Participants were first exposed to a color judgment task in which they were asked to decide whether the dominant color of a patch of dynamic random dots was yellow or blue (Figure 1a) over a range of difficulties (Mante et al., 2013; Bakkour et al., 2019; Kang et al., 2021). The difficulty of the color choice was conferred by the probability that a dot would be colored blue (${\displaystyle p_{{}_{\mathrm{b}\mathrm{l}\mathrm{u}\mathrm{e}}}}$) or yellow (${\displaystyle 1-p_{{}_{\mathrm{b}\mathrm{l}\mathrm{u}\mathrm{e}}}}$) on each frame. Therefore, the signed quantity ${\displaystyle C^{\pm }=2\left(p_{{}_{\mathrm{b}\mathrm{l}\mathrm{u}\mathrm{e}}}-0.5\right)}$, termed the color coherence, takes on positive values for blue dominant stimuli. We refer to the unsigned quantity $C=|C^{\pm }|$ as color strength and this took on six different levels $\{0,0.128,0.256,0.384,0.512,0.64\}$. The color strength determines the difficulty of the task with smaller strengths being more difficult. This task served to familiarize the participant with the task and instill the intuition that some decisions are more difficult than others. Only participants who performed above a set criterion (see Materials and methods) were invited to perform the main experiment. We refer to this task as a color judgment.

Figure 1

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Figure 1b shows choice accuracy and RTs averaged across ($N=20$) participants for color judgments. Participants’ choices and RTs varied according to color strength. As expected, participants chose the correct color more often, and they responded faster, when the strength increased. The data are well described by a standard drift diffusion model (black lines) that explains the choice and RT by applying a stopping bound to the accumulation of the noisy color evidence (see Materials and methods). We refer to the accumulation of color evidence (blue minus yellow) as the color decision variable, $DV$.

The main experiment required participants to make difficulty judgments (Figure 1c) of two stimuli simultaneously presented to the left and right of a central fixation cross. Participants had to select the stimulus for which they thought it would be easier to decide what the dominant color would be, regardless of whether the patch was yellow or blue dominant. Importantly, in the difficulty judgment task, participants were never asked to report color decisions for either stimulus. In this task all 12 × 12 coherence combinations of the two stimuli were presented in a randomized order (see Materials and methods). We refer to this task as a difficulty judgment.

In the difficulty task, choice and RTs were affected by both the left (S1) and right (S2) stimulus strengths (Figure 1d). Participants chose S1 more often with increasing strength of S1 (Figure 1d top, abscissa) and decreasing strength of S2, and vice versa. The RTs exhibit a more complex pattern. For a given S1 strength (Figure 1d bottom, abscissa), RTs tend to be slowest when the strength of both stimuli are the same (open symbols), and the RTs accompanying these same matched difficulties are shorter for higher strength (Figure 1d bottom, inset). Therefore, the RTs are not simply a function of the difference between the two strengths. For example, RTs were significantly longer in the hard-hard (0:0, leftmost point in inset) combination (mean =1.99 s, sd =0.6) compared to easy-easy (0.64:0.64, rightmost point in inset) conditions (mean 1.30 s, sd =0.36, difference in RT $t(19)=8.63$, ${\displaystyle p<0.001}$, $CI[0.52-0.85]$). These tendencies lead to a criss-cross pattern in the RTs. For example, when S1 is difficult the shortest RT occurs when S2 is easy, whereas when S1 is easy, the longest RT occurs when S2 is easy. This leads to an inversion in the order of colors in the leftmost and rightmost stacks of points.

We developed four models (Figure 2) of difficulty decisions and examined whether they could account for the choice and RT data. The models we consider assume that sensory evidence accumulates over time. This is a reasonable assumption because these models have been very successful in explaining choice, RT, and even confidence in many perceptual and mnemonic decision tasks, including color discrimination (Bakkour et al., 2019; Gold and Shadlen, 2007). All models assume that for each stimulus (S1 and S2), momentary color evidence (MCE) for blue vs. yellow (positive for blue) is extracted at each point in time. The models differ in how they use this MCE to determine which stimulus is easier.

Figure 2

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In each model, there are two separate streams of sensory evidence—one for each stimulus. For simplicity, we illustrate the models in Figure 2 as if the two streams of evidence were accumulated in parallel. However, in a previous study, we showed that decisions about two perceptual features (or two stimuli) are based on serial, time-multiplexed integration of decision evidence, causing longer RTs for double decisions compared to decisions about a single stimulus (Kang et al., 2021). We include this feature in our model, but its only consequence is to expand the decision times, without affecting distinguishing features of the models (see Materials and methods for additional explanation).

We fit all four models to each participant’s difficulty choices and RTs (see Materials and methods and Model recovery). For each model, five parameters were optimized: a drift rate $\kappa$ that relates coherence to the mean rate of accumulation of color evidence, three parameters controlling how each decision bound collapses over time, and a non-decision time $T_{nd}$ reflecting sensory and motor processing time that add to the decision time to yield the measured RT. For the two-step model, one additional parameter $B_{mini}$ was required for the bound height of the initial color choice. We fit the model by maximum likelihood to each participant’s data individually.

