# A reinforcement learning diffusion decision model for value-based decisions

## Abstract

Psychological models of value-based decision-making describe how subjective values are formed and mapped to single choices. Recently, additional efforts have been made to describe the temporal dynamics of these processes by adopting sequential sampling models from the perceptual decision-making tradition, such as the diffusion decision model (DDM). These models, when applied to value-based decision-making, allow mapping of subjective values not only to choices but also to response times. However, very few attempts have been made to adapt these models to situations in which decisions are followed by rewards, thereby producing learning effects. In this study, we propose a new combined reinforcement learning diffusion decision model (RLDDM) and test it on a learning task in which pairs of options differ with respect to both value difference and overall value. We found that participants became more accurate and faster with learning, responded faster and more accurately when options had more dissimilar values, and decided faster when confronted with more attractive (i.e., overall more valuable) pairs of options. We demonstrate that the suggested RLDDM can accommodate these effects and does so better than previously proposed models. To gain a better understanding of the model dynamics, we also compare it to standard DDMs and reinforcement learning models. Our work is a step forward towards bridging the gap between two traditions of decision-making research.

## Keywords

Decision-making Computational modeling Bayesian inference and parameter estimation Response time modelsResearch on value-based decisions investigates how individuals value options and make decisions between them. Every-day decisions can be based on descriptive information, such as when choosing a restaurant based on reviews, or on personal experience, such as when choosing a restaurant based on previous visits. Reinforcement learning (RL; Sutton & Barto, 1998) describes the processes involved in the latter case, and specifically how the value associated with an option is updated following reward or punishment.

In the past decades, substantial progresses in understanding the mechanisms of RL have been made both in psychology (e.g., Bechara, Damasio, Damasio, & Anderson, 1994; Erev, 1998; Estes, 1950; Luce, 1959; Rieskamp & Otto, 2006; Yechiam & Busemeyer, 2005) and neuroscience (e.g., Dayan & Daw, 2008; Frank, Seeberger, & O’Reilly, 2004; Holroyd & Coles, 2002; Niv, 2009; Schultz, Dayan, & Montague, 1997). Within this framework, computational models can be used to infer latent value representations and psychological constructs (Lewandowsky & Simon, 2010), for instance, the reliance on more recent or past feedback (often referred to as the *learning rate*). RL models usually have two components: a learning component, that describes how past information is integrated with newly received feedback to update options’ subjective values, and a choice model, that maps the subjective values associated with the options to the final choice probabilities. Despite providing a good fit to choice data, this mapping function (e.g., the soft-max choice rule) does not provide a description of the cognitive processes that lead to a specific decision. Fortunately, these mechanisms can be revealed by simultaneously inspecting choices and response times (RTs). For example, making the same choice faster or slower can indicate less or more decision conflict, respectively (Frank, Samanta, Moustafa, & Sherman, 2007). Furthermore, choices and RTs might be differently affected under different conditions, and those cognitive processes that only affect RTs would be overlooked by models based on choices alone.

Sequential sampling models (SSMs; for an overview, see Bogacz, Brown, Moehlis, Holmes, & Cohen, 2006; Smith & Ratcliff, 2004) are process models that aim to describe the cognitive computations underlying decisions and allow predicting choices and RTs in a combined fashion. SSMs define decision-making as an integration-to-bound process: When deciding between two options, noisy evidence in favor of one over the other option is integrated over time, and a response is initiated as soon as the evidence reaches a pre-set threshold. Cautious decision-makers increase their threshold to make more accurate, but at the same time slower, decisions. On the other hand, if the situation requires to respond as quickly as possible, the threshold can be lowered at the cost of accuracy. When confronted with an easy decision (i.e., between a very good and a very bad option), the integration (or *drift*) rate is higher, leading to faster and more accurate decisions. SSMs have been successfully applied in many psychological domains (for an overview, see Ratcliff, Smith, Brown, & McKoon, 2016), including both perceptual and value-based decision-making (e.g., Busemeyer & Townsend, 1993; Usher & McClelland, 2001). In particular, the diffusion decision model (DDM; Ratcliff, 1978), the dominant model in perceptual decision-making, has gained particular popularity in value-based decision-making research (Summerfield & Tsetsos, 2012). Thus, the DDM has been used to directly compare perceptual and value-based choices (Dutilh & Rieskamp, 2016), and it has been extended to account for and to model eye-movement data in consumer-choice behavior (Krajbich Krajbich, Armel, & Rangel, 2010; Krajbich, Lu, Camerer, & Rangel, 2012). Moreover, building on the discovery of a neural correlate of the integration-to-bound process during perceptual decisions in non-human primates (Gold & Shadlen, 2001), SSMs have also been used to link behavioral and neural measures, such as the decision threshold to activity in the striatum (Forstmann et al., 2008; Gluth, Rieskamp, & Büchel, 2012; van Maanen, Fontanesi, Hawkins, & Forstmann, 2016).

