm to simulate aDDM trials). The time step used for the estimation of aDDM was 10 ms. The set of parameters with lower negative log-likelihood (NLL) was used as centre of the grid for the next iteration. Therefore, the grid to search in the next iteration (t+1) was defined as [d_t − Δd_t/2, d_t, d_t + Δd_t/2], [θ_t − Δθ_t/2, θ_t, θ_t + Δθ_t/2], and [σ_t − Δσ_t/2, σ_t, σ_t+Δσ_t/2], considering the respective constrains of each parameter value. The iterative process finished once the improvement in the MLE of the proposed parameter solution was smaller than 0.05% (|minNNL_t+1 – minNNL_t| < 0.0005*minNNL_t). The convergence was reached after two iterations in our models. In our results Appendix 7—table 1), we found that for both, dislike and fewest conditions, the model fitted using goal-relevant evidence had better performance than the model using default value or number of dots, as indicated by a lower NLL value.

To test the capacity of the model to predict out-of- sample, the aDDM with the best fitted parameters using odd-numbered trials was used to predict the behaviour observed on the even-numbered trials. We generated 40,000 simulations for the Value Experiment and 48,000 trials for the Perceptual Experiment. Fixations, latencies and inter-fixations transitions were sampled from empirical distributions, obtained from the pooled even-numbered trials across participants following the procedure used by Tavares et al., 2017.

Appendix 7—figure 1

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Appendix 7—figure 2

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Appendix 7—figure 3

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Proof. Recall that, for ${i\ne i}_1,$ we have $V\left(i\right)=max\left\{{\mu }_{i_1},{\mu \text{'}}_i\right\}$ and ${\mu \text{'}}_i\sim N\left({\mu }_i,\theta \right)$. This means that the belief about $V\left(i\right)$, for ${i\ne i}_1$, coincides with $N\left({\mu }_i,\theta \right)$ for values above ${\mu }_{i_1}$, but has a mass point at ${\mu }_{i_1}$ equal to the probability that $N\left({\mu }_i,\theta \right)$ is below ${\mu }_{i_1}$. If we denote by $f_{\mu }$ the Probability Density Function of $N\left(\mathrm{\mu },\theta \right)$, it follows that we have

Recall also that $V\left(i_1\right)=max\left\{{\mu }_{i_2},{\mu \text{'}}_{i_1}\right\}$. The belief about $V\left(i_1\right)$ coincides with $N\left({\mu }_{i_1},\theta \right)$ above ${\mu }_{i_2}$, but has a mass point at ${\mu }_{i_2}$ equal to the probability that $N\left({\mu }_i,\theta \right)$ is below ${\mu }_{i_2}.$ Then,

Note that, by construction, we have

It follows that

and

But we also know that

Together with Equation A1 and A2, this proves the claim.■

Proof. Recall that, for ${i\ne i}_1$, we have ${\displaystyle V\left(i\right)=max\left\{{\mu }_{i_1},{\mu }_i^{\prime }\right\}}$, where ${\mathrm{\mu }}_{\mathrm{i}}^{\text{'}}\sim N\left({\mu }_{\mathrm{i}},\theta \right)$. It follows that the beliefs about both $V\left(i_2\right)$ and $V\left(i_j\right)$ (held before the second signal is acquired) has support ${[\mu }_{i_1},\infty )$. Denote by $F_i$ the Cumulative Density Function (CDF) of this belief. To prove the claim, we show that $F_{i_2}$ First Order Stochastically Dominates $F_{i_j}$ for all ${\displaystyle j>2}$, while the converse is not true: that is, we aim to show that for all $x$ in the support, $F_{i_2}\left(x\right)\le F_{i_j}\left(x\right)$, strictly for some x. (Recall that a distribution F first order stochastically dominates another distribution G if for all x, the probability that F returns at least x is not below the probability that G returns x or more.) This implies ${\displaystyle E\left[V\left(i_2\right)\right]>E\left[V\left(i_j\right)\right].}$

Let ${\displaystyle \delta :={\mu }_{i_2}-{\mu }_{i_j}}$ and note that we have ${\displaystyle \delta >0}$ and that $N\left({\mu }_{i_2},\theta \right)\left(x+\delta \right)=N\left({\mu }_{i_j},\theta \right)\left(x\right)$ for all ${\displaystyle x\in \mathbb{R}}$. Since $V\left(i\right)$ coincides with ${\mu }_i\text{'}$ whenever that lies above ${\mu }_{i_1}$ and since ${\mu }_i\text{'}\mathrm{}\sim N\left({\mu }_{\mathrm{i}},\theta \right)$, it follows that, for all ${\displaystyle x>{\mu }_{i_1}}$, we have ${(1-F}_{i_j}\left(x\right))={(1-F}_{i_2}\left(x+\delta \right)):$ the probability that $F_{i_j}$ assigns to $V\left(i_j\right)$ being $x$ or higher is the same that $F_{i_2}$ assigns to $V\left(i_2\right)$ being $x+\delta$ or higher. Then, $F_{i_j}\left(x\right)=F_{i_2}\left(x+\delta \right).$ Because CDFs are increasing and ${\displaystyle \delta >0}$, then $F_{i_2}\left(x\right)\le F_{i_2}\left(x+\delta \right)$, thus $F_{i_2}\left(x\right)\le F_{i_j}\left(x\right)$ for all ${\displaystyle x>{\mu }_{i_1}}$. Moreover, notice that we must have

That is, $F_{i_2}$ assigns to values below ${\mu }_{i_1}$ a lower probability than $F_{i_j}$ does. It follows that for all $x$ in the support ${[\mu }_{i_1},\infty )$,we have $F_{i_2}\left(x\right)\le F_{i_j}\left(x\right)$, strictly for some. Thus, $F_{i_2}$ First Order Stochastically Dominates $F_{i_j}$ for all ${\displaystyle j>2}$, while the converse is not true. The claim follows.

The two claims together prove Proposition 1.

The proof of Proposition 2 is identical once we replace $i_j$ by $i_{N+1-j}$ for $j=1,\dots ,N$. Intuitively, the problem of maximising the expected utility of the remaining items is strategically equivalent to the problem of choosing the lowest item, which, in turn, is symmetric to the problem of choosing the best item. $QED.$

• Pradyumna Sepulveda

• Benedetto De Martino