Introduction

Attention plays a key role in decision making, enabling individuals to focus on relevant information while ignoring distractions. It has also been hypothesized that attention biases decision makers’ preferences and choices. During decisions involving spatially distributed stimuli, gaze progressively shifts toward the option ultimately chosen, a phenomenon termed the gaze cascade effect (Shimojo et al., 2003; Glaholt and Reingold, 2009). Additionally, decision-makers often select the option they fixate on last before reporting their choice (a “last-fixation” bias; Krajbich et al. 2010). Experimental manipulations of gaze, including spatial cues, variations in exposure duration, and salience control, have demonstrated that options receiving more attention are more likely to be chosen (Nittono and Wada, 2009; Bhatnagar and Orquin, 2022; Pleskac et al., 2023). As gaze reflects spatial attention, these findings support the idea that attention biases choice (Shimojo et al., 2003; Armel et al., 2008; Zajonc, 1968).

The attentional drift diffusion model (aDDM) formalizes this hypothesis in a way that makes it suitable to quantitatively explain choice and response time (RT) (Krajbich et al., 2010). The aDDM builds on the drift-diffusion model of decision making (DDM), in which decisions are made by accumulating noisy samples of momentary evidence over time (Ratcliff, 1978). In preference-based decisions, the momentary evidence depends on the subjective value difference between the options. The decision is thought to terminate when the accumulated evidence crosses an upper or lower bound, simultaneously resolving the choice and the time it took to make it. The DDM and some of its variants, such as random walk, race, and attractor models, have been successful in explaining both choice and RT in a range of perceptual and cognitive decisions in which no role for attention is assumed (Link, 1975; Ratcliff and McKoon, 2008; Usher and McClelland, 2001; Vickers, 1979; Wang, 2002).

The aDDM extends the DDM by including a role for attention in the decision process. It proposes that attention can change the subjective value of decision alternatives: specifically, the value of unattended options is discounted by a multiplicative factor. This framework assumes that attention exerts its influence intra-decisionally—that is, during the deliberation process, before a choice is made. Originally developed for preference-based decisions between two options, the aDDM has been extended to multi-alternative (Krajbich and Rangel, 2011; Thomas et al., 2021), perceptual (Smith and Krajbich, 2019; Tavares et al., 2017), purchase (Krajbich et al., 2012), and attribute-based decisions (Fisher, 2021; Yang and Krajbich, 2023). The model provides an explanation for the association between choice, RT, subjective value, and gaze allocation, including the apparent causal influence of gaze on choice (for a review see Krajbich, 2019).

The allocation of attention in the aDDM is exogenous to the decision process—that is, attention shifts between items independently of the internal dynamics of decision making. In contrast, more recent studies model the control of attention as endogenous, arising from an optimization that balances the cost of delaying the decision and collecting additional evidence, with the expected benefit of improved accuracy. For instance, Callaway et al. (2021) formalized the decision process as a partially observable Markov decision process (POMDP), and approximated its solution by assuming that the value of additional deliberation can be expressed as a linear combination of factors such as the expected reward from acquiring an extra evidence sample and the expected reward if the item values were perfectly known. They fit this model jointly to datasets involving two- and three-alternative choices (from Krajbich et al. 2010 and Krajbich and Rangel 2011, respectively), capturing several aspects of the data.

A key assumption in Callaway’s model is that the mean of the prior probability over the value of the items is lower than the true mean. This assumption leads to a tendency to choose the most frequently sampled item, since the bias in the prior affects the least frequently sampled option more strongly. Jang et al. (2021) derived an optimal decision policy based on a similar assumption. Because they focused on two-alternative choices, they were able to solve the Bellman equations by backward induction without resorting to the kind of approximations required by Callaway et al. (2021).

Here we explore two novel behavioral predictions of models that posit a multiplicative effect of attention on value. While past analyses showed an overall-value dependent influence of dwell time on choice (Smith and Krajbich, 2019), our new predictions concern (i) how the link between last fixation and choice depends on the overall value of the alternatives, and (ii) how the difference in dwell time between items varies with the consistency between choices and the initial value ratings. We tested these predictions by reanalyzing data from previously published value-based decision tasks (Krajbich et al., 2010; Smith and Krajbich, 2018; Chen and Krajbich, 2016; Gwinn, 2016; Folke et al., 2016; Sepulveda et al., 2020). Our results reveal systematic deviations from the predictions of both the aDDM and the optimal inference models proposed by Callaway et al. (2021) and Jang et al. (2021). Instead, the data suggest that the association between gaze and choice is partially post-decisional—emerging after a covert commitment to a decision but before the overt response is made (Cavanagh et al., 2014; Westbrook et al., 2020).

Results

In the experiment of Krajbich et al. (2010), 39 hungry participants made decisions about snack items. The experiment consisted of two stages. In the first stage, participants viewed 70 items and rated how much they would like to consume each one on a numerical scale from –10 to 10 (Fig. 1A). In the second stage, participants were presented with pairs of items with non-negative ratings and asked to choose which one they would prefer to consume at the end of the experiment (Fig. 1B). Krajbich et al. (2010) tracked the participants’ gaze as they made the choices and identified the moments when the gaze was directed to the left and right items (Fig. 1C).

Figure 1.

The attentional drift-diffusion model

The decision-making process in the aDDM is governed by the state of a scalar decision variable, x, which takes the value zero at the start of each trial. The decision variable is updated according to

when the decision maker is looking at the left item, and according to

when looking at the right item. r_left and r_right represent the values assigned to the left and right items during the rating phase, respectively. If the decision variable reaches a bound at +1, the decision-making process terminates and the left item is selected, and if it reaches -1, the right item is selected. The parameter d controls the integration speed, θ is a parameter between 0 and 1 that determines how much the value of the unattended item is discounted, and ϵ_t is white Gaussian noise with variance σ². The difference between the aDDM and the standard drift-diffusion model is that gaze modulates the drift rate through the parameter θ.

The last-fixation bias on choice does not increase with overall value

A central feature of the aDDM is the multiplicative interaction between gaze and value. This multiplicative interaction accounts for faster response times when the overall value of the options is high—a phe-nomenon we refer to as the magnitude effect on response times (MERT) (Smith and Krajbich, 2019; Ratcliff et al., 2018). Given that attention has a greater influence when both options are highly valued, the last-fixation bias—the tendency to choose the item that was fixated last—should also be amplified under these conditions. We refer to this novel prediction of the aDDM as the magnitude effect on the last-fixation bias (MELFB).