While significant progress has been made in describing the processes underlying value-based decision-making, previous work mainly focused on situations in which rewards are not provided after each choice. SSMs typically assume that the subjective value associated with some option is stable in time and that repeated choices involving the same option do not affect its subjective valuation. The assumption of stable preferences might hold in many choice situations, but is presumably violated when some kind of feedback is received after every choice. In these cases, SSMs should be extended by adding a learning component.

To overcome the limitations of both SSMs (i.e., the absence of learning processes) and RL models (i.e., the absence of mechanistic decision processes), new models need to be developed. The goal of the present work is to propose a new computational cognitive model that describes both the processes underlying a single decision (by relying on the SSM approach of decision-making) and how these are influenced by the learning of subjective values of options over time (by relying on the RL framework). So far, only few attempts have been made to combine these two approaches (Frank et al., 2015; Pedersen, Frank, & Biele, 2017). In particular, these studies have proposed variants of the DDM in which an RL rule is used to update the subjective values, and these values in turn are mapped to trial-specific DDM parameters in a meaningful way (e.g., the difference in subjective values is mapped to the drift rate). Notably, in these studies, only the reward differences between options were manipulated, but not the mean values of different pairs of options. However, mean values have been reported to influence the speed of decisions (Palminteri, Khamassi, Joffily, & Coricelli, 2015; Pirrone, Azab, Hayden, Stafford, & Marshall, 2017; Polania, Krajbich, Grueschow, & Ruff, 2014), and could therefore be an important modulating factor of decisions during learning. Finally, an open question remains whether the subjective-value differences map linearly (as previously proposed) or non-linearly to the DDM parameters (more similarly to common decision rules in RL models, such as the soft-max choice rule).

In the present work, we propose a learning task in which not only value differences but also the mean values across different pairs of options are manipulated. We first test behavioral, cross-trial effects related to these manipulations, and develop a combined reinforcement learning diffusion decision model (RLDDM) that captures the observed learning and value-based behavioral effects. We then compare our model qualitatively and, whenever possible, quantitatively to other classes of models. We show that some of the value-based effects would have remained unnoticed if only choices but not RTs were taken into account—in particular those that are related to the mean value of pairs of options. Finally, we perform a rigorous model comparison analysis that illustrates the predictive advantages of the new model and provides insights into the cognitive processes underlying value-based decision-making during learning.

## Methods

### Participants and procedure

A total of 32 participants (24 female, age: 18–36, *M* = 22.36, *S**D* = 2.14) completed the experiment. Participants were mainly psychology students recruited through the subject pool of the Faculty of Psychology of the University of Basel. Participation in the experiment was possible for partial fulfillment of course credits or cash (20 Swiss francs per hour). In addition, a monetary bonus corresponding to the performance in the experiment was awarded. Before starting the experiment, participants gave informed consent, as approved by the institutional review board of the Faculty of Psychology, University of Basel. The instructions of the task were presented directly on the screen. Information about participants’ gender, age, handedness, and field of study were also requested on-screen before starting the task. Since an accuracy above 56% across 240 trials is unlikely due to random behavior alone, according to a binomial test (*p* < 0.05), only participants who surpassed this threshold were included in the analyses. Raw data and scripts will be made available upon publication of the manuscript at https://osf.io/95d4p/.

### Learning paradigm

The paradigm was a multi-armed bandit problem (Sutton & Barto, 1998). A total of four options per block were presented and participants chose between two of them in each trial. Options were randomly assigned either to the left or to the right of a fixation cross, and could be chosen by pressing either Q (for left) or P (for right) on the keyboard. After each choice, participants saw both options’ rewards (i.e., full feedback) and collected the chosen option’s reward. At the end of the experiment, the accumulated reward, divided by 1400, was paid in Swiss francs to the participants as a bonus (e.g., if they collected 7000 points, they received 5 Swiss francs). On average, participants gained a bonus of 8.10 Swiss francs.

