Peripheral Visual Information Halves Attentional Choice Biases

A growing body of research has shown that simple choices involve the construction and comparison of values at the time of decision. These processes are modulated by attention in a way that leaves decision makers susceptible to attentional biases. Here, we studied the role of peripheral visual information on the choice process and on attentional choice biases. We used an eye-tracking experiment in which participants (N = 50 adults) made binary choices between food items that were displayed in marked screen “shelves” in two conditions: (a) where both items were displayed, and (b) where items were displayed only when participants fixated within their shelves. We found that removing the nonfixated option approximately doubled the size of the attentional biases. The results show that peripheral visual information is crucial in facilitating good decisions and suggest that individuals might be influenceable by settings in which only one item is shown at a time, such as e-commerce.

algorithm when all options are presented simultaneously, as when we shop at the market, and when they are presented sequentially, as when we shop online?If not, does the absence of peripheral information change the fixation process and the magnitude of the attentional biases?
We studied these questions using an eye-tracking experiment in which participants made binary choices between foods that were displayed in marked screen "shelves" in two different conditions: (a) a simultaneous condition in which both items were displayed on the screen at the time of choice, and (b) a gaze-contingent condition in which items were displayed only when participants fixated within their shelves.Most previous studies have used choice tasks in which all options are displayed simultaneously, although a handful have used gaze-contingent presentation of stimuli (Folke et al., 2016;Franco-Watkins & Johnson, 2011;Sepulveda et al., 2020;Simion & Shimojo, 2006).However, none have compared the two situations directly, which is necessary to understand the effect of peripheral visual information on the choice process.
Based on what is known about choice in the simultaneous case, and the seemingly minor change involved in removing the nonfixated options from peripheral vision, it is natural to hypothesize that similar algorithms are at work in both conditions, albeit with some differences.In particular, the aDDM suggests two nonmutually exclusive mechanisms through which removing the nonfixated options from the visual field might affect choices.First, it might increase the overweighting of fixated relative to nonfixated items, which would result in an increased attentional bias.Second, it might change the fixation process in a way that exacerbates the attentional biases, for example, by increasing the asymmetry on fixation time across options.
Understanding the role of peripheral visual information in simple choice is important for multiple reasons.First, despite the robustness of the attentional biases identified in previous work, we do not know the channels through which covert and overt visual attention influences decisions or their relative contribution to choice.Decades of work in visual attention have shown that a substantial amount of information is processed through peripheral visual attention (Carrasco, 2011;Perkovic et al., 2022;Wästlund et al., 2018), which presents the puzzle of why and how fixations matter so much in economic choices, even when one is making decisions among familiar items.Second, the transition to e-commerce has increased the frequency with which our decisions are made in sequential presentation settings.We need to understand the impact that this has on the choice algorithms and their associated biases in order to design interfaces and nudges that enhance choice quality.
To preview the results, we found that removing the nonfixated options had little impact on the average quality of choices.However, we also found that it approximately doubled the magnitude of attentional choice biases, which makes decision makers more susceptible to marketing interventions (e.g., packaging) that affect attention independently of the value of products.We also found that the removal of the nonfixated options slowed down the fixation and decision process considerably but that the impact on attentional biases was driven mostly by an increase in the tendency to overweight the value of fixated options.

Task
We investigated the role of peripheral information about nonfixated stimuli using the task depicted in Figure 1.Participants made decisions in two conditions: (a) a visible condition in which both items were displayed on the screen at the time of choice, and (b) a hidden condition in which items were displayed only when participants fixated within the location associated with the stimulus.Participants were asked to refrain from eating for 2 hr before the start of the experiment and to refrain from eating any foods afterward during a 1-hr waiting period, except for the snack that they chose in a randomly selected trial, which was given to them at the end of the experiment.
Participants completed two tasks.First, they were asked to provide liking ratings for 60 snack foods available at local stores ("How much would you LIKE to eat