We illustrate the MELFB prediction through simulations of the aDDM using the best-fitting parameters identified by Krajbich et al. (2010) and Smith and Krajbich (2019). To evaluate the impact of attention on choice, we apply logistic regression with the following model:

where Δr represents the difference in value between the left and right items. I_s is an indicator variable that equals 1 for trials completed by subject s and 0 otherwise. I_LastFix is another indicator variable, set to 1 if the left item was fixated last and 0 if the right item was fixated last. We categorize trials into quintiles of Σr, the sum of the rating assigned to the left and right items; the variable I_g identifies trials belonging to quintile g ∈ {1..5}. The βs are the regression coefficients.

The first set of terms on the right-hand side of the equation (the summation over s) captures the effect of Δr (along with a participant-specific bias) on the probability of choosing the left item. The second set of terms (the summation over g) reflects the influence of attention, measured by whether the left or right item was fixated last, with this effect estimated for each quintile of Σr.

As mentioned, the aDDM predicts that the effect of attention on choice increases with the total value of the items. Fig. 2A (top) shows the regression coefficient β_LastFix,g for the different quintiles of overall value, for simulations of the aDDM model. The dashed black line is not a fit to these data points, but is obtained from a model similar to that of Eq. 3 except that the term associated with the quantiles of overall value is replaced by an interaction between I_LastFix and Σr (Eq. 12). The analysis shows that the influence of the item attended last on choice increases with the overall value of the items (Fig. 2A, top).

Figure 2.

A potentially more intuitive way of showing the effect of Σr as a modulator of the effect of attention on choice is shown in Fig. 2A (bottom). It shows the proportion of trials for which the participants chose the item they looked at last, as a function of the difference in rating between the item looked at last and the other item. The analysis was performed independently for trials with low and high Σr (relative to the median). For the simulations of the aDDM, the tendency to choose the last fixated item is lower when Σr is small than when Σr is large.

We tested these predictions of the aDDM with data from the food-choice task. We fit the logistic regression model (Eq. 3) to the data of Krajbich et al. (2010). No significant effect was observed for the effect of Σr on the strength of the association between gaze and choice (Fig. 2B, top). A similar conclusion is reached by analyzing the choice functions median-split by the Σr: the influence of the last fixation on choice is not stronger when Σr is high (Fig. 2B, bottom). That is, in contrast with model’s prediction, the last-fixation bias does not increase with the overall value of the alternatives.

The difference in dwell time is independent of choice consistency

People tend to look longer at the item they end up choosing than at the one they do not (Krajbich et al., 2010; Krajbich and Rangel, 2011). We refer to the difference in total fixation time between the chosen and unchosen items as ΔDwell. The aDDM predicts how ΔDwell should depend on the consistency of the choice with the item ratings. We define inconsistent choices as those in which the lower-rated item is selected over the higher-rated one. According to the aDDM, if individual dwells are not influenced by item value, then ΔDwell should be greater for inconsistent choices than for consistent ones. This prediction is confirmed in our simulations of the aDDM (Fig. 3A; p < 10⁻⁷, Eq. 13, H₀: β_c = 0).

Figure 3.

This prediction can be understood as follows. If, in a given trial, the decision maker had primarily attended to the higher-rated item, the likelihood of making an inconsistent choice would be lower than if attention had been equally divided between the options, because attention would have increased the extent to which the drift rate favored the higher-rated item. Thus, when a choice is known to be inconsistent, it is more likely that gaze was predominantly directed to the lower-rated item.

This logic also applies to consistent choices, but to a lesser extent. Even if attention is primarily allocated to the lower-valued item during a trial, it is still highly likely that the decision-maker will ultimately choose the higher-valued item. This is because, while attending to the lower-valued item reduces the effective drift rate in favor of the higher-valued item, it does not usually result in a change in the sign of the drift rate. Therefore, knowing that the decision was consistent does not provide as much information about the allocation of attention as knowing that the decision was inconsistent.

A similar logic explains why ΔDwell should depend on response time. The aDDM predicts that ΔDwell increases with RT for inconsistent choices (Fig. 3A). This is because the response time sets an upper limit on ΔDwell, so the longer the RT, the greater the value that ΔDwell can be. In contrast, ΔDwell decreases—and even becomes negative—with RT for consistent choices (Fig. 3A). This is because if the response time is longer than expected—given the values of the items being compared—then it is likely that attention was mostly focused on the lower-valued item. The effect is not as strong for inconsistent choices, because even if attention were primarily focused on the lower-valued item, the drift rate would still usually favor the unselected, higher-valued item.

We tested this prediction on the data from Krajbich et al. (2010). We found no significant differences in ΔDwell for consistent and inconsistent choices (Fig. 3B) (p = 0.12, Eq. 13, H₀ ∶ β_c = 0). This is incompatible with the prediction of the aDDM. This incompatibility is not specific to the aDDM, but as we show next, applies to other instantiations of models that posit a causal influence of attention on choice, including those in which gaze is optimally allocated.

Optimal models of decision making

We asked whether the optimal attentional allocation model of Callaway et al. (2021) would show the same discrepancies with the behavioral data as the aDDM. To this end, we reanalyzed the simulations of the optimal model performed by Callaway and colleagues. Like the aDDM, the model of Callaway et al. (2021) incorrectly predicts a strong effect of overall value on the strength of the association between attention and choice (Fig. 2C). This is clearly counter to what is observed in the data (Fig. 2B).

We also analyzed the association between ΔDwell and choice consistency. The results of this analysis are shown in Fig. 3C. The model of Callaway et al. (2021) predicts that ΔDwell depends on choice consistency, with greater ΔDwell for inconsistent choices. The explanation is the same as for the aDDM: if attention has a causal influence on choice, then for inconsistent choices it is highly likely that attention was directed to the item of lower value for a substantial fraction of the trial. Again, this is in clear contrast to what we observe in the data (Fig. 3B).

We also performed simulations of the ‘optimal’ model of Jang et al. (2021), using the best-fitting parameters reported in their study. The model predicts that ΔDwell changes sign as a function of response time (Fig. 3D), contrary to what is observed in the data (Fig. 3B). For example, at response times of ∼1.5 s, it incorrectly predicts that people will tend to choose the option they looked at the least. The model of Jang et al. (2021) also predicts that the influence of the last fixation on choice depends on Σr (Fig. 2D), contrary to what is observed in the data (Fig. 2B). We conclude that neither the aDDM nor the POMDP-based models account for the aspects of the data analyzed in Figs. 2 and 3.

Is the gaze-choice association purely post-decisional?