### Design

In each learning block, only four of the six possible pairs of options were presented: AB, AC, BD, and CD (but not AD and BC). The order was pseudo-randomized so that the same pair would not be presented more than three times in a row. Presenting these four couples of options allowed us to test whether our model can predict two established behavioral effects of reward-based decision-making in addition to the learning effects. Previous studies have shown that, when deciding among options that have similar values (i.e., difficult choices), people tend to be slower and less accurate (e.g., Dutilh & Rieskamp, 2016; Oud et al., 2016; Polania et al., 2014). We will refer to this effect as the *difficulty* effect. In our study, difficulty, given by the mean value difference, was low in pairs AC and BD (the difference was 14 on average), and high in pairs AB and CD (the difference was 4 on average). Previous studies have also shown that absolute shifts in value can affect decision speed without necessarily changing accuracy (e.g., Palminteri et al., 2015; Pirrone et al., 2017; Polania et al., 2014): Participants tend to be faster when deciding between two higher-valued options as compared to two lower-valued options. We will refer to this effect as the *magnitude* effect. In our study, magnitude, given by the mean value of the pairs of options, was lowest in pair AB (38), followed by AC (43), BD (47), and CD (52). Finally, we refer to the *learning* effect as the improvement in performance throughout the trials. In this study, each pair was presented for 20 trials per block, and each option was presented in 40 trials per block (since each option is included in two different pairs).

### Stimuli

During the experiment, each participant saw a total of twelve different figures (four in each block) representing the options. The figures were matrices of 5×5 squares of which 17 were colored, arranged symmetrically along the vertical axis. To control for visual salience, we selected 12 evenly spaced colors in the HSL_{UV} space. A black rectangle was drawn around the chosen option at feedback presentation to highlight the collected reward. The experiment was programmed and presented using PsychoPy (Peirce, 2007).

### Cognitive models

In total, we estimated three classes of computational models: RL models, the DDM, and combinations of the two, RLDDM (some of which were previously proposed by Pedersen, Frank, and Biele (2017)). In the next sections, we present each class of models in detail.

#### Reinforcement learning models

RL models assume that the subjective values associated with the options are updated in each trial after experiencing a new reward (i.e., the reward feedback). These subjective values are then mapped to the probability of choosing one option over the other: Options with higher subjective values are chosen more often. Participants can differ in how much weight they give to new compared to old information: When more weight is given to old information, they are less affected by sudden changes in the rewards. They can also differ in how sensitive they are to subjective value differences: When they are very sensitive, their choices become more deterministic as they tend to always choose the option with the highest value. These two constructs, the stability of the subjective values and the deterministic nature of choices, are formalized in RL models by two parameters. The learning rate *η* (with 0 ≤ *η* ≤ 1), and the sensitivity *𝜃* (with *𝜃* ≥ 0). The learning rate is the weight that is given to new information when updating the subjective value. When *η* is close to 0, the old subjective value remains almost unchanged (implying that even observations dating far back are taken into account), whereas when *η* is close to 1, the new subjective value almost coincides with the new information (implying that earlier observations are heavily discounted). The sensitivity parameter regulates how deterministic the choices are. With a higher *𝜃*, choices are more sensitive to value differences, meaning that subjectively higher-valued options will be chosen over lower-valued options with higher probability.

On each trial, the subjective values *Q* of the presented options are updated following the so-called *delta learning rule*:

*t*is the trial number, and

*f*is the experienced feedback. In the first learning block, Q-values were initialized at 27.5. This value was the average value shown in the task instructions at the beginning of the experiment, which was the same for all participants. In the subsequent learning blocks, the Q-values were initialized at the mean values learned in the previous blocks. We reasoned that adjusting initial Q-values according to prior knowledge is more realistic than simply initializing them at zero. Indeed, preliminary model estimations revealed that all models provided better fits when adjusting Q-values to prior knowledge. Choices in each trial are predicted by the soft-max choice rule:

*p*is the probability of choosing the option with the highest mean reward, and

*Q*

_{cor}and

*Q*

_{inc}are the subjective values of the options with a higher and lower mean reward, respectively.