Figure 3 shows model fits, averaged across participants (for parameters, see Supplementary files 2–5). To compare the models we computed the Bayesian information criterion (BIC) for each model. This showed that the difference model provided the best fit overall (group-level $\mathrm{\Delta }\text{BIC}=148.7$ compared to the two-step model, which was the second best). Compared to all other models, the difference model was the preferred model for 12 out of 20 participants (Figure 3, bottom row). We also calculated the exceedance probability (Stephan et al., 2009; Rigoux et al., 2014), which measures how likely it is that any given model is more frequent than all other models in the comparison set, which across the four models gave $[0,0.97,0.03,0]$, showing that the difference model has a probability of 0.97 of being the most frequent model, compared to 0.03 for the two-step model.

Figure 3

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The race model fails to capture the crossing of RTs for easy-easy vs. easy-hard stimuli since it predicts that decisions are fastest when both stimuli are easy. This model also fails systematically to explain the choice behavior (Figure 3a, top). Overall, the race model provides the poorest fit to participants’ data (group-level $\mathrm{\Delta }\text{BIC}=930.2$ from difference model).

The absolute momentary evidence model provides a better fit to participants’ choices and correctly predicts the crossing of RTs. However, it underestimates RTs when both stimuli are hard. This is because in this model, any momentary evidence (regardless of sign) contributes to the difficulty decision. Consequently, this model also did not provide a good fit to the data overall (group-level $\mathrm{\Delta }\text{BIC}=509.0$ from best model).

Finally, the fits obtained with the difference model and two-step model were very similar, and both provided a good fit to participants’ data. However, the two-step model requires an extra parameter for the initial color bound. Consequently, compared to the two-step model, the difference model was the preferred model for 14 out of 20 participants. Because of the similarity of these models, we verified that model recovery could distinguish between these models. We generated 200 synthetic data sets using each model and then fit them with both models. The fitting procedure correctly classified (${\displaystyle \mathrm{\Delta }\text{BIC}>10}$) the correct model with 100% and 77% accuracy for the data generated by the difference and the two-step models, respectively (see Materials and methods).

In summary, we found support for a model in which difficulty is based on a moment-by-moment comparison of the absolute accumulated evidence from two stimuli.

The difference model predicts that behavior should change if the color dominance of each patch is known. Figure 4a contrast the difference model for an easy-easy vs. a hard-hard trial. When the drift component is small (hard-hard) the density of DVs across trials (orange circle) remains near the origin. When the drift component is large (easy-easy) the density moves into one of the channels (purple circle). Therefore, more density will cross the bound in the high drift case (gray hatched area for purple vs. orange circle), leading to RTs being shorter for the easy-easy condition compared to hard-hard condition (as in Figure 3). However, if the color dominance of both stimuli is known then this changes the bounds as shown in Figure 4b. In this example, both patches are known to be blue (positive) dominant. In this case the DVs can be compared in a signed manner where positive evidence corresponds to information in support of the known color, leading to a bound $D{V}_{S1}-D{V}_{S2}=\pm B$, that is a simple channel (as in the two-step model). In this case both hard-hard and easy-easy trials will on average cross the bound at the same rate and faster in general than in the unknown color condition. When the dominant colors are known, (i) reaction times should be faster overall, and (ii) reaction times should only depend on the difference in strengths between the stimuli, that is on $\mathrm{\Delta }\text{C}=\left|{\text{C}}_{S1}-{\text{C}}_{S2}\right|$ (i.e., reaction times for 0:0 and 0.64:0.64 strengths should be the same). Note that the two-step model becomes identical to the difference model if the color dominance of each patch is known, obviating the need for the first step of the model.

Figure 4

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We test both of these predictions in an additional experiment on three participants, with the same basic paradigm as Experiment 1. However, in some blocks of trials, participants were informed that both stimuli would be blue dominant or both yellow dominant (‘known color’ condition). In the other blocks they did not receive any instructions about the stimulus color so that, as in Experiment 1, each stimulus could either be blue or yellow dominant (‘unknown color’ condition). For the unknown color condition, only trials in which both patches had the same color dominance were included in the analyses. This ensured better comparability with performance in the known-color blocks (where both stimuli by definition always had the same color).

We first replicate the findings from Experiment 1, as shown in the left column of Figure 5a. To evaluate the predictions, we also plot the data as a function of the difference in strength (Figure 5a, right column). This way of plotting the RT highlights the fact that the RTs depend on more than the difference in strengths (i.e., also the strengths of S2, colors). The curves are fits of the difference model.

Figure 5

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Figure 5b shows the results for the same participants when the color dominance was known. The choice behavior (top row) is only subtly different from the unknown condition (Figure 5a) for reasons explained below. However, there is a striking, qualitative difference in the pattern of RTs. Consistent with prediction, the RTs appear to depend only on the difference in strengths (Figure 5b, bottom right), and they are faster when the color dominance is known (Figure 5c). Note that all solid curves accompanying the known color data in panels Figure 5b and c are not fits but predictions of the best fitting model to the unknown color condition.