The aspects of the data we analyzed in Figs. 2 and 3 are suggestive of a non-multiplicative interaction between attention and value (Cavanagh et al., 2014; Westbrook et al., 2020). We explore a model in which the link between gaze and choice arises only after a choice has been covertly made. Attention has no effect on the choice itself or on the time taken to make the choice. We refer to this model as the Post-Decision-Gaze (PDG) model.

In the PDG model, the decision is made by accumulating momentary evidence over time. The momentary evidence is represented by samples from a Gaussian distribution. The mean of the sampling distribution is a linear function of Δr, such that the drift rate of the drift-diffusion process (μ) is given by:

where κ is a signal-to-noise parameter. Unlike the aDDM, the drift-rate is not modulated by the focus of attention.

Unlike most implementations of the DDM, in the PDG model the variance of the momentary evidence depends on Σr. This assumption is needed to explain why response times depend on both Δr and Σr (Ratcliff et al., 2018). We parameterize the variance as

where γ is fit to the data. The assumption has empirical support in neurobiology (Discussion).

The evidence accumulation process begins after a short sensory delay, τ_s (Fig. 4A). For the purposes of our analysis, we fixed τ_s at 0.3 s for all participants. Note that while neurophysiological studies in monkeys have estimated that τ_s is on the order of 0.2 s (Roitman and Shadlen, 2002; Steinemann et al., 2022), τ_s is likely to be longer in the food-choice task since participants start each trial fixating on a central spot before directing their gaze to one of the choice alternatives (Krajbich et al., 2010).

Figure 4.

The model also includes a non-decision latency, τ_m, between the time when the decision maker commits to a choice, signaled by crossing the decision threshold, and the time when a key is pressed to report the choice (Fig. 4A). τ_m accounts for latencies related to motor preparation and is independent of decision difficulty. The response time is given by the sum of τ_m, τ_s, and the decision time (Fig. 4A).

In the model, attention has no causal effect on the decision process. Therefore, there is a 50% chance that the gaze will be directed to either item at the time a threshold is crossed. The key assumption of the model is that once the decision variable crosses the decision threshold, the gaze is directed to the chosen item. To account for eye-movement related latencies, we assume that directing gaze to the chosen item occurs with a latency of τ_e from the time of bound crossing (Fig. 4A), after which the gaze is held on the chosen item until the response. If the decision maker was already looking at the chosen item after τ_e has elapsed from the time of bound crossing, no additional gaze shift occurs (Fig. 4B).

Because the time it takes to make an eye movement is usually less than the time it takes to complete the manual response, τ_m, the decision maker is more likely to be looking at the chosen item when a key is pressed (Fig. 4A,B). However, due to variability in τ_m and in τ_e, the time at which gaze is directed to the chosen item after choice commitment varies from trial to trial, and may even occur after the key press, as in Fig. 4C. Crucially, in contrast with the aDDM and related models (Krajbich et al., 2010; Thomas et al., 2019; Krajbich and Rangel, 2011), it is the choice that affects gaze allocation, not the other way around.

Fits of the PDG model to the behavioral data

We fit the PDG model to the choice and response time data from Krajbich et al. (2010). Fig. 5A and C show the proportion of trials in which participants chose the item on the right and the average RT as a function of Δr. The PDG model provides a good fit to the choice and RT data.

Figure 5.

The model also accounts for MERT—the tendency to make faster decisions when the items being compared are overall more desirable, even when the value difference between them is the same (Smith and Krajbich, 2019; Sepulveda et al., 2020; Ratcliff et al., 2018). To illustrate this effect in Krajbich’s data, we fit a bell-shaped curve to the relation between RT and Δr, and computed residuals of RT by subtracting from each trial’s RT the expectation given by the best-fitting bell-shaped curve (see Fig. S1 for an illustration of the method). We then analyzed how the residuals of RT correlated with Σr. This correlation was negative, indicating that responses were faster when Σr was higher. That is, the data show a magnitude effect on RT (Fig. 5D). Note that the analysis of RT residuals rather than raw RT is necessary because of the positive correlation between Δr and Σr present in the data.

The DDM has been considered incapable of explaining the magnitude effect on RT because in the most common version of the DDM the drift rate depends only on the difference in value between options and ignores their absolute value, leading to the erroneous prediction that the response time for a given Δr would be independent of Σr. The PDG model captures the magnitude effect because the variance of the momentary evidence is allowed to change with Σr (Eq. 5) (Ratcliff et al., 2018). For the best-fitting model, the variance of the momentary evidence increased with Σr (Table S1). Because higher variance leads to faster responses (Zylberberg et al., 2016), the model displays a magnitude effect on RT, similar to the data (Fig. 5D).

In the PDG model, attention has no causal influence on choice. Yet, it successfully accounts for several features of the observed association between gaze and choice. We compared the probability of choosing the right item—as a function of Δr —on trials in which the last fixation was on the right versus the left item. The model predicts a systematic relation between the last fixation and choice that closely mirrors what is observed in the data (Fig. 5B).

In the PDG model, a gaze bias toward the chosen item can only occur during the non-decision time between choice commitment and report, giving rise to a gaze cascade effect—the observation that the probability of looking at the ultimately chosen item aligned to the response increases gradually over time (Fig. 5E). This gradual increase is explained as the average of step-like events (saccades to the chosen item) that occur at different times with respect to the response time. The variation in timing is due to trial-to-trial variability in non-decision time (captured by the model parameter σ_nd, see Methods) and in τ_e. Since the model was fit to maximize the likelihood of its parameters based on the choice and RT data, without using gaze information, the gaze cascade effect can be considered a prediction of the model.

The PDG model also correctly predicts the two novel behavioral observations shown in Fig. 2 and Fig. 3. In the PDG model, there is no significant change in the strength of the last-fixation bias as a function of the overall value of the alternatives, consistent with the experimental data (Fig. 5F). This is because, in the model, the probability of directing gaze to the covertly chosen item is independent of variables affecting the decision process, like Δr or Σr.

The model also correctly predicts that ΔDwell does not depend on the consistency of the choice. In the model, ΔDwell is often positive because gaze is directed to the covertly chosen item and because τ_e is usually smaller than τ_m. These factors do not depend on the consistency of the choice, resulting in similar values of ΔDwell for consistent and inconsistent choices (p = 0.27, Eq. 13, H₀ ∶ β_c = 0)(Fig. 5G).

The gaze cascade effect continues after the choice report

In the PDG model, the bound crossing may have occurred hundreds of milliseconds before the choice is reported, and the two events are not time-locked due to variability in τ_m and τ_e. The model posits that the gaze cascade arises because shifting gaze to the chosen item takes time, and the likelihood of having completed that shift increases with time elapsed since boundary crossing. By this logic, the model predicts an even further increase in the probability of looking at the chosen item immediately after the choice report.