Building on the simplest RL model, we took into account models that incorporate all possible combinations of two additional mechanisms, one concerning the learning rule and one concerning the choice rule. The first alternative mechanism allows *η* to differ depending on the sign of the reward prediction error. The reward prediction error is the difference between the feedback *f*_{t} and the previous reward expectation *Q*_{t− 1}. Previous studies have found differences in learning rates for positive and negative reward prediction errors (Gershman, 2015) and have related this feature to optimism bias (Lefebvre, Lebreton, Meyniel, Bourgeois-Gironde, & Palminteri, 2017). The second mechanism allows *𝜃* to increase as a power function of how many times an option has been encountered before (as in Yechiam & Busemeyer, 2005) so that choices become more deterministic throughout a learning block:

*n*is the number of times an option has been presented,

*b*(with

*b*> 0) is a scaling parameter, and

*c*(with

*c*≥ 0) is the consistency parameter. When

*c*is close to 0,

*𝜃*reduces to 1 and is fixed in time, while higher values of

*c*lead to steeper increase of sensitivity throughout learning.

#### Diffusion decision model

The DDM assumes that, when making a decision between two options, noisy evidence in favor of one over the other option is integrated over time until a pre-set threshold is reached. This threshold indicates how much of this relative evidence is enough to initiate a response. Since the incoming evidence is noisy, the integrated evidence becomes more reliable as time passes. Therefore, higher thresholds lead to more accurate decisions. However, the cost of increasing the threshold is an increase of decision time. In addition, difficulty affects decisions: When confronted with an easier choice (e.g., between a very good and a very bad option), the integration process reaches the threshold faster, meaning that less time is needed to make a decision and that decisions are more accurate. The DDM also assumes that a portion of the RTs reflects processes that are unrelated to the decision time itself, such as motor processes, and that can differ across participants. Because of this dependency between noise in the information, accuracy, and speed of decisions, the DDM is able to simultaneously predict the probability of choosing one option over the other (i.e., accuracy) and the shape of the two RT distributions corresponding to the two choice options. Importantly, by fitting the standard DDM, we assume that repeated choices are independent of each other, and discard information about the order of the choices and the feedback after each choice. To formalize the described cognitive processes, the simple DDM (Ratcliff, 1978) has four core parameters: The drift rate *v*, which describes how fast the integration of evidence is, the threshold *a* (with *a* > 0), that is the amount of integrated evidence necessary to initiate a response, the starting-point bias, that is the evidence in favor of one option prior to evidence accumulation, and the non-decision time *T*_{er} (with 0 ≤ *T*_{er} < RT_{min}), the part of the response time that is not strictly related to the decision process (RT = decision time + *T*_{er}). Because, in our case, the position of the options was randomized to either the left or the right screen position, we assumed no starting-point bias and only considered drift rate, threshold, and non-decision time. Within a trial, evidence is accumulated according to the diffusion process, which is discretized in finite time steps according to:

*i*is the iteration within a trial, and a response is initiated as soon as

*x*≥

*a*or

*x*≤ 0 (i.e., the evidence reaches the upper or the lower thresholds, respectively). The time unit

*dt*is assumed to approach 0 in the limit (when

*d*

*t*= 0, the integration process is continuous in time). Choices are given by the value of

*x*at the moment of the response (e.g., correct if

*x*≥

*a*, incorrect if

*x*≤ 0).

In total, we fit three versions of the DDM, varying in the number of free between-condition parameters. The first DDM had separate *v* s for difficult and easy choices, to allow accounting for the difficulty effect: Higher *v* s lead to faster and more accurate responses. The second model is as the first, but also has separate *a* s for option pairs with a higher or lower mean reward. This model variant allows accounting for the magnitude effect: Lower *a* s lead to faster, but not much more accurate decisions (Forstmann et al., 2011). This would explain the magnitude effect as a reduction of cautiousness: When confronted with more attractive options, individuals reduce their decision times (and therefore the time to the reward) by setting a lower threshold. The third model is as the second, but has also separate *v* s for option pairs with higher or lower mean reward, to check whether the magnitude effect is attributed only to a modulation of the threshold (i.e., cautiousness) or also to a modulation of the drift rate (i.e., individuals are better at discriminating two good options compared to two bad options).