To quantify the extent to which the difference in strengths ($\mathrm{\Delta }\text{C}$) explains RTs in the known vs. unknown conditions, we compared the variance explained by the six unique $\mathrm{\Delta }\text{C}$ levels for the two conditions. The increase in variance explained under the known color dominance is highly significant for all participants ($F_{176,1290}=1.31,1.41,1.39$ all ${\displaystyle p<{10}^{-6}}$). Further, in the known color condition only one of the three participants showed a significant explanatory effect of ${\text{C}}_{S2}$ on RT beyond its contribution to $\mathrm{\Delta }\text{C}$ (p=0.002, 0.15, and 0.68 for the three participants; ANOVA of RT as additive in $\mathrm{\Delta }\text{C}$ and ${\text{C}}_{S2}$, six levels each).

Examining choice as a function of the difference in strengths (Figure 5c) shows that accuracy of difficulty choices was similar in the two conditions (mean $P(\text{Correct})=0.87$, sd $=0.014$ for known and 0.86, sd $=0.018$ for unknown condition; Fisher’s exact test, p>0.12 for each participant). However, RTs were significantly faster in known vs. unknown condition (mean $=1.08$ s, sd $=0.19$ vs. 1.15 s, sd $=0.31$). A two-way ANOVA of RT by condition (two levels: known vs. unknown) $\times \mathrm{\Delta }\text{C}$ (six levels) gave a main effect of condition ${\displaystyle p<{10}^{-8}}$ for all participants. Taken together, this suggests that similar bounds are used in the unknown and known color conditions as accuracy is similar. However, in the unknown color condition it takes longer, on average, to reach the bound (as in Figure 4a vs. Figure 4b) leading to longer RTs.

Difference in confidence model

Up to now we have considered models that base difficulty on the color evidence, without requiring a decision about color dominance. We next consider the possibility that the difficulty comparison is actually a confidence judgment in disguise—that is the confidence one would assign to the two color dominance decisions were they made at each moment in time. Confidence is a mapping from DV and time into the log-odds of being correct if one were to choose the color (Kiani and Shadlen, 2009; Kiani et al., 2014). Our known color condition makes this seem unlikely as participants have no uncertainty about the color of each patch and, therefore, should have full confidence in both color estimates (so that the difference in confidence should be zero). It is not inconceivable, however, that participants evaluate a counterfactual form of confidence even in the known color condition—something to the effect of ‘How confident would I be in the color choice if I did not know the color?’

We fit difficulty judgments to the unknown color condition by calculating the confidence for each decision, that is the probability (log-odds) of being correct if one were to choose the color dominance based on the sign of the DV. In the confidence model, the difficulty decision is made when the difference in absolute confidence for each stimulus reaches a bound. The fits of the confidence model are very poor ($\mathrm{\Delta }\text{BIC}=38,56,47$ for participants 1–3, in favor of the difference model; fit parameters: Supplementary file 8), and the model fails to explain the known color condition (Figure 6). The reason for this failure may be understood as follows. In the difference model the decision only depends on the difference in the two DVs and not on the level of either DV. In contrast, the difference in confidence depends on the difference in DVs as well as the actual level of each DV. Indeed, confidence increases supra-linearly with DV such that a 0.64:0.64 coherence trial will tend to have a bigger difference in confidence than a 0:0 trial, which would lead to much shorter RTs for the former. This leads to the confidence model failing to explain the range of RTs apparent in the data (Figure 6).

Figure 6

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Controlled duration task

In addition to the RT version of the task, the participants performed a version of the task in which viewing duration was controlled by the experimenter. Stimulus duration was sampled from a truncated exponential distribution leading to an equal probability of six discrete durations: $\{0.1,0.15,0.25,0.45,0.85,1.65\}$ s (which leads to roughly equal changes in accuracy between consecutive duration). We focus on short duration stimuli as the model predicts that differences in performance between the known and unknown color dominance conditions are largest for short durations. Participants had to wait for the stimuli to disappear before indicating their response. Again there were blocks with unknown color dominance and blocks with known color dominance. The difference model predicts that accuracy should be better in the known color dominance condition.

Figure 7 shows the decision accuracy as a function of stimulus duration for the difficulty tasks, for both known (solid red) and unknown (hollow red) color. As expected, accuracy improved with longer stimulus durations. More importantly, choice accuracy was generally higher in blocks with known color (accuracy across durations: mean $P(\text{Correct})=0.82$, sd $=0.01$) compared to blocks with unknown color (mean $P(\text{Correct})=0.76$, sd $=0.01$; Fisher’s exact test, ${\displaystyle p<0.001}$ for each participants).

Figure 7

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We fit a single DDM model simultaneously to each participant’s choices for both difficulty tasks (solid and dashed lines in Figure 7). For the unknown color task we used the difference model as in Experiment 1 (i.e., using the equation shown in Figure 4a). For the known color task, we set the signs of the DVs according to the true color dominance (as in Figure 4b).

The model has four parameters (fit parameters Supplementary file 7): a drift rate coefficient ($\kappa$) and three parameters controlling how each decision bound collapses over time (as in the RT experiments). Previous work has shown that although evidence integration is serial, there is a buffer that can store a limited amount sensory information from both streams of evidence (Pashler, 1994). This means information can be acquired in parallel until the buffer is full and then the accumulation happens in a multiplexed manner (Kang et al., 2021). We included a 80 ms buffer in the model, that represents the duration of the two stimulus streams that can be held in short-term memory. The buffer simply accommodates the observation that decision makers use all the information from both stimuli when they are presented for very short durations. Indeed, we found that the model with buffer was superior to a model without a buffer for all three participants (${\log }_{10}\mathrm{BF}=2.0,2.9,5.9$ in favor of the model that includes the buffer).