To test this prediction, we examined the gaze allocation after the choice report. Fig. 6A shows the probability of looking at the chosen item, as a function of time, aligned to the response. The probability of looking at the chosen item continues to increase after the choice report. This can be seen more clearly in Fig. 6B, which shows the probability of looking at the chosen item during the 200 ms immediately before and after the choice report. For most participants, the probability is greater after the choice report (p < 10⁻⁷, Wilcoxon signed-rank test).

Figure 6.

That is, a common process—directing the gaze to the chosen item—may explain the allocation of gaze both immediately before and after the choice report. In contrast, models that posit an exclusively intra-decision effect of attention on choice must then explain the allocation of gaze after the response as a separate process (e.g., directing gaze to the chosen item), or by assuming that individuals continue to evaluate the decision alternatives even after the choice report.

Limitations of the PDG model

Despite the ability of the PDG model to explain many aspects of the data, other aspects are not as well captured. In the PDG model, choice accuracy is unaffected by fixation patterns prior to crossing the decision threshold. That is, whether decision-makers focus more on the higher-value item or distribute their attention evenly between the options, the model’s predictions for choice and RT remain unchanged. This invariance arises from the lack of a causal relation between gaze and the decision-making process. As a result, the PDG model makes no concrete predictions for the gaze pattern before choice commitment.

In simulated data—where dwell durations are independent of the value of the alternatives—the PDG model fails to explain certain experimental findings that demonstrate an association between dwell duration and choice (Krajbich et al., 2010; Callaway et al., 2021). We highlight this limitation with two analyses. The first examines the likelihood of choosing the item on the right as a function of the difference in dwell time between the right and left items. Both empirical data and PDG model simulations reveal a positive association, but this relation is steeper in the data (Fig. 5H). That is, ΔDwell is more predictive of choice in the experimental data than in the model, suggesting that decision-makers may fixate longer on the eventually chosen item even before covert choice commitment.

The PDG model also fails to capture the relation between the duration of the first dwell and choice. Fig. 5I shows the proportion of trials in which decision-makers selected the initially fixated option as a function of the first dwell duration. This relation is steeper in the data than in the model. The PDG model does predict a positive relation between the duration of the first dwell and choice, but this is only because the bound is crossed during the first dwell on some trials. On trials with more than one dwell, the PDG model predicted no significant association between first-dwell duration and choice probability (all p > .05, likelihood-ratio tests; Eq. 14, Fig. 7A). In contrast, the empirical data revealed a significant positive association for shorter dwell-count conditions. Specifically, longer first-dwell durations significantly increased the probability of choosing the first-fixated item in 2-dwell (p = .013) and 3-dwell trials (p = .026), whereas this relationship was not significant for 4-dwell (p = .72) or 5-dwell trials (p = .96; Fig. 7B).

Figure 7.

Additive intra-decision attention also fails to explain the gaze–choice association

Alternative formulations of the aDDM posit that attention modulates value additively, rather than multi-plicatively. We therefore asked whether a model with an additive, within-decision influence of attention can explain the novel empirical observations. In this additive intra-decisional attention model, the decision variable evolves according to

where s = +1 when attention is directed to the left item and s = −1 when it is directed to the right item. In this formulation, the drift rate shifts by +ω or −ω depending on which item is attended, in contrast to the aDDM, where the unattended item’s value is discounted multiplicatively.

We fit both the multiplicative (aDDM) and additive model variants to the data from Krajbich et al. (2010), using identical fitting procedures. Model fitting was performed using the empirically observed sequence and duration of fixations on each trial (as in Li and Ma, 2021). To assess parameter identifiability, we simulated data using the best-fitting parameters and re-fit the models to these simulated datasets. The recovered parameters closely matched the generating values, regardless of whether attention acted additively or multiplicatively (Figs. S3 and S4). Across participants, both models provided comparable fits to the data: the additive and multiplicative models yielded similar BIC values (Fig. S5).

Both the multiplicative and additive variants reproduced the main behavioral patterns, including the psychometric function (Fig. 8A; Fig. 9A), the chronometric function (Fig. 8C; Fig. 9C), and the gaze cascade effect (Fig. 8E; Fig. 9E). However, only the multiplicative model captured the observed depen-dence of response time on overall value (Fig. 8D; Fig. 9D), as has been shown by Smith and Krajbich (2019). Some aspects of the data were less accurately reproduced. In particular, both multiplicative and additive models underestimated the influence of the last fixation on choice (Fig. 8B; Fig. 9B). This discrepancy is larger than that reported by Krajbich et al. (2010). This is explained by the fact that their simulations assumed a uniform distribution of item-value pairs, resulting in a higher proportion of high-value trials than in the empirical dataset. Such a design amplifies value-dependent attentional effects, thereby increasing the apparent influence of the last fixation. As noted by Li and Ma (2021) and Callaway et al. (2021), assumptions about value distributions can substantially affect model behavior. Importantly, this limitation is not specific to our fitting approach: even when simulating the original aDDM using parameters from Krajbich et al. (2010), the model underestimates the effect of the last fixation when empirical value distributions are used (Fig. S7B). It is worth noting that Li and Ma (2021) analyzed the same dataset using a similar framework. In their report, the aDDM appears to reproduce the psychometric function conditioned on the last fixation more accurately, without the underestimation observed here (Fig. 8B). However, this difference arises from an analysis error in Li and Ma (2021), which led to misidentification of the last fixation in approximately 25% of trials (Wei Ji Ma, personal communication with Ariel Zylberberg). Overall, we conclude that neither the additive nor the multiplicative variants of the aDDM fully capture all empirical effects. Most notably, both fail to reproduce the observed association between choice consistency and ΔDwell.

Figure 8.

Figure 9.

Combined intra- and post-decision attention

Neither a purely intra-decision nor a post-decision account of the gaze-choice association fully captures the behavioral data. We therefore considered the possibility that a combination of the two accounts provides a better explanation. Westbrook et al. (2020) proposed that the influence of attention on choice evolves within each trial: from an initial multiplicative influence, to a subsequent additive influence indicative of attention shifting toward the covertly chosen item.

We assessed whether a model incorporating both attentional mechanisms better explains the data, specifically regarding the effect of magnitude on the last fixation bias (MELFB) and the difference in ΔDwell between consistent and inconsistent choices. We combined elements from the aDDM and PDG models. Prior to committing to a choice, decision dynamics follow the aDDM framework—the value of the unattended option is multiplicatively discounted and evidence accumulation terminates upon reaching a threshold at ±B. Following threshold crossing, the gaze shifts to the covertly chosen item after a delay τ_e, as described in the PDG model.