Statement of Relevance
When consumers make shopping decisions, does their behavior depend on whether items are displayed in close proximity (so that alternative options can be seen in the periphery) or whether they are shown one at a time?This question helps us understand how point-of-sale marketing in the growing domain of e-commerce might be influencing decisions differently from traditional retail settings.We have found that when only one option is shown at a time, consumers are more biased toward selecting the option that attracts more attention, compared with when all options are shown at the same time.These results suggest that peripheral visual information plays a critical role in facilitating good decision-making and suggest a potential mechanism for future use in nudges.this food?"from 1 = don't like to 5 = like a lot, 0.25 intervals).Each item was rated twice, in random order, using a slider bar controlled by the arrow keys and initialized to a random location to reduce anchoring effects.We used the average of the two ratings as a measure of each item's value.
Second, participants made choices between two food items, shown on the left and right sides of the screen, in two separate conditions: (a) a visible condition in which both items were shown simultaneously, and (b) a hidden condition in which items were shown only when participants fixated within their region of interest (ROI).The ROIs were indicated in both treatments with a white box (Fig. 1).Trials started with an enforced 500-ms central fixation.Participants indicated their choices with the left and right arrow keys and responded at their own pace.The selected option was highlighted for 1 s, and trials were separated by a 1-s blank screen.Participants made 360 choices in the exploratory sample and 400 in the confirmatory sample, half on each experimental condition.The task was divided into four equal-size blocks, two with the hidden condition and two with the visible condition, in random order.
The choice pairs in the exploratory data set were randomly selected from all 60 food items.In the confirmatory sample, they were constructed as follows.We used each participant's ratings to prune the stimulus set down to the 40 food items that resulted in the most uniform distribution of ratings to maximize the spread of rating differences across choice pairs.Stimuli for each trial were then randomly selected, subject to the constraint that they be used 4 times per block in the exploratory data set and 5 times per block in the confirmatory data set.All 60 foods were shown once every 30 trials in the exploratory data set, and all 40 foods were shown once every 20 trials in the confirmatory data set.

Participants
Fifty participants (age: M = 30.6years, range = 18-50 years; 34 female) were recruited from Caltech and the surrounding community using flyers.Additional demographic information was not collected.We prescreened participants for a self-reported liking for snack foods (e.g., candy and potato chips) and against requiring glasses for vision correction that might interfere with eye tracking.Participants were paid a $35 participation fee.The experiment was approved by Caltech's Institutional Review Board.
To obtain high quality data, we implemented a participant filter at the data collection stage.Immediately after data collection, we deleted participants who failed any of the following criteria: (a) correlation between

Binary Choice Task
Fig. 1.Task.Participants had to fixate on a center fixation cross for 500 ms for the trial to start.In the visible condition, participants were presented with two snack food items simultaneously, each located within a white box on the left and right sides of the screen.In the hidden condition, participants had to fixate within the white boxes to reveal the snack food item inside.Participants indicated their response at their own pace with a keyboard press.After a choice was made, a blue box highlighted the selection for 1 s, followed by a 1-s intertrial interval.RT = reaction time.
the two liking ratings of at least 70%, (b) mean RT in choice trials between 0.7 and 6 s, (c) probability of choosing the best item significantly different from chance (based on a binomial test), and (d) at most 10% missing fixation data.Data collection continued until 50 participants passed the data quality criteria.The first 25 participants were allocated to the exploratory sample, and the other 25 to the confirmatory sample.The number of trials per participant and the number of participants were chosen on the basis of related studies that have shown that this sample size provides reliable estimates of the parameters and effects of interest.

Eye tracking
Participants' fixation patterns were recorded using an EyeLink 1000 eye tracker (SR Research, Kanata, ON, Canada) at 500 Hz.Participants sat approximately 60 cm from a 1,920 pixel × 1,080 pixel monitor.Food image sizes were 403 pixels × 302 pixels.Fixations within the ROI for the left food were classified as "left," those within the right food's ROI were classified as "right," and those outside the two ROIs were classified as "blank."If a sequence of blank fixations was recorded between two fixations of the same type (e.g., left-blankblank-left), they were recoded as a fixation of the same type (e.g., left-left-left-left) because blank fixations of this type are typically due to eye-tracking noise and tend to be quite short.Blank fixations recorded between two fixations of different types (e.g., left-blank-right) were coded as a saccade period between fixations.Trials in which any eye-tracking information was missing were dropped from further analysis (mean of 6 and 4 trials per participant in the exploratory and confirmatory data sets, respectively).