An alternative explanation for the accuracy improvements is that the difficulty judgment is always based on the absolute DVs (as in Figure 4a) but that the drift rate might be higher when participants know the correct color. For example, integration might be more efficient when focusing on evidence that is consistent with the instructed color. We therefore compared our model with an alternative model in which we allowed two different $\kappa$ parameters—one for the known color and one for the unknown color condition. Model comparison reveals that the more parsimonious model with a single $\kappa$ across both conditions was preferred overall and strongly for two of the participants: ${\log }_{10}\mathrm{BF}=8.5$ (group level) and $3.7,4.1,0.75$ (participants) in favor of the single $\kappa$ model.

In summary, Experiment 2 provides evidence that knowing the correct color improves the accuracy and speed of difficulty judgments despite the fact that color identity is not relevant for the final difficulty choice. This effect is explained by the difference model: when the correct color is known, difficulty judgments are based on a comparison of appropriately signed DVs, which provide more information than the unsigned absolute strength of evidence, as explained above.

We model the decision process on of Experiment 1 (RT) as a partially observable Markovian decision process (POMDP), and transform it into a fully observable Markov decision process (MDP) over the decision maker’s belief states. The belief states are uniquely defined by the tuple $⟨t_{S1},t_{S2},D{V}_{S1},D{V}_{S2}⟩$ where t_x is the time elapsed sampling stimulus $x$ and $D{V}_x$ is the accumulated evidence for stimulus $x$. Three actions are available in each belief state: choose the option $S1$, option $S2$, or continue gathering evidence (with evidence time shared equally between $S1$ and $S2$). The assignment of policies to belief states is deterministic: only one action is chosen in each state. Transitions between states, however, are stochastic, and depend on the coherences of the stimuli and noise, which is assumed Gaussian.

For the unknown color dominance condition (Experiment 1) behavior derived from the optimal decision policy (see Materials and methods) is qualitatively similar to that of the participants (Figure 8a). Optimal performance shows a clear modulation by the strength of each stimulus. The proportion of correct responses and the decision speed are higher when strength is higher. The optimal model also captures the crossing of RTs for easy-easy vs. easy-hard stimuli that we observed in the data.

Figure 8

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To determine which of the four models presented above is most similar to the optimal model, we performed simulations of the optimal model and fit the four models to the simulated data. The model that best accounted for the simulated data was the difference model, followed by the absolute momentary evidence model ($\mathrm{\Delta }BIC=2,847$ relative to the difference model), and the two-step and race models ($\mathrm{\Delta }BIC=6,949$ and $15,465$ respectively).

An analysis of the decision space of the optimal model allows us to identify similarities with the four models. In Figure 8B we show the decision space of the optimal model for four different times ($t_{S1}+t_{S2}$). At early times, the decision space resembles that of the difference model, in that the decision variable can diffuse along four paths defined by the possible signs of the color coherences. However, later in a trial, the regions where it is optimal to continue sampling evidence become disjoint. That is, the center of the graph is no longer a region where it is optimal to keep sampling information. This is because, if after prolonged deliberation the decision variable is still in the central region of the decision space, then it is reasonable to infer that the sensory evidence is weak, in which case it may be optimal to hasten the decision and move to the next trial rather than continue deliberating on a decision that has a high probability of being wrong. The logic is the same as why it is optimal to collapse decision boundaries over time in binary decisions (Frazier and Yu, 2007; Drugowitsch et al., 2012). Interestingly, such a disjoint decision space approximates an internal commitment to a decision about the sign of the color coherence, as in the two-step model.

We also derived the optimal model for the known color RT condition of Experiment 2. To do this we limited the distribution of coherences so that both were positive. The optimal model was in qualitative agreement with the experimental data (Figure 8c) and its bounds (Figure 8d) are similar to the difference model. Response times were largely determined by the difference between the coherences, such that RTs were faster when the absolute value of difference was higher. However, there is a notable exception. In contrast to what we observed in the experimental data, the RTs of the optimal model also depended on the absolute value of the coherences, such that coherences closer to the extremes of the range led to faster RTs. This can be understood as follows. Near the range limit of coherences, the decision maker can obtain evidence samples that are very informative about the coherence of a patch. For example, if a very negative sample is obtained for one of the stimuli, then, since the decisions maker knows that all coherences are positive, then this evidence value is most likely due to a very low coherence, from which one could conclude that the other stimulus is likely to be of higher coherence. In contrast, for coherences that are in the middle of the range, there is no single sample that can be as informative. The optimal model—but not the participants—seems to exploit the knowledge of the distribution over coherences to make better decisions.