Incorporating these post-decisional gaze shifts into the aDDM (Fig. 5) improves the model’s ability to explain the dependency between the last dwell, overall value, and choice (compare Fig. 10B,H and Fig. 8B,H). However, this hybrid approach still (i) inaccurately predicts that the gaze bias is stronger for inconsistent than for consistent choices (Fig. 10G), and (ii) displays a weaker association between the duration of the first fixation and choice, compared to what is observed in the data (Fig. 10I). From this we conclude that the combination of intra- and post-decision attentional mechanisms does not fully account for the behavioral data.

Figure 10.

Drift-rate variability across trials

None of the models that treat attention as an intra-decisional process were able to account for the difference in gaze bias between choices consistent versus inconsistent with stated preferences (e.g., Fig. 8G). We considered the possibility that inter-trial variability in the drift rate (e.g., Ratcliff and McKoon, 2008) could explain this discrepancy.

To test this, we fit a variant of the aDDM in which additive Gaussian noise corrupts the drift rate on each trial. The behavioral data and model fits are shown in Fig. S6. The model still incorrectly predicts a larger gaze bias for inconsistent choices than for consistent choices (Fig. S6G). The model also failed to capture the relation between first-dwell duration and choice (Fig. S6I).

To assess whether post-decisional processes might improve the model’s match to the data, we in-corporated a post-decision gaze shift mechanism, similar to that in the PDG model. Even with this post-decisional mechanism the model predicts a larger gaze bias for inconsistent than for consistent choices, unlike what is observed in the data (Fig. 11G), and still fails to capture the observed association between first-dwell duration and choice (Fig. 11I).

Figure 11.

We also explored an alternative implementation of inter-trial drift-rate variability, in which the item values—rather than the drift rate—are perturbed by additive Gaussian noise that remains constant within a trial. This approach is intended to capture the possibility that the values reported during the rating phase may differ from those used in the choice phase (c.f. Zylberberg et al., 2024). However, this variant did not provide a better fit to the behavioral data than the previous implementation (Fig. S8B,F-I).

For thoroughness, we also illustrate predictions from the models of Callaway et al. (2021) (Fig. S9) and Jang et al. (2021) (Fig. S10). Direct comparisons between these models should be interpreted cautiously due to variations in fitting methods: Callaway et al. (2021) fit their model simultaneously to two- and three-choice datasets, and Jang et al. (2021) manually selected parameters to approximate Krajbich et al. (2010)’s findings. Despite these methodological differences, it is evident that none of these models fully captures the comprehensive set of behaviors observed in this simple binary-choice task.

Generalization to other food-choice datasets

We examined whether the absence of a difference in dwell time for consistent versus inconsistent choices (Fig. 3) generalizes to other food-choice datasets. We analyzed six publicly available datasets from tasks closely resembling that of Krajbich et al. (2010), reported in Smith and Krajbich (2018), Chen and Krajbich (2016), Gwinn (2016), Folke et al. (2016), and Sepulveda et al. (2020). Across all datasets, we found no significant difference in ΔDwell between choices that were consistent versus inconsistent with participants’ stated preferences (p > 0.12 for all datasets, one-tailed t-test; Eq. 13, Fig. 12), replicating the pattern observed in the data of Krajbich et al. (2010) (Fig. 3B).

Figure 12.

Discussion

We found post-decision signatures in the choice-RT-gaze data, suggesting that the association between gaze and choice, especially late in the trial, is not exclusively formative. Models positing an intra-decision (i.e., formative or constructive) multiplicative effect of attention on value—such as the attentional drift-diffusion model (aDDM)—predict that: (i) the last-fixation bias should be stronger when the items under consideration are overall more desirable, and (ii) ΔDwell—the difference in time spent looking at the chosen versus unchosen item—should be greater for inconsistent than for consistent choices. This is because inconsistent choices, in such models, benefit disproportionately from the attentional amplification of value. The data do not support these predictions (Fig. 2 and Fig. 3).

Instead, these observations are better explained by a post-decision account of the gaze-choice association—that is, one in which gaze shifts to the selected item after a covert commitment to a choice. The PDG model captures many empirical observations, including the last-fixation bias and the gaze cascade effect.

To account for the effect of overall value (Σr) on RT (Smith and Krajbich, 2019), the PDG model relies on the assumption that the variability of value representations increases with the values themselves (Ratcliff et al., 2018). This assumption is supported by neurobiological evidence. In the cortex, more desirable options tend to evoke higher firing rates (Platt and Glimcher, 1999; Padoa-Schioppa and Assad, 2006), and neural signal variance typically scales with its mean (Tolhurst et al., 1981). If the two value representations are independent, the variance of the momentary evidence, Δr, should therefore increase in proportion to Σr. In contrast, the aDDM explains the same relation between overall value and RT more directly—as a consequence of the multiplicative effect of attention on value—and thus does so more parsimoniously (Krajbich et al., 2010). Moreover, empirical results show that higher overall value leads to both faster and more accurate choices, arguing against the notion that sensitivity decreases with value. Instead, participants appear to invest additional effort in high-value decisions, which may offset any increase in variability (Shevlin et al., 2022).

Further, the PDG model underestimates the observed association between both early dwell-time and choice probability, as well as the association between ΔDwell and choice probability. Inspired by Westbrook et al. (2020), we therefore examined a hybrid model in which attention initially exerts a causal influence on the choice process and subsequently reflects the chosen option. This mixed model reintroduced some of the same limitations seen in the aDDM—most notably, its inability to account for the null effect of overall value on the last-fixation bias (MELFB) and the similarity in gaze bias across consistent and inconsistent decisions. We conclude that neither of these models is fully able to account for the data from the food-choice task.

Similar shortcomings to those observed in the aDDM were observed in models that derive attention-choice associations from optimal or near-optimal policies. Specifically, Callaway et al. (2021) and Jang et al. (2021) formalized the decision process as a partially observable Markov decision process (POMDP), in which attention enhances either the quantity (Callaway et al., 2021) or quality (Jang et al., 2021) of evidence about item value. To explain the gaze-choice association, both models assume that priors over item values are miscalibrated, such that decision-makers underestimate the true values. As a result, less-attended items are more biased toward zero, since less-attended items are more influenced by the miscalibrated prior. However, simulations based on these models fail to reproduce key empirical findings, including the association between ΔDwell, decision consistency, and RT (Figs. 2, 3, S9 and S10).