#### Reinforcement learning diffusion decision models

The goal of our work is to propose a new model that overcomes the limitation of both the SSM and the RL frameworks. Therefore, we propose an RLDDM that is a combination of these two classes of models. The RLDDM simultaneously predicts choices and response times and describes how learning affects the decision process. Here, the DDM is tightly constrained by the assumed learning process: Instead of considering all choices as independent and interchangeable, the relationship between each choice, the experienced reward feedback, and the next choice is taken into account. The RLDDM assumes that, as in the RL framework, the subjective values associated with the options are updated after experiencing a reward feedback. The decision process itself is described by the DDM. In particular, the difference between the updated subjective values influences the speed of evidence integration in the next trial: When the difference is higher, as it might happen after experiencing several feedback, the integration becomes faster, leading to more accurate and faster responses. To formalize these concepts, we built a DDM in which the drift rate parameter is defined on each trial as the difference between the subjective values that are updated via the learning rule of RL models. The first and simplest RLDDM has four parameters (similarly to Model 1 in Pedersen et al.,, 2017): one learning rate *η* to update the subjective values following Eq. 1, a scaling parameter *v*_{mod} to scale the difference between values, one threshold *a*, and one non-decision time *T*_{er}. On each trial, the drift rate is defined as:

*v*is defined as the difference of subjective values, the difficulty effect naturally emerges from the model without assuming separate

*v*s for easy and difficult choices.

We considered three additional mechanisms and fit different combinations of them, resulting in a total of eight different models. The first variation is similar to one considered for RL models and includes two separate *η*s for positive and negative prediction errors (as in Pedersen et al.,). The second variation is similar to one considered in the DDM to account for the magnitude effect. However, because subjective values are learned in time, instead of fitting separate *a* s for different pairs of options (as we do in the DDM), we propose a trial-by-trial modulating mechanism:

*a*

_{fix}is the fixed threshold,

*a*

_{mod}is the threshold modulation parameter, and \(\overline {Q}_{\text {pres}}\) is the average subjective value of the presented options. When

*a*

_{mod}= 0, this model reduces to the simplest model. The third variation is to make the mapping between subjective values and choices in the RLDDM more similar to the mapping in the soft-max choice rule. In Eq. 5,

*v*is linearly related to the difference in values. Since different pairs of options can have very similar or very different values (e.g., in Fig. 1, pairs AB and AC), participants might differ in how sensitive they are to these differences. In RL models, this is regulated by the sensitivity parameter

*𝜃*. We therefore propose a very similar, nonlinear transformation of the value differences in the definition of

*v*:

*S*(

*z*) is an S-shaped function centered at 0, and

*v*

_{max}is the maximum absolute value that

*S*(

*z*) can take on: \(\lim _{z\to \pm \infty } S(z) = \pm v_{\max }\). While

*v*

_{max}only affects the maximum and minimum values that the drift rate can take,

*v*

_{mod}affects the curvature of the function. Smaller values of

*v*

_{mod}lead to more linear mapping between the value difference and the drift rate, and therefore less sensitivity to value differences. Note that this model only resembles the previous models in the limit (i.e., when

*v*

_{max}has higher values).

### Analysis of the behavioral effects

To assess the difficulty and the magnitude effects, we fit two separate Bayesian hierarchical models: a logistic regression on accuracy and a linear regression on log-transformed RTs. Accuracy was coded as 0 if the option with the lower mean reward was chosen (e.g., A is chosen over B), and as 1 if the option with higher mean reward was chosen (e.g., B is chosen over A). For both models, we included magnitude and difficulty as predictors and tested main effects and the interaction. Magnitude was defined as the true mean reward in each pair of options, and was standardized before fitting. Easy trials were coded as 1 and difficult trials as -1. For simplicity, and because we were interested in cross-trial effects, even though we were dealing with time-series data, no information about trial number was included in the models.

All models were fit using PyStan 2.18, a Python interface to Stan (Carpenter et al., 2017). We ran four parallel chains for 8000 iterations each. The first halves of each chain were warm-up samples and were discarded. To assess convergence, we computed the Gelman-Rubin convergence diagnostic \(\hat {R}\) (Gelman & Rubin, 1992). As an \(\hat {R}\) close to 1 indicates convergence, we considered a model successfully converged when \(\hat {R}\leq 1.01\). Weakly informative priors were chosen for both models. For a graphical representation of the Bayesian hierarchical models and for the exact prior distributions, see Appendix A.

To assess whether difficulty and magnitude had an effect on the behavioral data, we calculated the 95% Bayesian credible interval (BCI) on the posterior mean group distribution of the regression coefficients. If the BCI included 0, we concluded that there was no effect of a manipulation on either RT or choices. Finally, to assess model fit, we computed posterior predictive checks (Gelman, Meng, & Stern, 1996) for mean accuracy and mean RT for each pair of options and looked whether the 95% BCIs of the posterior predictive distributions included the observed mean accuracies and RTs for AB, AC, BD, and CD. Posterior predictive distributions are useful to assess the quality of the models in their ability to predict patterns observed in the data. To approximate the posterior predictive distributions, we drew 500 samples from the posterior distribution, generated 500 independent datasets, and then computed the mean accuracy and mean RTs in each dataset, separately for choice pairs.