The RT for difficulty judgments did not just depend on the difference in difficulty between the two color decisions. Specifically, responses were faster, when the sum of the color strengths was larger (i.e., a ‘magnitude’ effect) even if the difference in difficulty was the same. This effect has a parallel in value-based decision making. For instance, when people choose between two highly desirable items, the decision is faster than when the items are both less desirable, even if the decisions are difficult because the pairs comprise items of approximately equal value (Smith and Krajbich, 2019). This effect can be explained by arguing that attention fluctuates between both items under comparison and that attention has a multiplicative effect on the subjective value of the unattended item (Smith and Krajbich, 2019; Sepulveda et al., 2020). Because of this multiplicative effect, the greater the value of the items under comparison, the greater the discount of the unattended item, leading to faster decisions (i.e., the magnitude effect).

This explanation does not apply in our case, since we could largely abolish the magnitude effect by informing the participants about the dominant color in each stimulus patch. In our task, the magnitude effect occurs, because DVs on hard-hard trials (Figure 4a, orange circle) cross the bounds later than on easy-easy trials (purple circle). This arises because the DVs for easy-easy trials move into one of the channels, whereas weak-weak DVs linger near the origin making them less likely to cross a bound. When the color dominance is known (Figure 4b), the bounds change so that the DV can reach a bound when lingering near the origin. Therefore, the RTs now only depend on difference in difficulty and not the magnitude.

The model that maximizes reward rate in our unknown color dominance task (Experiment 1) qualitatively reproduces the behavior of the participants. In particular, the optimal model shows a similar ‘magnitude effect’ to that observed in the data and the same criss-cross pattern in response times. Therefore, these features of the data do not signify suboptimal decision making. We fit the four models (Figure 2) to simulations of the optimal model and found that the difference model best accounts for the data from the optimal model. That is, the model that most closely resembles the optimal model is also the one that best explains the participants’ data. The decision space derived from the optimal policy was similar to that of the difference model, with one notable exception. The optimal model predicts that the speed at which the bounds collapse depend both on elapsed time and the magnitude of the decision variables (Figure 8b). The same was observed in the optimal model (Figure 8d) derived for the known color dominance task (Experiment 2).

In contrast, in the difference model the speed at which the bounds collapse depends only on elapsed time. It remains to be determined the extent to which this difference has a meaningful impact on behavior. Conducting a variant of our experiment in which participants are incentivized to maximize reward rate could be informative.

In our models we assume that participants sample the two stimuli in difficulty judgment sequentially through alternation. However, our results are not affected by whether participants sample the stimulus sequentially through alternation (which we assume is fast and has equal times for both stimulus) or in a parallel manner (cf., Kang et al., 2021). What does change is the parameters of the model (but not their predictions/fits). In the parallel model information is acquired at twice the rate of the serial model. We can, therefore, obtain the parameters of parallel models (that had identical predictions to the serial model) directly from the parameters of the current sequential models simply by adjusting the parameters that depend on the time scale (subscripts $s$ and $p$ for serial and parallel models): ${\displaystyle {\kappa }_p={\kappa }_s/\sqrt{2}}$, $u_p=u_s\sqrt{2}$, $a_p=a_s/2$, and $d_p=2d_s$ (Equation 2).

We acknowledge several limitation to our study. First, we only consider accumulation models as these have been very successful in explaining choice, RT, and even confidence in many perceptual and mnemonic decision tasks, including color discrimination (Bakkour et al., 2019; Gold and Shadlen, 2007). Using such a model we were able to account for difficulty judgment in both RT and controlled duration tasks in which the color dominance was either unknown or known. Given that simple perceptual decision of the type we study have been shown to involve accumulation, we chose not to consider non-accumulation models (Stine et al., 2020; Cisek et al., 2009). Second, for Experiment 2 we required considerable data from each participant to be able to test our hypothesis. Therefore we ran 3 participants over 18 sessions each. While this provided a large data set we accept that the number of participants is small. Third, it is an open question whether the results we obtain from our simple perceptual decision task would generalize to more naturalistic real-world tasks (e.g., choosing an instrument to learn or a recipe to cook). We chose to study judgments of difficulty in simple tasks amenable to quantitative modeling and, eventually, neurophysiological investigations in nonhuman animals. Moreover, the feedback we provided on each trial—which of the two tasks was actually easier—is likely to differ in the real world, where difficulty is rarely directly indicated. Importantly, accumulation models have been extended to more complex tasks such as decision making in a game of chess (Fernandez Slezak et al., 2018) and therefore in principle the same types of algorithm could operate for more complex difficulty judgment tasks.

It has been proposed that confidence judgments about the accuracy of a decision require having learned a mapping between (i) the state of a decision variable ($DV$) and elapsed time ($t$) and (ii) the probability that the decision is correct (Kiani and Shadlen, 2009; Kiani et al., 2014; van den Berg et al., 2016). This mapping is used to define a policy; for instance, respond with high confidence if the estimated probability of being correct is greater than a criterion. An explicit calculation of confidence in the color decisions is not necessary in our task, because the difficulty decision is based on quantity derived directly from the two color decision variables. Indeed a model that compares confidence to make the difficulty judgment provides a poor fit to the data (Figure 6). That said, there may be situations where the difficulty determinations involve calculation of confidence as an intermediate step. For example, in a difficulty comparison of a color dominance in one patch and motion direction in another patch, the decision variables may not be directly comparable and may therefore require a conversion to confidence (de Gardelle et al., 2016).