The possibility that the gaze-choice association is partially post-decisional may help reconcile discrep-ancies between studies using free-viewing paradigms and those employing causal manipulations of attention. Common approaches to manipulating attention include limiting exposure duration (Frömer et al., 2022), interrupting trials based on gaze duration (Pärnamets et al., 2015; Newell and Le Pelley, 2018; Tavares et al., 2017; Pleskac et al., 2023), and cueing spatial attention (Störmer and Alvarez, 2016; Gwinn et al., 2019). A meta-analysis of the effects of visual attention on binary consumer choice (Bhatnagar and Orquin, 2022) found that these manipulations typically shift choice probabilities by ∼2–4% from a 50% baseline (cf., Tavares et al., 2017). These small effects contrast sharply with the large effects estimated by fits of the aDDM, which often posit a 70% discount of unattended items (Krajbich et al., 2010). The discrepancy between model predictions and the results of the attention-manipulation studies may arise from attempting to account for post-decision gaze-choice correlations using models that assume that the gaze-choice association is exclusively intra-decisional.

A post-decision gaze-choice association may offer a parsimonious explanation for differences in gaze behavior between tasks in which items are either chosen or rejected. In “choose” tasks, participants select the preferred item; in “reject” tasks, they exclude the less preferred item. Although logically equivalent in binary choice, the gaze is directed more to the preferred item in “choose” tasks and to the non-preferred item in “reject” tasks (Sepulveda et al., 2020; van der Laan et al., 2015; Mitsuda and Glaholt, 2014; Nittono and Wada, 2009). This difference has been explained by a variant of the aDDM in which attention modulates the integration of goal-relevant evidence (Sepulveda et al., 2020). Without ruling out this possibility, our results suggest that the difference in gaze allocation between “choose” and “reject” tasks arises because toward the end of the trial the gaze is directed to the selected option, regardless of whether it is to be accepted or rejected.

In this work, we focused on a simple yet widely used class of models in which decisions are based on comparing noisy value signals assigned to each item, and where explicit ratings are assumed to reflect the true underlying value of those items. An alternative class of models proposes that decisions arise from comparisons along individual feature dimensions, rather than at the level of the item as a whole (Tversky, 1972; Roe et al., 2001; Usher and McClelland, 2004; Busemeyer and Townsend, 1993; Summerfield and Tsetsos, 2015; Shadlen and Shohamy, 2016; Lichtenstein and Slovic, 2006; Johnson et al., 2007; Lee and Hare, 2023; Yang and Krajbich, 2023; Fisher, 2021). In food choice tasks, for example, relevant features might include expected satiety, caloric content, tastiness, saltiness, and so on (Rangel and Hare, 2010; Sullivan et al., 2015; Rramani et al., 2020; Suzuki et al., 2017). Attention is thought to fluctuate across these features, updating a decision variable for each alternative as different dimensions are sampled. Which dimensions are evaluated—and the weight assigned to each—may depend on the items being compared, past experience, or the broader decision context (Noguchi and Stewart, 2018; Roe et al., 2001; Juechems and Summerfield, 2019; Zylberberg et al., 2024; Lee and Pezzulo, 2022; Trueblood et al., 2014; Bhatia, 2013). As a result, the desirability of an item during the decision process may differ from that reported during the rating phase. A choice may appear inconsistent relative to initial ratings, but not relative to the specific features that were actually attended during deliberation. Similarly, if the ratings themselves are noisy and do not perfectly reflect subjective value, then some choices labeled as inconsistent may, in fact, be consistent with the decision-maker’s true preferences. Incorporating noise into the rating process would blur the distinction between consistent and inconsistent choices, potentially improving the alignment between model and data. These speculative ideas remain to be tested in future work.

Overall, our findings suggest that the association between attention and choice is not fully captured by models in which attention plays a purely causal role during decision formation. The data suggest an additional post-decision association, in which attention reflects, rather than shapes, the covert choice. While this does not preclude a formative influence of attention at earlier stages, it highlights the importance of considering the temporal dynamics of commitment when interpreting gaze patterns.

Methods

Food-choice task

We reanalyzed the data from the experiment reported in Krajbich et al. (2010). The task was completed by 39 participants. The experiment consisted of two phases. In the first (‘rating’) phase, participants were asked to indicate how much they would like to consume each item after the experiment. This was done using an on-screen slider bar (scale -10 to 10). 70 items were rated in this phase. In the next (‘choice’) phase, participants were presented with pairs of previously rated items and had to choose which one they would prefer to eat at the end of the experiment. Items that received a negative rating were excluded from the choice phase. Each item could be repeated up to six times. Choice trials were created such that the difference in value between items was less than or equal to five (except for five participants for whom this condition could not be met, see Krajbich et al. 2010). Choices were made by pressing the left or right arrow keys on the keyboard. Prior to the presentation of the pair of items, participants had to maintain fixation on a central spot for two seconds. After participants indicated their choice, a box was drawn around the chosen item, which remained visible for one second. Each participant completed 100 trials. Details of the experiment can be found in the original publication.

PDG model

In the PDG model, choice and decision time are determined by the state of a scalar decision variable, x. The decision variable is the cumulative sum of samples from a normal distribution with mean μdt and variance σ²dt, and thus its evolution is described by the stochastic differential equation,

where W is the standard Wiener process and x(t = 0) = 0. The drift rate μ is given by

where r_left and r_right are the ratings assigned to the items presented on the left and right of the screen. The variance of the momentary evidence,σ², scales linearly with the sum of the ratings (Eq. 5).

The accumulation process ends when the decision variable, x(t), reaches one of two bounds positioned symmetrically around zero, at ±B. The left (right) item is chosen when the decision variable reaches the upper (lower) bound. This first passage time establishes the decision time. The RT is the sum of the decision time plus the non-decision latencies, τ_nd. We assume that τ_nd is Normally distributed with mean μ_nd and standard deviation σ_nd.

The accumulation process starts after a short sensory delay, τ_s (Fig. 4), which is part of the non-decision latency. We used a fixed value of τ_s = 0.3 s for all participants. The remaining non-decision time (τ_m = τ_nd −τ_s) is assigned to the time between crossing a decision bound and reporting the choice.

After crossing a bound, gaze is directed to the selected item. The time taken to switch gaze to the chosen item has an associated non-decision latency of τ_e. This latency is assumed to follow a truncated Normal distribution with a mean of μ_e = 0.35 seconds and a standard deviation σ_e of one-third of the mean, truncated to ensure non-negative values. Since τ_e is usually smaller than τ_m, the gaze is informative about the choice.

