Behavioral signatures of post-decisional attention in preferential choice

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).

Here, we examine two behavioral predictions of models that posit a multiplicative influence of attention on subjective value. These predictions concern how gaze allocation (i.e., time spent viewing each item) relates to the overall value of the alternatives and to whether choices are consistent with stated preferences. One prediction of the aDDM and related models is that gaze allocation at the end of the decision should be more predictive of the choice when the overall value of the alternatives is higher (Ting and Gluth, 2025). A second prediction is that the difference in dwell time between items should depend strongly on choice consistency—specifically, that the lower-valued item should be looked at for longer for choices that are inconsistent with stated preferences. We tested the predictions by reanalyzing data from several previously published value-based decision tasks (Krajbich et al., 2010; Smith and Krajbich, 2018; Chen and Krajbich, 2016; Gwinn and Krajbich, 2016; Folke et al., 2016; Sepulveda et al., 2020). Contrary to these predictions, the data show systematic deviations from the aDDM and related frameworks (Callaway et al., 2021; Jang et al., 2021). Instead, the results suggest that the association between gaze and choice is at least partly post-decisional, arising after a covert commitment to a decision but before the overt response is executed (Cavanagh et al., 2014; Westbrook et al., 2020).

We analyzed data from previously published studies that used the food-choice paradigm introduced by Krajbich et al., 2010. In this paradigm, hungry participants performed two stages. In the first stage, participants viewed snack food items one at a time and rated how much they would like to consume each one (Figure 1A); most of the studies we reanalyze used a numerical liking scale (Krajbich et al., 2010; Smith and Krajbich, 2018; Chen and Krajbich, 2016; Gwinn and Krajbich, 2016), while others elicited subjective value through an incentive-compatible Becker–DeGroot–Marschak auction (Folke et al., 2016; Sepulveda et al., 2020). In the second stage, participants were presented with pairs of items and asked to choose which one they would prefer to consume at the end of the experiment (Figure 1B). Throughout the choice stage, participants’ gaze was recorded, allowing identification of the moments when attention was directed to the left versus right item (Figure 1C).

Figure 1

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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 phenomenon 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. A similar observation has recently been made by Ting and Gluth, 2025. We refer to this 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:

(3) ${\displaystyle {\displaystyle {\displaystyle \text{logit}[p_{\text{left}}]=\sum _s\left({\beta }_{\mathrm{\Delta }r,s}\mathrm{\Delta }r+{\beta }_{0,s}\right)I_s+\sum _g{\beta }_{\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{F}\mathrm{i}\mathrm{x},g}{I}_g{I}_{\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{F}\mathrm{i}\mathrm{x}},}}}$

where $\mathrm{\Delta }r$ 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. $I_{\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{F}\mathrm{i}\mathrm{x}}$ 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 $\mathrm{\Sigma }r$, the sum of the rating assigned to the left and right items; the variable $I_g$ identifies trials belonging to quintile $g\in \{1\dots 5\}$. 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 $\mathrm{\Delta }r$ (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 $\mathrm{\Sigma }r$.

Figure 2A shows the regression coefficient ${\beta }_{\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{F}\mathrm{i}\mathrm{x},g}$ for the different quintiles of overall value, for the 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 Equation 3 except that the term associated with the quantiles of overall value is replaced by an interaction between $I_{\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{F}\mathrm{i}\mathrm{x}}$ and $\mathrm{\Sigma }r$ (Equation 12). The analysis shows that the influence of the item attended last on choice increases with the overall value of the items (Figure 2A).

Figure 2

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Testing the predictions in data from the food-choice task

We tested these predictions of the aDDM with data from the food-choice task. To test the prediction regarding the MELFB, we fit the logistic regression model (Equation 3) to the data of Krajbich et al., 2010 and other datasets of the food-choice task (Smith and Krajbich, 2018; Chen and Krajbich, 2016; Gwinn and Krajbich, 2016). In contrast with the prediction of the aDDM, the estimated strength of the association between last-dwell focus and choice was either non-significant or significantly negative (Figure 3; p-values and βs indicated in the figure). A similar result was recently reported by Ting and Gluth, 2025. That is, in contrast with the model’s prediction, the last-fixation bias does not increase with the overall value of the alternatives.