In summary, we have shown that a difference model can explain difficulty judgments in both an RT and controlled duration task and when the dominant color of both patches is either unknown or known. The results extend decision-making models, which have been used to explain choice, RT, and confidence to judgments of difficulty.

For Experiment 1 (Difficulty judgments in an RT task), 51 participants were recruited on Amazon Mechanical Turk. Only participants who completed the entire study and met performance-based inclusion criteria (see below) were included in the final sample of 20 participants (12 male and 8 female; 19 right-handed; age 20–51, mean = 33.6, sd = 9.1). Participants completed two 1 hr sessions each. They received $1 for each session, plus a performance-based bonus of up to $5. Participants who successfully completed both sessions of the task within 48 hr received an additional bonus of $1. These experiments were an early foray into using Mechanical Turk and we simply matched payment with the typical payment for similar online experiments. We have since become aware, and agree with, advocates who feel the pay is too low and have since moved to using the Prolific platform and ensure we pay $8/hr minimum.

For Experiment 2 (Difficulty judgments with unknown vs. known color dominance), four participants were recruited via a Sona Systems participant pool. After initial training, three participants (one male and two female; all right-handed; age 20–24, mean = 21.7, sd = 2.1) were selected for the main experiment based on their performance (see below). Participants completed a total of eighteen 1 hr sessions and received $17/hr and an additional performance-based bonus.

All participants had normal or corrected-to-normal vision and were naïve about the hypotheses of the experiment. Participants provided written informed consent prior to the study. The study was approved by the local ethics committee (IRB-AAAR9148, Institutional Review Board of Columbia University Medical Center).

Both experiments were conducted remotely during the SARS-CoV-2 pandemic (summer 2021). Participants completed the task online using a Google Chrome browser. The task was programmed in JavaScript and jsPsych (de Leeuw, 2015). During the task, two dynamic random dot patches with yellow and blue dots were presented in rectangular apertures ($3\times 5^{\circ }$, horizontal × vertical) to the left and right of a red fixation cross, separated by a central gray bar ($2\times 5^{\circ }$; Figure 1a). Visual stimuli were presented with a screen refresh rate of 60 Hz and stimulus density of 16 dots/deg²/s (i.e., 4 dots displayed in each aperture on each frame). On each video frame, each dot was displayed in a location chosen from a uniform distribution over the aperture. The color of each dot was determined independent of its location, according to one of six color strength levels (see below).

Prior to the experiment, participants completed a virtual chin-rest procedure in order to estimate viewing distance and calibrate the screen pixels per degree (Li et al., 2020). This procedure involves first adjusting objects of known size displayed on the screen to match their physical size and then measuring the horizontal distance from fixation to the blind spot on the screen (taken as ${13.5}^{\circ }$).

On each trial, two patches of random dots were presented after an onset delay of 400–800 ms. Participants were asked to decide for which one of two patches of dots it is easier to decide what the dominant color is (yellow/blue). The instructions given the participants was ‘Your task is to judge which patch has a stronger majority of yellow or blue dots. In other words: For which patch do you find it easier to decide what the dominant color is? It does not matter what the dominant color of the easier patch is (i.e., whether it is yellow or blue). All that matters is whether the left or right patch is easier to decide.’

Participants were instructed to press the F or J key with their left/right index finger to indicate whether the left or right stimulus was the easier one, respectively. Participants did not have to report the individual color decision (yellow or blue) for either stimulus. Instead, they were required to indicate which patch (left or right) was the easier one, regardless of whether the majority of dots in that patch was yellow or blue.

The difficulty of the color choice was conferred by the probability that a dot would be colored blue or yellow on each frame. We refer to the signed quantity, $C^{\pm }=2\left(p_{\mathrm{blue}}-0.5\right)$, as the color coherence where positive coherences refer to blue dominant stimuli. The dominant color of each stimulus (yellow/blue) and its difficulty (color strength $C=|C^{\pm }|$) was fixed during a trial but randomized independently across trials. We used six different coherence levels $\pm \{0,0.128,0.256,0.384,0.512,0.64\}$, resulting in 12 signed coherence levels for the two colors. All 12 × 12 color-coherence combinations for the two patches were presented in a pseudo-random, counterbalanced manner during the task.

Visual feedback was provided at the end of each trial. For correct responses, participants won 1 point. After errors and miss trials (too early/late), participants lost 1 point. For trials in which both stimuli were equally difficult (i.e., same coherence level), half of the trials were randomly designated ‘correct’. Miss trials were repeated later during the same block. Participants were instructed to try and gain as many points as possible and at the end of the experiment they received an extra bonus of 1 cent for every point they accumulated. Their point score was shown in the corner of the screen throughout the task and additional feedback about percent accuracy was provided at the end of every block.

Prior to completing the task with difficulty judgments, all participants were first trained on a task in which they had to make color judgments (blue/yellow) about a single patch of dots presented to the left or right of the central fixation cross. Participants used the M and K keys with their right index/middle finger to indicate their response. Throughout the task, the response mapping (M=yellow, K=blue) was shown on the screen. The ± sign of the 0 coherence level determined which response would be rewarded.

Participants were instructed to keep their eyes fixated on the central fixation cross throughout the task.