We fit the model parameters λ = {κ, B, γ, μ_nd, σ_nd} using maximum likelihood to the choice and RT data from each trial. Fits were performed independently for each participant. The log-likelihood of the parameters is given by:

where the summation is over trials. The joint probability of choice and decision time was obtained by numerically solving the Fokker-Planck equation associated with the drift-diffusion process using the Chang-Cooper fully implicit method (Chang and Cooper, 1970; Zylberberg et al., 2016; Kiani and Shadlen, 2009). To obtain the joint probability distribution over choice and RT, we convolve the probability distribution of decision times with the distribution of non-decision times, given by a truncated normal distribution with parameters μ_nd and σ_nd (the truncation constrains the non-decision times to be positive).

Attentional drift-diffusion model

We simulated the aDDM (Eqs. 1 and 2) using the best-fitting parameters reported by Krajbich et al. (2010): d = 0.0002 ms⁻¹, σ = 0.02, θ = 0.3 and bounds B = ±1. The response time was calculated as the sum of the decision time from the drift-diffusion process and a fixed non-decision time of t_nd = 0.355 s, as in Smith and Krajbich (2019).

In the aDDM (as well as in the DDM), it is necessary to set one of the parameters d, σ, or B to a fixed value for the model parameters to be identifiable. In the original description of the aDDM (Krajbich et al., 2010), the upper and lower bounds were set to a fixed value of ±1. For consistency with the PDG model, we reformulated the aDDM so that the variance of the noise accumulated during one second of unbounded accumulation is equal to 1. Then the decision variable of the aDDM, x(t), evolves as in Eq. 7 with σ = 1. Time is discretized in steps of dt = 0.001 s. The drift rate μ is equal to κ(r_right − θr_left) when looking at the item on the right, and to κ(θr_right − r_left) when looking at the item on the left, where κ is the signal-to-noise. A model equivalent to the original aDDM is obtained with the following parameters:

Simulating the aDDM requires modeling how attention alternates between the two items. We fit the empirically-observed dwell durations with log-normal distributions. Separate fits were conducted for the first and middle dwells. Middle dwells are those that were neither the first nor the last of the stimulus-viewing epoch. The fits provide a good match to the experimental data (Fig. S2). Each simulated trial of the aDDM begins by sampling from the first dwell distribution, followed by sampling from the middle dwell distribution until a decision threshold is reached. Similar to the experimental data, the first fixation has a 0.74 probability of being directed to the left item, and attention alternates between the two items thereafter. We simulated the same trials (same value pairs) as those completed by the participants, repeating each trial 10 times.

We fit three variants of the aDDM to individual participant data. In the first variant, attention has an additive effect on value, rather than a multiplicative effect (Fig. 9; see equations for the drift rate in the main text). In the second variant (Fig. S6), the drift rate exhibits inter-trial variability. Specifically, on each trial, the drift rate μ is given by κ(r_left − θr_right) + 𝜈 when fixating the left item, and by κ(θr_left − r_right) + 𝜈 when fixating the right item. Here, 𝜈 is trial-specific noise drawn from a normal distribution with mean 0 and standard deviation σ_d, which is estimated from the data. Importantly, 𝜈 is constant within a trial but varies across trials.

The third aDDM variant can be interpreted as alternative implementation of the one with inter-trial drift-rate variability. The noise does not directly affect the drift rate, but the items’ value. Specifically, the drift rate μ is given by κ((r_left + 𝜈₁) − θ(r_right + 𝜈₂)) when fixating the left item, and by κ(θ(r_left + 𝜈₁) − (r_right + 𝜈₂)) when fixating the right item. Here, 𝜈_x represents trial-specific noise, drawn from a normal distribution with mean 0 and standard deviation σ_d. This formulation reflects the idea that the value reported in the rating phase may deviate from the item’s “true” underlying value (c.f., Zylberberg et al. (2024); Polania et al. (2019)).

Additionally, we simulated the aDDM using the best-fitting parameters reported in previous studies by Krajbich et al. (2010) and Smith and Krajbich (2019) (Fig. S7).

The core parameters of the aDDM are λ = {κ, B, θ}, corresponding respectively to the signal-to-noise ratio, the decision bound height, and the value-scaling factor applied to the unattended option. In the model with inter-trial drift-rate variability, σ_d is added to capture the standard deviation of the drift-rate noise. The models are fit to maximize the likelihood of the parameters given the choice and decision time (DT), and given the sequence and duration of the dwells observed on each trial:

where D⁽ⁱ⁾ is the sequence and duration of dwells observed on trial i, and DT⁽ⁱ⁾ is the decision time on trial i that is assumed to be equal to the sum of the dwells on either of the two items. The joint probability of choice and decision time was computed via numerical approximations of the corresponding Fokker-Planck (FP) equation. Unlike the special case where θ = 1, here the drift rate varies with the focus of attention. As a result, solving the FP equation becomes more computationally demanding, since the probability density of the decision variable must be computed separately for each trial, given the trial-specific stochastic fluctuations in gaze. Numerical solutions were obtained using a fully implicit method (Chang and Cooper, 1970), propagating the probability density of the decision variable over time adjusting the drift rate depending on the focus of gaze.

For the model with inter-trial variability in the drift-rate, we discretized the distribution of drift pertur-bations into a finite number of bins. Specifically, we drew n_bins=11 quantile-based samples from a zero-mean Gaussian distribution with standard deviation σ_drift, using the midpoint of each quantile as a representative value. We numerically solved the Fokker–Planck equation independently for each of these bins. The final probability distribution over choice and decision times was computed as the average (uniform-weighted) across the solutions for each bin, effectively marginalizing over the distribution of drift perturbations.

We use the best-fitting parameters to simulate the aDDM independently for each participant. From the simulations we obtain a choice and decision time for each trial. Response times are estimated as the decision time plus mean non decision time, μ_nd, defined as the trial-average RT of each participant minus the trial-average decision time obtained from the model simulations.

Model simulations

Simulations of the PDG model (Fig. 5) and aDDM (Figs. 8 to 10) were made using the same trials (same value pairs) that participants completed, with each trial repeated 10 times. Dwell durations were randomly sampled from log-Normal distributions fit to the duration of the dwells (Fig. S2). First and subsequent dwells were fit separately. The probability of sampling the left item first was set to 0.74 to match the value obtained from the experimental data.

Combined aDDM-PDG model

The combined aDDM-PDG model (Fig. 10) builds on the aDDM fit to single-participant data. We add to the data simulated with the aDDM (Fig. 8) two aspects of the PDG model: (i) a sensory delay between the onset of the food items and the start of the evidence accumulation process of τ_s, and (ii) another delay τ_e between the time at which a bound is crossed and the time at which the gaze is directed to the chosen item (truncated such that τ_e is non-negative). Parameters τ_s, μ_e and σ_e were set to 0.25s, 0.2s and 0.05s respectively. The same approach and parameter values were used in the aDDM model with inter-trial variability in the drift rate (Fig. 11).