Data analysis strategy
To be able to explore the data in detail, while avoiding the type of statistical problems that have raised questions about the validity of some published research, we collected two separate data sets with 25 participants each.We used the first one to carry out exploratory analyses until we understood the data-generating process in sufficient detail.On the basis of this, we pinned down a set of analyses and tests that were carried out in a second confirmatory data set of equal size.Thus, the confirmatory data set serves as a replication of our findings and provides unbiased statistics for hypothesis testing.Given the similarity of the estimates and findings in both samples, and in the spirit of meta-analysis, we also provide results on the pooled sample and describe summary statistics in terms of the pooled estimates.

Computational model
As illustrated in Figure 2, the aDDM is a version of the DDM of binary choice (Gold & Shadlen, 2007;Ratcliff et al., 2016;Ratcliff & McKoon, 2008) in which value sampling is affected by fixation location.Participants integrate noisy value signals into an evolving evidence process.Evidence starts every trial at an initial location b, which may include some bias toward one of the options if b ≠ 0. A choice is made the first time evidence crosses one of two prespecified barriers, which are fixed at 1 for the left item and −1 for the right item.The identity of the barrier crossed determines which option is chosen.Critically, evidence evolves as the following diffusion process: where e t is independent and identically distributed white Gaussian noise with variance σ 2 , and the slope of the process depends on the fixation location.In particular, when the left item is fixated, the slope of integration is µ θ ) , and when the right item is fixated, it is ) , where d is a parameter controlling the speed of integration and θ is a parameter controlling the attentional bias.When θ = 1, the fixations do not affect choices, there are no attentional biases, and the process reduces to a standard DDM.In contrast, when θ < 1, the value of the fixated item is overweighted relative to the nonfixated value, which results in an attentional bias that increases as θ gets smaller.
Importantly, the aDDM assumes that the fixation process is orthogonal to the state of evidence in any given trial.Thus, when simulating the model, we sample fixations from the observed fixation distributions, separately for first and middle fixations.

aDDM fitting
We fitted the aDDM using a hierarchical Bayesian model, separately for the visible and hidden conditions, using the methods and associated toolbox developed by Lombardi and Hare (2021).We estimated the model separately for the exploratory, confirmatory, and pooled data sets.In every case, the model was fitted using only the odd trials because the even trials were reserved for out-of-sample predictions.As described in Figure S1 in the Supplemental Material available online, the model has the following free parameters, at both the group and individual levels: the evidence accumulation drift rate (d ), the standard deviation of the Gaussian noise for the drift process (σ), the attentional bias parameter (θ), and the initial bias of the drift process (b).
Posterior distributions were estimated using Markov chain Monte Carlo methods with three chains for a total of 55,000 burn-in samples and 30,000 samples from each of the posteriors.Gelman-Rubin statistics for all estimates are at or below 1.1, indicating convergence.
The model that we estimated and report in the article specifies priors without any correlation of parameters across the hidden and visible conditions.We did this to maximize the extent to which our posterior estimates were driven by the data.However, to investigate the role of the uncorrelated priors on our model fits, we also estimated a version of the model in which the priors for the same parameter in the visible and hidden conditions are correlated.In particular, each parameter (x ) consists of two parts: a baseline ( x ) and a hiddencondition deviation (∆ x ).See Figure S2 in the Supplemental Material for details.As discussed further below, both models generated very similar parameter estimates.