Participants performed two separate sessions with a total of 432 trials of the color judgment task and 1152 trials of the difficulty task. In session 1, participants were first trained on the color judgment task (see above). They performed 6 blocks of 72 trials each, in which all 12 signed coherence levels were presented in random order. They then completed 3 blocks of the difficulty judgment task (96 trials/block), in which all 12 × 12 coherence combinations for the two patches were presented in random order. In a second session there were 9 blocks (96 trials/block) of the difficulty task.

In both sessions, participants performed an RT task in which they were instructed to respond as soon as they had made their decision. The stimuli disappeared as soon as participants made a response. Participants were instructed to try to be both as fast and as accurate as possible in order to maximize their score. Warning messages were presented if participants initiated a response before stimulus onset or within 200 ms of stimulus onset (‘too early’) or when RTs exceeded 5 s (‘too slow!’).

Participants completed a total of 18 sessions on separate days. The first 4 sessions were regarded as training (see below) and were not included in the analysis. During sessions 5–17, participants performed different versions of the difficulty judgment task that alternated every 3–4 blocks of 96 trials each (order counterbalanced across participants): (i) controlled duration task with unknown color (total of 3888 trials per participant), (ii) controlled duration task with known color (1944 trials, half of which were blocks with blue stimuli while the other half of blocks had yellow stimuli), (iii) RT task with unknown color (2592 trials), (iv) RT task with known color (1296 trials, half of which were blocks with blue stimuli while the other half of blocks had yellow stimuli). In the final session, participants completed 600 trials of a controlled duration task in which they had to judge the dominant color of a single stimulus. In all versions of the controlled duration task, stimuli were presented with one of six stimulus durations $\{0.1,0.15,0.25,0.45,0.85,1.65\}$ s, which were randomized within blocks. This gives a discrete sample from a truncate exponential distribution. Participants were instructed to wait for the stimuli to disappear before indicating their response. Warning messages were presented if participants initiated a response before stimulus offset (‘too early’) or later than 5 s after stimulus offset (‘too slow!’).

In the difficulty task with unknown color, all 12 × 12 signed coherence combinations for the two patches were presented in random order. Participants were instructed that each color patch could be either yellow or blue and that the two patches would not necessarily be of the same color. In the blocks with known color, only coherence combinations with the same sign (6 × 6 combinations) were presented. Participants received instructions that all patches in the following blocks would be dominantly blue (or yellow). Throughout the task, participants were shown a brief instruction in the corner of the screen reminding them of the condition of the current block.

In all versions of the task, coherence combinations where both stimuli had the same difficulty (i.e., same strength) were presented with one third the frequency compared to other coherence combinations in order to reduce overall trial numbers. The rationale behind this was that choice accuracy was determined randomly in this condition, and thus, it is not informative with regard to participants’ actual choice performance. Consequently, for analyses of choice accuracy in the controlled duration task, all trials with equal coherence combinations were excluded. All other coherence combinations were presented with equal frequency and in a randomized order.

Although all possible 12 × 12 coherence combinations were presented in the unknown color condition, only trials in which both patches had the same color were included in the analyses. This ensured better comparability with performance in the known color blocks (where both stimuli by definition always had the same color). The task was designed with this in mind due to the fact that previous pilot studies from our lab revealed that performance tends to be worse when the two patches are of different color compared to when they are of the same color. Thus, participants performed twice the number of total trials with the unknown color condition, but only trials with equal color combinations were included in the main analyses, resulting in the same trial numbers for the known vs. unknown condition (1944 trials each for controlled duration and 1296 trials each for RT).

We fit a standard drift diffusion model to participants’ color choices and RTs. The model assumes that MCE is accumulated into a DV. The process terminates when the DV reaches an upper or lower bound ($\pm B$), which corresponds to making a blue vs. yellow choice, respectively. The DV is determined by a Wiener process with drift, which starts at 0 and then evolves according to the sum of a deterministic and a stochastic component (with discrete updates every $\mathrm{\Delta }t$):

The deterministic term depends on the drift $\mu =\kappa (C^{\pm }+C_0)$, where $C^{\pm }$ is the signed color coherence (positive for blue; negative for yellow), $\kappa$ converts coherence to drift rate, and C₀ allows for a color bias. Here, the bias term is modeled as an offset in the coherence, rather than a shift in the starting point of the accumulation process. This approximates the optimal way of implementing a bias when coherence levels vary across trials (Hanks et al., 2011; Zylberberg et al., 2018).

The second term of Equation 1 describes the stochastic component, which captures the variability introduced by the noise in the stimulus and in the neural response. This variability is modeled as independent samples from a Normal distribution with mean 0 and standard deviation $\sqrt{\mathrm{\Delta }t}$, which results in the variance of the DV being equal to 1 after accumulating evidence for 1 s. This was done by convention, since for any other scaling of the variance, it would be possible to define an equivalent model in which the variance is 1 and the other parameters are a scaled version of the original ones (Palmer et al., 2005).