Model fitting

Parameter optimization was performed using the Bayesian Adaptive Direct Search (BADS) method (Acerbi and Ma, 2017). Table S1, Table S2, Table S3 and Table S4 show the best-fitting parameters for the PDG model, the aDDM, the model with intra-decisional additive attention, and the aDDM with inter-trial drift-rate variability, respectively.

Optimal decision models

The optimal model of Jang et al. (2021) has 5 free parameters: the cost of switching attention between items (c_s), the cost, per second, of accumulating evidence (c), the variance of the evidence sampling distribution (σ_X²), the variance of the prior distribution (σ_Z²), and the relative information gain for attended vs. unattended items (κ). See Jang et al. (2021) for a detailed explanation of the model. We simulated 1280 trials per participant with the parameters that Jang et al. (2021) reported best replicated the human behavioral data: c_s = 0.0065, c = 0.23, σ_x² = 27, σ_z² = 18, κ = 0.004. As in Jang et al., the prior mean over the items’ values, z̄, was set to zero, which is lower than the true mean value of the items. The model requires this feature to produce a gaze bias (Jang et al., 2021).

We also analyzed the simulations of the optimal model developed by Callaway et al. (2021). The model has five free parameters: the standard deviation of the evidence sampling distribution, the cost of obtaining a sample, the cost of switching attention between items, the degree to which the prior over the item values is biased toward zero, and a ‘temperature’ parameter of a Boltzmann distribution, which controls the degree of stochasticity in the selection of the optimal policy. The simulations (N = 4,550,400 trials) with the parameters that best fit the human behavioral data were kindly provided by Frederick Callaway and are available at https://github.com/fredcallaway/optimal-fixations-simple-choice. A detailed explanation of the model and the fitting procedure can be found in the original publication (Callaway et al., 2021).

Data analysis

The dashed lines in Fig. 2 (top) were derived from the following logistic regression model:

I_LastFix takes a value of 1 if the left item was fixated on last and 0 if the right item was fixated last. The rightmost term captures the interaction of I_LastFix with Σr; the associated β is the slope of the dashed line in Fig. 2 (top).

We fit the following linear regression model to test for an association between choice consistency (c) and the difference in looking time between the chosen and unchosen items (ΔDwell):

where N is the number of participants, I_i is an indicator variable that takes the value 1 if the trial was completed by subject i and 0 otherwise, and c is equal to 1 for trials in which the higher-rated item was chosen, defined as a consistent choice, and and 0 for trials in which the lower-rated item was chosen, defined as an inconsistent choice. We used a one-tailed t -test to test whether β_c is negative, that is, whether ΔDwell is larger for inconsistent than for consistent choices. When the use of a one-tailed test is not explicitly mentioned, we used two-tailed t -tests to test whether β_c differed significantly from zero.

To test the hypothesis that there was no difference in the probability of looking at the chosen item before and after the choice report (Fig. 6), we used a Wilcoxon signed-rank test. We define two time epochs, one from -200 ms to 0 ms, and the other from 0 ms to 200 ms, relative to the time of the choice report. For each time epoch, we calculated the time each participant spent looking at the chosen item and divided it by the time spent looking at either item. We obtain two proportions per participant (Fig. 6B), which we subjected to a two-tailed Wilcoxon signed-rank test.

For the plots showing the probability of looking at the chosen item aligned to stimulus onset (e.g., Fig. 5E, left), we eliminated the gaze information from the 500 ms prior to the choice report. This step was taken to eliminate,from the stimulus-aligned plots, gaze effects that might be related to the response.

For the plots showing the psychometric function split by whether the last fixated item was the one on the left or right (e.g., Fig. 5B), we classified trials as ‘left item fixated on last’ and ‘right item fixated on last’ depending on which item was being looked at at the time of the choice report. We excluded trials in which the participant either was not fixating on one of the two relevant items at the time of the choice report or the direction of gaze could not be resolved (e.g., eye blinks). We repeated the analyses reclassifying trials according to which item was last looked at before the choice report regardless of when it occurred during the trial, and obtained nearly identical results.

To test the association between first-dwell duration and the probability of choosing the option that was fixated first (Fig. 7), we fit the following logistic regression model, separately for each total–dwell–count condition (2–5 total dwells):

where p_match is the probability of choosing the item fixated first, D is the standardized first-dwell duration, and r_first and r_other represent the value ratings of the first-fixated and alternative items, respectively. Statistical significance of the first-dwell-duration coefficient was assessed using a likelihood-ratio test, comparing the full model to a reduced model that excluded the β₁ term.

To illustrate the gaze cascade effect (e.g., Fig. 5E), we filled a matrix of dimensions NumberOfTrials × NumberOfTimeSteps, with a 1 when participants were looking at the right item, a 0 when they were looking at the left item, and a NaN otherwise. The time step was 1 millisecond. We averaged this matrix across trials, ignoring the NaNs, first within participants and then across participants. For the RT aligned plots, we followed the same procedure after aligning each trial to the response time. For the aDDM and and Callaway’s optimal model, we followed a slightly different procedure, because in these models the drift-rate is undefined when participants are not looking at either of the two items. Therefore, we removed from each trial the times when the participant’s gaze was not directed to either snack item, and aligned the responses to the total fixation time rather than to the response time.

Supplemental information

Figure S1.

Figure S2.

Figure S3.

Figure S4.

Figure S5.

Figure S6.

Figure S7.

Figure S8.

Figure S9.

Figure S10.

Table S1.

Table S2.

Table S3.

Table S4.

Data availability

The analysis code will be made available upon publication

Acknowledgements

We are grateful to Fred Callaway and Yaniv Abir for helpful comments on an earlier version of the manuscript, to Fred Callaway for kindly sharing the simulations of his model, and to Antonio Rangel for sharing data and for helpful and stimulating discussions.

Additional information

Author contributions

AZ conceived, designed and conducted the research; AZ wrote the original draft of the manuscript; All authors revised the manuscript.

Funding

This work was supported by the National Institutes of Health (R01NS113113 to M.N.S.), the Air Force Office of Scientific Research (FA9550-22-1-0337 to M.N.S), the Howard Hughes Medical Institute (M.N.S.) and the National Science Foundation (grant 2333979 to I.K.).

Funding

Howard Hughes Medical Institute (HHMI)

• Michael N Shadlen