Figure 3

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We also tested the prediction regarding the difference in ΔDwell between consistent and inconsistent choices. Across several datasets, we found no significant differences in ΔDwell for consistent and inconsistent choices (Figure 4; Equation 13, $H_0:{\beta }_c=0$; $p>0.13$ for every dataset). This is incompatible with the prediction of the aDDM. This incompatibility is not specific to the aDDM, but as we show below, applies to other instantiations of models, including those in which gaze is optimally allocated. (Note that some of these datasets were excluded from Figure 3 because last-dwell focus was unavailable in the public data.)

Figure 4

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Is the gaze–choice association purely post-decisional?

The aspects of the data we analyzed in Figures 3 and 4 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 $\mathrm{\Delta }r$, such that the drift rate of the drift-diffusion process (µ) is given by

(4) ${\displaystyle {\displaystyle \mu =\kappa (r_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}-r_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}),}}$

where $\kappa$ 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 $\mathrm{\Sigma }r$. This assumption is needed to explain why response times depend on both $\mathrm{\Delta }r$ and $\mathrm{\Sigma }r$ (Ratcliff et al., 2018). We parameterize the variance as

(5) ${\sigma }^2=1+\gamma \left(r_{\mathrm{left}}+r_{\mathrm{right}}\right),$

where $\gamma$ is fit to the data. The assumption has empirical support in neurobiology (‘Discussion’).

The evidence accumulation process begins after a short sensory delay, ${\tau }_{\mathrm{s}}$ (Figure 5A). For the purposes of our analysis, we fixed ${\tau }_{\mathrm{s}}$ at 0.3 s for all participants. Note that while neurophysiological studies in monkeys have estimated that ${\tau }_{\mathrm{s}}$ is on the order of 0.2 s (Roitman and Shadlen, 2002; Steinemann et al., 2022), ${\tau }_{\mathrm{s}}$ 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 5

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The model also includes a non-decision latency, ${\tau }_{\mathrm{m}}$, 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 (Figure 5A). ${\tau }_{\mathrm{m}}$ accounts for latencies related to motor preparation and is independent of decision difficulty. The response time is given by the sum of ${\tau }_{\mathrm{m}}$, ${\tau }_{\mathrm{s}}$, and the decision time (Figure 5A).

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 ${\tau }_{\mathrm{e}}$ from the time of bound crossing (Figure 5A), 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 ${\tau }_{\mathrm{e}}$ has elapsed from the time of bound crossing, no additional gaze shift occurs (Figure 5B).

Because the time it takes to make an eye movement is usually less than the time it takes to complete the manual response, ${\tau }_{\mathrm{m}}$, the decision-maker is more likely to be looking at the chosen item when a key is pressed (Figure 5A and B). However, due to variability in ${\tau }_{\mathrm{m}}$ and in ${\tau }_{\mathrm{e}}$, 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 Figure 5C. 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. Figure 6A and B show the proportion of trials in which participants chose the item on the left and the average RT as a function of $\mathrm{\Delta }r$. The PDG model provides a good fit to the choice and RT data.