The accumulation process terminates when the DV crosses one of two bounds ($\pm B$), resulting in a blue or yellow choice. The decision time $T_d$ is the time it takes for the DV to reach a bound. To account for the observation that RTs tend to be slower in erroneous, compared to correct choices, for a given coherence level (data not shown here), we implemented bounds that collapse over time (Rapoport and Burkheimer, 1971; Drugowitsch et al., 2012; Shadlen and Kiani, 2013). Collapsing bounds result in an increased probability that the $DV$ will reach the wrong bound the longer the accumulation process takes. We parameterized the upper collapsing bound as a logistic function:

with three parameters $a,u,d$. Therefore, the bound collapses (slope related to $a$) and reaches a value of $u/2$ at $t=d$ and approaches 0 as $t\to \mathrm{\infty }$. The lower bound is simply the negative of the upper bound ($u\to -u$).

Given a set of parameters ($\mathrm{\Phi }=[\kappa ,C_0,u,a,d]$), we can estimate the joint probability density function for choices and decision times $T_d$ as a function of the signed color coherence ${\displaystyle C\pm }$. We used a $\mathrm{\Delta }t$ of 0.5 ms and obtained the probability density function by numerically solving the Fokker-Planck equation associated with a Wiener process with drift (Kiani and Shadlen, 2009), using the finite difference method of Chang and Cooper, 1970. Finally, the RT is determined by the sum of the decision time $T_d$ and a non-decision time, which is assumed to be Gaussian with mean $T_{nd}$ and a standard deviation that was fixed to ${\sigma }_{Tnd}=0.05$ s.

We fit the model parameters to the mean choice-RT data of individual participants by maximizing the likelihood of observing the data given the model parameters ($\mathrm{\Phi }=[\kappa ,C_0,u,a,d,T_{nd}]$) and the signed color coherence ${\displaystyle C\pm }$. We used Bayesian adaptive direct search (BADS; Acerbi and Ma, 2017) to optimize the model parameters. All model fits were obtained by performing several iterations of the optimization procedure with 10 different sets of starting parameters.

We developed four alternative models of difficulty judgments. All models assume that MCE is integrated into a DV, as in the drift diffusion model for color judgments. However, for difficulty judgments, there are two DVs: $D{V}_{S1}$ and $D{V}_{S2}$, representing the accumulated momentary evidence from the left and right stimulus, respectively. In all models, we assumed that accumulation proceeds in a serial, time-multiplexed manner where evidence integration switches back and forth between the two stimuli (Kang et al., 2021). We assumed perfect time sharing between $D{V}_{S1}$ and $D{V}_{S2}$, resulting in decision times that are twice as long compared to parallel integration.

The models differ in (i) how MCE is accumulated into a decision variable and (ii) the decision criterion that is used to make the difficulty choice (Figure 2).

In order to model choice accuracy in the unknown and known color condition, we used a difference model in which difficulty choice was based on either the difference in absolute DVs (unknown color) or appropriately signed DVs (known color). To fit choices in difficulty judgments, we excluded trials in unknown condition in which both stimuli had the same strength (i.e., $\mathrm{\Delta }\text{C}=0$), and trials in which the two stimuli had different dominant colors (so as to make comparable to the known color condition).

For the RT task in Experiment 2, we fit the difference model as in Experiment 1 to the data from the unknown color condition. We then used the optimized parameters to predict choices and RTs in the known color condition using the same model, but with appropriately signed DVs instead of absolute DVs.

To fit the choices in the controlled duration task we simulated 2000 trials for each unique combination of signed coherence for S1 × S2 (using a $\mathrm{\Delta }t$ of 5 ms) with a collapsing bound as in the RT task. Choice was based on crossing a bound or on the the sign of the difference in DVs at the end of evidence accumulation if no bound had been crossed. We fit both the known and unknown color conditions simultaneously using the appropriate decision rules for each (difference model with absolute DVs for the unknown and appropriately signed DVs instead of absolute DVs for the known color condition).

We included a buffer $T_{\mathrm{buf}}$ in the model that controls the duration of the stimulus streams that can be stored while the accumulation process alternates between the two stimuli and this was set to 80 ms based on our previous work (Kang et al., 2021). If the stimulus duration is shorter than the buffer, the amount of time that each stimulus is sampled, $T_{\mathrm{dur}}$, equals the stimulus duration (i.e., all the information can be stored in the buffer and can be integrated into the DV). If the stimulus duration is longer than the buffer, $T_{\mathrm{dur}}$ equals the buffer plus the remaining stimulus duration, time shared between the two stimuli:

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

This work was supported by the National Institutes of Health (R01NS117699 to DMW; R01NS113113 to MNS), and the Air Force Office of Scientific Research under award (FA9550-22-1-0337 to DMW and MNS), the Howard Hughes Medical Institute (MNS), and Kavli Institute for Brain Science (AL).

Human subjects: Participants provided written informed consent prior to the study. The study was approved by the local ethics committee (IRB-AAAR9148, Institutional Review Board of Columbia University Medical Center).

1. Sent for peer review: February 13, 2023
2. Preprint posted: February 14, 2023
3. Reviewed Preprint version 1: May 5, 2023
4. Reviewed Preprint version 2: September 12, 2023
5. Version of Record published: November 17, 2023

You can cite all versions using the DOI https://doi.org/10.7554/eLife.86892. This DOI represents all versions, and will always resolve to the latest one.

© 2023, Löffler et al.

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