Out-of-sample simulations
Even-numbered trials were set aside as out-of-sample data to compare them with the predictions of the aDDM model fitted on the odd trials.We simulated 10 data sets for each participant and condition, using the same rating pairs encountered in the experiment.For each simulated data set, we sampled a set of parameters from the joint posterior distribution for that participant and condition.Then, we simulated each trial as follows.We sampled all fixation duration statistics from their observed empirical distributions in the even trials, conditional on the hidden or visible condition.For example, when we simulated a hidden-condition trial, evidence for the trial was initialized at the bias parameter.Evidence evolved on the basis of the noise only up to the duration of the sampled latency to first fixation.Afterward, a maximum first fixation duration was sampled from the distribution of first fixations in the hidden condition, and evidence evolved according to the drift rate, noise, and attentional bias parameters depending on the fixation location, as described in the Computational Model section.If a barrier was crossed before the maximum fixation duration was reached, the process was terminated and the choice and RT were recorded.Otherwise, a new saccadic duration and maximum fixation duration were sampled from the distributions of saccades and middle fixations in the hidden condition, respectively.The process was repeated until a choice was made.Note that this assumes that the value of the nonfixated item was known during the first fixation, which is unrealistic and interferes with the quality of our fits.

Hierarchical regressions
All the logistic and linear regressions reported in this article are based on standard hierarchical models with random coefficients for all parameters.The regressions were implemented using the brms package (Bürkner, 2017(Bürkner, , 2018) ) in the R programming environment (Version 4.1.3;R Core Team, 2022) and used the default weakly informative priors, occasionally scaled depending on the units of the independent variable.Posterior distributions were estimated using three chains for a total of 9,000 burn-in samples and 9,000 samples from each of the posteriors.See the companion data and code package for details (https://www.rnl.caltech.edu/publications/).

Basic psychometrics
The top row of Figure 3 depicts the psychometric choice curve, separately for each experimental condition and data set.See Table S1 in the Supplemental Material for the associated regression estimates and test statistics.We found a small but significant increase in the responsivity of choices to value differences in the hidden condition.The middle row of Figure 3 depicts RTs as a function of choice difficulty.We found that RT increases with choice difficulty, that average RTs are about 32% (520 ms) slower in the hidden condition, and that this slowdown does not vary significantly with choice difficulty.The bottom row of Figure 3 depicts the number of fixations as a function of choice difficulty.We found that the number of fixations increases with choice difficulty, and they are approximately similar in both conditions, except for a small flattening in the slope of the fixation curve in the hidden condition.
Together, these results show that removing the nonfixated items slows down the choice process but has a negligible effect on the quality of average choices: probability best chosen visible = 0.865 ± 0.007 (± standard errors), probability best chosen hidden = 0.876 ± 0.007, d = 0.19, t(49) = 1.81, p = .08.

Fixation process
Figure 4 as well as Table S2 in the Supplemental Material explore the fixation process in more detail.The goal here was to understand the impact that removing nonfixated items has on the fixation process, which is essential to understanding how it affects attentional biases.
The first row of Figure 4 depicts the probability that the first fixation is to the best item, as a function of choice difficulty.The first fixation location is at chance in both conditions.Figure S3 (top row) in the Supplemental Material shows that there is no difference between conditions on the probability of first fixating left: probability first fixating left visible = 0.759 ± 0.042, probability first fixating left hidden = 0.803 ± 0.048, d = 0.14, t(49) = 1.64, p = .11. Figure S3 (bottom row) and Table S3 in the Supplemental Material show that there is also no difference between conditions on the latency to the start of the first fixation.
The second row of Figure 4 depicts the mean duration of first, middle, and last fixations, separately for the two conditions.We found that the three types of fixations were longer in the hidden condition by about 40%, on average: ∆ first = 160 ms, d = 1.43, t( 49 An average trial has three fixations, and each fixation was, on average, 165 ms longer in the hidden condition, which implies that decisions should take 495 ms longer, just shy of the observed RT difference.The third row of Figure 4 depicts middle fixation durations as a function of choice difficulty.We found that middle fixation durations increase with choice difficulty.Figure S4 and Table S4 in the Supplemental Material show that this difference was driven by the value of the fixated item: In the hidden condition, middle fixation durations increased with the value of the fixated item, whereas the opposite occurred in the visible condition.Interestingly, Figure S4 also shows that middle fixation durations decreased with the value of the nonfixated item, even in the hidden condition. The fourth row of Figure 4 depicts the first fixation duration as a function of choice difficulty.We found that duration was independent of value in both conditions and about 46% (160 ms) longer in the hidden condition.See Figure S4 and Table S4 for additional results.
The bottom row of Figure 4 shows the relationship between relative value and relative fixation time.In both conditions, the relationship exhibits an S-shape.Both items were fixated the same amount when they had equal value, but otherwise, the better item was fixated longer, with the asymmetry on fixation time increasing in the value advantage.In addition, the effect was stronger in the hidden condition, and as a result, the distribution of net fixation times was more asymmetric in favor of the better item in this condition.Note that because the fixated item is overweighted in the aDDM, this asymmetry in relative fixation time facilitates choosing the better option.