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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 $\mathrm{\Delta }r$, and computed residuals of RT by subtracting from each trial’s RT the expectation given by the best-fitting bell-shaped curve (see Figure 6—figure supplement 1 for an illustration of the method). We then analyzed how the residuals of RT correlated with $\mathrm{\Sigma }r$. This correlation was negative, indicating that responses were faster when $\mathrm{\Sigma }r$ was higher. That is, the data show a magnitude effect on RT (Figure 6C). Note that the analysis of RT residuals rather than raw RT is necessary because of the positive correlation between $\mathrm{\Delta }r$ and $\mathrm{\Sigma }r$ 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 $\mathrm{\Delta }r$ would be independent of $\mathrm{\Sigma }r$. The PDG model captures the magnitude effect because the variance of the momentary evidence is allowed to change with $\mathrm{\Sigma }r$ (Equation 5; Ratcliff et al., 2018). For the best-fitting model, the variance of the momentary evidence increased with $\mathrm{\Sigma }r$ (Appendix 1—table 1). Because higher variance leads to faster responses (Zylberberg et al., 2016), the model displays a magnitude effect on RT, similar to the data (Figure 6C).

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 $\mathrm{\Delta }r$—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 (Figure 6D).

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 (Figure 6E). 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 ${\sigma }_{\mathrm{n}\mathrm{d}}$, see ‘Methods’) and in ${\tau }_{\mathrm{e}}$. 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 Figures 3 and 4. 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 (Figure 6F). 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 $\mathrm{\Delta }r$ or $\mathrm{\Sigma }r$.

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 ${\tau }_{\mathrm{e}}$ is usually smaller than ${\tau }_{\mathrm{m}}$. 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, Equation 13, $H_0:{\beta }_c=0$; Figure 6G).

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 ${\tau }_{\mathrm{m}}$ and ${\tau }_{\mathrm{e}}$. 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. Figure 7A 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 Figure 7B, 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 ($\mathrm{p}<{10}^{-7}$, Wilcoxon signed-rank test).

Figure 7

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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.

Parameter-sensitivity analysis

In the PDG model, the gaze–choice association arises from the temporal gap between manual and oculomotor response latencies, that is, from the difference ${\tau }_{\mathrm{m}}$- ${\tau }_{\mathrm{e}}$. In our simulations, ${\tau }_{\mathrm{e}}$ is modeled as a normal distribution with mean ${\mu }_e=0.35$ s and standard deviation ${\sigma }_e={\mu }_e/3$, truncated at zero to exclude negative values. To assess the sensitivity of the model to this assumption, we varied ${\mu }_e$ across a range and recomputed the key behavioral signatures. Figure 8 shows the resulting gaze-cascade effect, the association between last-fixation focus and choice, and the interaction between ΔDwell and consistency, paralleling the analyses reported in Figure 6E–G.

Figure 8

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As ${\mu }_e$ increases, the probability that gaze is already directed toward the chosen item at the time of response decreases (Figure 8A). Because both ${\tau }_{\mathrm{e}}$ and ${\tau }_{\mathrm{m}}$ are variable, longer mean eye-movement latencies reduce the likelihood that the eye movement is completed before the manual response. Accordingly, the predicted association between gaze and choice weakens as ${\mu }_e$ increases (Figure 8B). The magnitude of ΔDwell also declines with increasing ${\mu }_e$ (Figure 8C). However, for all values of ${\mu }_e$, ΔDwell remains positive and does not differentiate between consistent and inconsistent choices. Thus, the qualitative predictions of the PDG model are robust to the precise value assumed for ${\tau }_{\mathrm{e}}$. In the ‘Discussion’, we explain why the comparatively large best-fitting values of ${\tau }_{\mathrm{e}}$ do not necessarily conflict with shorter post-decisional eye-movement latencies reported in neurophysiological studies of monkeys (Roitman and Shadlen, 2002).

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 left as a function of the difference in dwell time between the left and right items. Both empirical data and PDG model simulations reveal a positive association, but this relation is steeper in the data (Figure 6H). 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. Figure 6I 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>0.05, likelihood-ratio tests; Equation 14, Figure 9A). 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 two-dwell (p=0.013) and three-dwell trials (p=0.026), whereas this relationship was not significant for four-dwell (p=0.72) or five-dwell trials (p=0.96; Figure 9B).

Figure 9

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Predictions of other decision-making models

For comparison with the PDG model, we analyze different decision-making models fit to the data of Krajbich et al., 2010. These models include variants of the aDDM (additive attention, inter-trial drift-rate variability, and post-decisional latencies) as well as pseudo-optimal models of decision-making.

Additive intra-decision attention

We fit an additive variant of the aDDM to the data from Krajbich et al., 2010, using identical fitting procedures. In this additive intra-decisional attention model, the decision variable evolves according to

(6) $x_t=x_{t-1}+d(r_{\mathrm{left}}-r_{\mathrm{right}}+s\omega )+{ϵ}_t,$

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 $+\omega$ or $-\omega$ depending on which item is attended, in contrast to the aDDM, where the unattended item’s value is discounted multiplicatively.

The additive model, just as the multiplicative variant (Figure 10A), fails to capture qualitative aspects of the data, most notably the interaction between choice consistency and ΔDwell (Figure 10B).

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Combined intra- and post-decision attention

Because neither a purely intra-decision nor a post-decision account of the gaze–choice association fully captures the behavioral data, we considered the possibility that a combination of the two accounts provides a better explanation (Westbrook et al., 2020). To this end, we combined elements from the aDDM and PDG models. In this variant of the model, 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 $\pm B$. Following threshold crossing, the gaze shifts to the covertly chosen item after a delay ${\tau }_{\mathrm{e}}$, as described in the PDG model.

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

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. 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 Figure 10D. The model still incorrectly predicts a larger gaze bias for inconsistent choices than for consistent choices. It also failed to capture the relation between first-dwell duration and choice.

To assess whether post-decisional processes might improve the model’s match to the data, we incorporated 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, and still fails to capture the observed association between first-dwell duration and choice (Figure 10E).

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 (Zylberberg et al., 2024). However, this variant did not provide a better fit to the behavioral data than the previous implementation (Figure 10—figure supplement 1).

Optimal gaze allocation

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 (Callaway et al., 2021; Jang et al., 2021; Song et al., 2019; Hébert and Woodford, 2018; Gluth et al., 2026; Gluth et al., 2020; Zhu, 2022). 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.

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 (Figure 10F). This is clearly counter to what is observed in the data (Figure 3).

We also analyzed the association between ΔDwell and choice consistency (Figure 10F). 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 (Figure 4).

We also performed simulations of the ‘optimal’ model of Jang et al., 2021, using the best-fitting parameters reported in their study (Figure 10G). The model predicts that ΔDwell changes sign as a function of response time, contrary to what is observed in the data. 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 $\mathrm{\Sigma }r$, unlike what we observe in the data. We conclude that neither the aDDM nor the POMDP-based models account for the aspects of the data analyzed in Figures 3 and 4.

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 (Figures 3 and 4).

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. We argue that directing gaze to the chosen item after a covert choice commitment is sensible, as the benefits of attending to a stimulus do not end with the decision itself. In naturalistic settings, for instance, selecting a food item is typically followed by the action of reaching toward it, where visual attention supports spatial localization and motor planning for the upcoming action. Although participants in our computerized task did not physically act on their choices, these sensorimotor processes are likely highly automatized and may still be engaged by default, even when not strictly required. Beyond motor preparation, post-decisional attention may also serve additional functions, such as facilitating sensory anticipation of the reward, supporting metacognitive evaluation of the decision, and contributing to value updating for future choices. From this perspective, a degree of attentional ‘stickiness’—whereby the chosen item remains preferentially attended after commitment—could emerge as an effectively optimal policy once these post-decisional processes are taken into account. Moreover, a specific feature of the task design may further reinforce this tendency: in the snacks paradigm, the unchosen item typically disappears from the screen immediately after a response is registered. It is therefore plausible that directing gaze to the chosen item after commitment partly reflects anticipation of the imminent disappearance of the unchosen option. To disentangle these mechanisms, it would be interesting for future work to test whether this attentional bias persists when the chosen item, rather than the unchosen one, is the stimulus that disappears upon response.