Choice biases
Figure 5 as well as Table S5 in the Supplemental Material depict the attentional bias in both conditions.The goal here was to provide a model-free test of the extent to which removing nonfixated items affects attentional biases.
The top row depicts the probability of choosing the left item as a function of its relative rating and the location of the last fixation.In the absence of an attentional bias, the location of the last fixation should not matter, and the choice curves should lie on top of each other.In contrast, and consistent with previous studies (Fisher, 2017;Krajbich et al., 2010Krajbich et al., , 2012;;Krajbich & Rangel, 2011;Smith & Krajbich, 2018, 2019;Tavares et al., 2017), we found a substantial attentional bias in the visible condition: On average, when the left and right items are equally valued, the left item is 2.5 times more likely to be chosen when the last fixation is to the left than when it is to the right.The bias was substantially larger in the hidden condition, where the left item was 5 times more likely to be chosen when the last fixation was to the left than when it was to the right.
The middle row depicts the relationship between net fixation time and the corrected probability of choice.The choice measure is corrected by subtracting from each choice observation (coded as 1 if left chosen, and 0 otherwise) the proportion with which the left is chosen at each relative value.As a result, in the absence of an attentional bias, the corrected probability of choice should be 0, independent of net fixation time.In contrast, we found that shifting net fixation time toward the left item by 1 s increased its choice probability by 24% in both conditions.
The bottom row depicts the relationship between excess first fixation durations and the corrected choice probability of the first seen item, using the same correction described above.Excess first fixation duration is defined as first fixation duration minus mean first fixation duration (computed for each participant).In the absence of an attentional bias, the corrected probability should be 0 regardless of excess first fixation duration.In contrast, we found that an increase in the excess first fixation duration by 1 s increased the choice probability by about 22% in the visible condition but that there was no such effect in the hidden condition.aDDM Given that the aDDM has been shown to provide good quantitative accounts of the relationship between fixations, choices, and RTs, we fitted a hierarchical approximation of this model to our data, separately for the visible and hidden conditions.The goal was to investigate the impact of removing the nonfixated items on the parameters of the aDDM and the attentional biases that they predict.
Table 1 summarizes the maximum a posteriori (MAP) estimates for group-level mean parameters.We found that θ group V = 0 52 .
0 12 0 3 confidence interval or CI 5 5]), which means that the attentional bias parameter in the hidden condition worsened by a factor of two, consistent with the results described above.We also found differences in the estimated parameters for the slope ( ).As shown in Table S6 in the Supplemental Material, the estimates using the model with correlated priors led to very similar conclusions.
The hierarchical model also provides individual parameter estimates for each participant, which are shown in Figure 6.Except for bias, the parameters in the visible condition were larger for most participants.
We estimated θ without the typical bounds at 0 and 1.In the visible condition, none of the 50 participant-level MAP estimates for θ fell below 0 and one fell above 1.In the hidden condition, seven out of 50 fell below 0 and none fell above 1.However, in each of these cases, the 95% highest density intervals include the traditional boundaries.See Figures S5 and S6  shown in Figure S9 in the Supplemental Material, the model with correlated priors leads to very similar individual parameter estimates.