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 ($\mathrm{\Sigma }r$) 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, $\mathrm{\Delta }r$, should therefore increase in proportion to $\mathrm{\Sigma }r$. 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; Ting and Gluth, 2025).

A limitation of the PDG model is that it 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.

The key assumption of the PDG model is that there is a delay between the moment a choice is internally committed and the moment it is externally reported with a key press. Because eye movements are typically faster than manual responses (${\tau }_{\mathrm{e}}$ < ${\tau }_{\mathrm{m}}$ in our simulations), this delay creates a window during which gaze can already be directed toward the covertly chosen item before the response is formally registered. We do not interpret these non-decision latencies as irreducible physiological minima for moving the eyes or pressing a button (Bompas et al., 2025). Rather, they are inferred indirectly by fitting an additive non-decision-time parameter to the behavioral data, which we decompose into a sensory delay (${\tau }_{\mathrm{s}}$) and a manual execution delay (${\tau }_{\mathrm{m}}$). Values of ${\tau }_{\mathrm{e}}$ are then chosen so that the model reproduces the observed magnitude of the behavioral effects. This estimation procedure has important limitations. Some participants show relatively ‘flat’ chronometric functions: response times vary little with value despite otherwise normal psychometric performance. Such patterns likely reflect processes not explicitly represented in the model, including procrastination, reduced motivation, task-unrelated thought, or noise in item ratings. Within a drift-diffusion framework, however, these cases are accommodated by assigning a long non-decision time together with a short evidence-accumulation period (Appendix 1—table 1). Consequently, some estimated non-decision times are substantially longer than would be expected if they represented only sensory and motor delays. A further limitation is conceptual. We model non-decision time as occurring either before or after evidence accumulation, whereas in reality decisional and non-decisional components are likely temporally interleaved (Graziano et al., 2011). This simplification may also inflate the recovered latency estimates. With these caveats in mind, sensory and oculomotor delays on the order of 300 ms remain broadly plausible, although they likely lie near the upper end of a realistic range. The estimated eye-movement latency is especially long. For instance, in monkeys trained to report simple perceptual decisions with a saccade, roughly 100 ms elapses between the threshold-crossing signal in parietal cortex (or the superior colliculus) and the executed eye movement (Roitman and Shadlen, 2002; Stine et al., 2023). Crucially, however, varying the assumed non-decision latencies across a reasonable range does not alter the qualitative predictions of the model (Figure 8).

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 (Figure 10F and G).

The possibility that the gaze–choice association is partially post-decisional may help reconcile discrepancies 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.

Code and data required to reproduce the model fitting, simulations, and figures presented in this paper are available at https://github.com/arielzylberberg/PostDecisionalAttention_eLife2026, copy archived at Zylberberg, 2026.

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Author details

1. Ariel Zylberberg

   1. Mortimer B Zuckerman Mind Brain Behavior Institute, Columbia University, New York, United States
   2. Virtual Confidence and Metacognition Laboratory, Los Angeles, United States

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Validation, Investigation, Visualization, Methodology, Writing – original draft, Project administration, Writing – review and editing

   For correspondence

   ariel.zylberberg@gmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-2572-4748

2. Ian Krajbich

   Department of Psychology, University of California, Los Angeles, Los Angeles, United States

   Contribution

   Resources, Data curation, Funding acquisition, Validation, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-6618-5675

3. Michael N Shadlen

   1. Mortimer B Zuckerman Mind Brain Behavior Institute, Columbia University, New York, United States
   2. Department of Neuroscience, Columbia University, New York, United States
   3. The Kavli Institute for Brain Science, Columbia University, New York, United States
   4. Howard Hughes Medical Institute, Chevy Chase, United States
   5. Grossman Center for the Statistics of the Mind, Columbia University, New York, United States

   Contribution

   Resources, Funding acquisition, Validation, Project administration, Writing – review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-2002-2210

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