Mechanisms of choice bias
Our results show that attentional biases are approximately twice as large in the hidden condition and that this is accompanied by a change in fixation durations, a change in the key attentional bias parameter θ, and changes in other aDDM parameters.In this section, we use out-of-sample simulations to investigate the extent  to which the attentional biases are driven by changes in the fixation process, changes in θ, or changes in nonattentional model parameters.
The simulations are shown in Figure 7.We started the analysis by comparing the observed and simulated attentional bias in the visible condition.To do this, we simulated 10 data sets for every participant in the outof-sample even trials, using the empirical fixation patterns from the even trials and the aDDM parameters fitted in the odd trials of the visible condition (see the Method section for details).As shown in the top panel, we found a good quantitative match between the observed and simulated data.
In Row 2, we repeated the exercise by changing one component of the simulations at a time.Figure 7b depicts data simulated using the fixations from the visible condition but using the values of θ fitted in the hidden trials.The panel shows that this change by itself generates a good qualitative account of the increased attentional bias in hidden trials.
Figure 7c depicts data simulated using the parameters fitted in the visible condition but using the fixation process from the hidden condition (∆Fix).For clarification, when we use the fixation process from the hidden condition, we mean that we are sampling properties of the fixation process (probability of first fixation to the left, latency to first fixation, first fixation duration, middle fixation duration, saccadic duration) from their empirical distributions across the hidden trials, separately for each participant.All properties of the fixation process are independently sampled once per trial, except for middle fixation durations and saccade durations, which are independently sampled until the driftdiffusion process terminates.We found that this change, by itself, had a negligible impact on the attentional bias and thus cannot account for observed data in the hidden condition.
Figure 7d depicts data simulated using the fixations from the visible condition but using the values of ( , , ) d b σ fitted in the hidden trials.Again, this change, by itself, was unable to provide a good qualitative account of the increased attentional bias in hidden trials.
Row 3 depicts simulations in which two of the components are changed at a time.In Figure 7e, we used the θ parameters and the fixation process from the hidden condition, but the values of the other parameters were taken from the visible condition.In Figure 7f, we used the values of the other parameters ( , , ) d b σ and the fixation process from the hidden condition, but the value of θ was taken from the visible condition.
Finally, the bottom row depicts a simulation in which all parameters as well as fixations were taken from the hidden condition.
A comparison of these plots shows that the model can account for the large differences in attentional choice biases as long as the change in the θ parameter is taken into account, but not otherwise.This shows that the impact on attentional biases is driven mostly by an increase in the tendency to overweight the value of fixated options.
One natural concern with this simulation analysis is that changes in the θ parameter might be correlated with changes in fixation durations, across participants.Figure S10 in the Supplemental Material shows that this is not the case.
For completeness, Figure S11 and Table S8 in the Supplemental Material show that the estimated model parameters were able to qualitatively account for the observed choice biases associated with net fixation time and excess first fixation duration (in the visible condition) in the bottom rows of Figure 5.However, they were unable to account for the observed disappearance of excess first fixation bias in the hidden condition.

Discussion
Our experiment was designed to study the impact of peripheral visual information on the decision algorithm and its performance.Removing the nonfixated option had little impact on the quality of average choices, although it slowed down the choice process by about 32% (or 520 ms).More importantly, we found that attentional choice biases were approximately twice as large when the nonfixated option was not shown.
The conclusion about the relative magnitude of the attentional biases in the two conditions was based on two different sets of analyses.A model-free way of measuring the size of the attentional bias, which does not depend on the assumption that the aDDM is a good description of the data-generating process (Mormann & Russo, 2021), is to ask what is the probability of choosing the last fixated item when decisions have equal value (Fig. 5, top row).In the absence of an attentional bias, both items should be chosen with equal probability.In contrast, the last seen item is 2.5 times more likely than the other item to be chosen when all items are shown simultaneously, and 5 times more likely when nonfixated items are hidden.Another way of measuring the attentional bias is based on the aDDM.In this model, the value of nonfixated options at any given time is downweighted by a parameter θ.When θ = 1, there is no attentional bias.When θ < 1, there is an attentional bias in favor of the fixated item, which is stronger for lower values of θ.Our mean estimates are θ = 0 52 . when all items are shown and θ = 0 29 .when nonfixated options are hidden.In both cases, the results show that removing peripheral visual information doubles the size of the attentional biases.
We found that middle fixations slow down by about 23% and first fixations slow down by about 46% in the hidden condition, independently of the stimuli's value.There are two natural hypotheses for this change.One hypothesis, based on bottom-up control of the fixation process, is that the removal of peripheral stimuli changes Finally, in panel g, the simulated data uses fixations and all parameters from the hidden condition.The figures show that the simulations provide a good qualitative match for the difference between the visible and hidden conditions when the attentional bias parameter is modified, but not otherwise.The simulations include 10 observations per trial, per participant.
the priority map that controls fixation durations and locations (Itti & Koch, 2000;Towal et al., 2013).This is consistent with findings from the visual search literature, which have found that a decrease in the saliency of peripheral stimuli, of which removal is an extreme case, increases fixation durations (Machner et al., 2020), as well as with the finding that fixation durations increase in patients with hemispatial neglect (Machner et al., 2012).An alternative hypothesis, based on top-down control of the fixation process, is that fixations slow down to accommodate the increased difficulty of generating value samples for the nonfixated stimuli in the absence of peripheral visual information.
Beyond showing that attentional choice biases increase substantially when only one item is shown at a time, our findings also provide some novel clues about the mechanisms at work in simple choice.
First, we found that in the absence of peripheral stimuli, the attentional bias parameter (θ) was greater than zero on average, which means that the values of nonfixated items were still being processed by some, even if they were underweighted.This suggests that foveation facilitates the extraction of value samples but that it is not necessary, at least after the second fixation when the identity of both stimuli becomes known.This also implies that covert attention is paid to the nonfixated item, at least after the second fixation.In fact, one interpretation of our results is that removing the nonfixated item reduces the amount of covert attention that it receives (for an outstanding review of the role of covert visual attention, see Carrasco, 2011).
Second, the estimated parameters in the visible condition, and specifically the attentional bias parameter, are consistent with related literature.When fitting the aDDM to individuals in their data set, Krajbich, Armel, and Rangel (2010) found that the average value of attentional bias among participants was θ = ± 0 52 0 3 . ., although their best-fitting model had θ = 0 3 . .Our estimates of attentional bias in both conditions are similar to the estimated influences of gaze on choice in the study by Weilbächer and co-authors (2021), where all options were hidden and had to be recalled from memory at the time of decision-making (attentional discounting parameter: hidden mean = 0.12, visible mean = 0.42).Interestingly, our estimate of θ in the hidden condition is larger than the one in the Weilbächer et al. study, which suggests that the attentional bias is stronger when all information about the choice stimuli has to be recalled from memory (in their study) than when it is available conditional on foveation (as in the hidden condition in this study).
Third, Bayesian models of information sampling in simple choice have proposed that fixations matter because they control which value samples are obtained and that samples matter because they shift the value estimates from a common initial prior to posteriors that are closer to the true value of each stimulus.As a result, the value estimates of better-than-average items tend to increase with additional fixation time, and the opposite is true for worse-than-average items (Armel & Rangel, 2008;Callaway et al., 2021;Jang et al., 2021;Li & Ma, 2021).This Bayesian perspective could account for the increased attentional bias when nonfixated items are hidden.Value samples must be taken in parallel from both choice options, and either the rate of sampling must be slower or the sampled information must be noisier for the nonfixated item.These variations should be even more exaggerated when nonfixated items are not present in peripheral vision.Existing Bayesian models do not account for the former, although they do account for the latter ( Jang et al., 2021).
Finally, our results also have implications for the growing field of choice architecture, which investigates how seemingly minor changes in the choice environment affect decisions and how to apply this information to help individuals make better decisions ( Johnson et al., 2012).We found substantially larger attentional biases in settings where only one option is shown at a time-as is done on many shopping websites-than in settings where all options are presented simultaneously-such as supermarket shelves.This suggests that individuals might be more susceptible to marketing influences that attract attention (e.g., salient packaging or point-of-sale ads) in the growing domain of e-commerce than in traditional retail settings.Although our experiments measured only the effect of removing peripheral stimuli, similar issues could arise in contexts where choice options are described sequentially using other sensory modalities (e.g., when a waiter describes the menu specials).Extrapolating from our results, we also hypothesized that similar increases in attentional biases could be induced simply by increasing the spatial separation between stimuli, so that it becomes difficult to process nonfixated options using peripheral vision.Consistent with this hypothesis, other researchers have found that participants with a narrower spatial attention tend to exhibit larger attentional choice biases than those with broader spatial attention (Smith & Krajbich, 2018).
Several aspects of our study might limit the generalizability of the findings.First, our results are limited to the context of binary choice, whereas in many decision contexts, more than two options are available for selection.The impact of peripheral visual information on the choice process might depend on the complexity of the environment.Second, on the basis of previous work, we used food stimuli as a basis for understanding attentional effects on value-based choices (Krajbich et al., 2010;Krajbich & Rangel, 2011).However, it is possible that the quantitative influence of peripheral information might depend on the nature of the stimuli (e.g., lotteries, toys, concert tickets), especially if it differs on how easily it can be processed in peripheral vision.Third, in the real world, it may be more costly and slower for consumers to switch between different options than it is in our simple gaze-contingent paradigm.For instance, consumers may have to walk between two different shelves at a supermarket or click through a list online, whereas in our paradigm, they simply needed to fixate between two ROIs.Fourth, our participants included young to middle-age adults from Caltech and its surrounding community who self-reported an affinity for snack foods.Further work is needed to investigate the generalizability of our findings to other cultures, ages, or populations with different preferences.

Declaration of Conflicting Interests
The author(s) declared that there were no conflicts of interest with respect to the authorship or the publication of this article.

Fig. 2 .
Fig. 2.An illustration of how the attentional drift-diffusion-model (aDDM) makes decisions in a sample trial.Colored vertical bands denote fixation locations.

Fig. 3 .Fig. 4 .
Fig. 3. Basic psychometrics.The top row shows the probability of choosing the left item as a function of its relative value.The middle row shows reaction time as a function of trial difficulty, as measured by the rating difference between the best and worst items.The bottom row shows the number of fixations as a function of trial difficulty.Columns indicate which data set generated the figures.Error bars show standard errors of the mean across participants.

Fig. 5 .
Fig.5.The top row shows the probability of choosing the left item as a function of its relative value, conditional on last fixation location.The middle row shows the corrected probability of choosing the left item as a function of the net fixation time to the left item.The corrected probability is computed by subtracting from each choice observation (coded as 1 if left chosen, and 0 otherwise) the proportion with which left is chosen at each relative value.The bottom row shows the corrected probability that the first seen item is chosen as a function of the excess first fixation duration, defined as first fixation duration minus mean first fixation duration (computed for each subject).Columns indicate which data set generated the figures.Error bars show standard errors of the mean across participants.

Fig. 6 .
Fig. 6.Comparison of participant-level attentional drift-diffusion model (aDDM) parameter estimates from model with uncorrelated priors.Points denote maximum a posteriori (MAP) estimates in the visible and hidden conditions.Colored lines denote 95% highest density intervals..

Fig. 7 .
Fig. 7. Mechanisms of choice bias.In each panel, we plot the probability of choosing the left item, as a function of its relative value and last fixation location, in observed (blue, red) and simulated (black) data.Each panel differs on the assumptions that were used to simulate the data.In panel a, the simulated data for out-ofsample even trials use the empirical fixation patterns and MAP parameters fitted in the visible condition.In panel b, the simulated data now uses the θ MAP parameter fitted out-of-sample in the hidden condition.In panel c, the simulated data uses the empirical fixation patterns from the hidden condition (ΔFix.).In panel d, the simulated data uses the (d,σ,b) MAP parameters fitted from the hidden condition.In panel e, the simulated data now uses the empirical fixation patterns and the θ MAP parameter from the hidden condition.In panel f, the simulated data uses the empirical fixation patterns and the (d,σ,b) MAP parameters fitted from the hidden condition.Finally, in panel g, the simulated data uses fixations and all parameters from the hidden condition.The figures show that the simulations provide a good qualitative match for the difference between the visible and hidden conditions when the attentional bias parameter is modified, but not otherwise.The simulations include 10 observations per trial, per participant.

Table 1 .
Group-Level Maximum a Posteriori (MAP) Parameter Estimates for Model With Uncorrelated Priors Across Data Sets and Conditions