### Model fitting and model comparison

For all classes of cognitive models, parameters were estimated using a Bayesian hierarchical modeling approach. Again, all models were fit using PyStan. Since the models vary in their complexity, the sampler was run for a different number of iterations. We first started with few samples (i.e., 1000) and checked for convergence, reflected in \(\hat {R}\leq 1.01\). If the model did not converge, more samples were collected. We also checked for saturation of the maximum tree depth (considered satisfactory if less than .1%), energy Bayesian Fraction of Missing Information, and for divergences (considered satisfactory if less than .1%). Four parallel chains were run for all models and only the second half of each chain was kept for later analyses.

To assess the predictive accuracy of the models, we computed the widely applicable information criterion (WAIC; Watanabe, 2013). To compute the WAIC, we used the variance of individual terms in the log predictive density summed over the data points to correct for model complexity, as it approximates best the results of leave-one-out cross-validation (Gelman, Carlin, Stern, & Rubin, 2014). We also computed the standard error of the difference in the predictive accuracy of the best RLDDM, DDM, and among the models of Pedersen et al.,, using the R package loo (Vehtari, Gelman, & Gabry, 2017). This measure provides a better understanding of the uncertainty around the difference in WAIC scores. We then proceeded with the posterior predictive checks: Posterior predictives were calculated for mean accuracy and mean RT across learning by binning the trials within the learning blocks in eight groups of ten trials and across the pairs of options AB, AC, BD, and CD. As for the regression analyses, we sampled 500 parameter sets from the joint posterior distribution and generated 500 independent full datasets using those parameters. We then computed the mean accuracy and RTs in each dataset, separately for choice pairs and trial bins.

For a graphical representation of the Bayesian hierarchical models, and details about the prior distributions, see Appendix A. It has been shown that RL models can suffer from poor identifiability due to low information content in the data (Spektor & Kellen, 2018). To alleviate this concern, we conducted a parameter recovery study whose results can be found in Appendix D.

## Results

Five participants were excluded for not reaching the minimum criterion of accuracy (see Methods section), so that the data of 27 participants were included in the following analyses. The mean accuracy ranged from .43 to .53 (*M* = .49, *S**D* = .04) for the excluded participants, and from .62 to .94 (*M* = .81, *S**D* = .08) for the remaining ones.

### Behavioral results

*M*= .71 in the first 20 trials, to

*M*= .86 in the last 20 trials), while at the same time responses became faster (from

*M*= 1.51 s in the first 20 trials, to

*M*= 1.36 s in the last 20 trials). They also showed difficulty and magnitude effects (Fig. 3c and d): They were more accurate in easier compared to difficult choices (

*M*= .89 compared to

*M*= .74), while at the same time being faster (

*M*= 1.38 s compared to

*M*= 1.46 s); they were not more accurate in higher valued choice pairs compared to lower valued ones (

*M*= .81 compared to

*M*= .81), but they were faster (

*M*= 1.35 s compared to

*M*= 1.48 s).

*M*= 0.56) for the mean group difficulty coefficient (meaning that easier decisions were more accurate), but it was around 0 for the magnitude coefficient (−0.27 to 0.16,

*M*= − 0.05) and for the interaction coefficient (−0.26 to 0.15,

*M*= − 0.05). To check whether the regression model predictions fit the data well, we used posterior predictive checks. In particular, we checked whether the regression model correctly predicts the mean accuracy across different pairs of options. As can be seen in Fig. 4a, the regression model correctly predicts the observed pattern.

*M*= − 0.10), for the difficulty coefficient (−0.04 to −0.02,

*M*= − 0.03) and for the interaction coefficient (−0.06 to −0.02,

*M*= − 0.04). Similarly to the previous regression model, we also checked whether the regression model correctly predicts the mean RTs across the different pairs of options. As can be seen in Fig. 4b, the regression model correctly predicts the observed pattern.

Widely applicable information criteria of the reinforcement learning models

Model | | | | −lppd | WAIC |
---|---|---|---|---|---|

RL 1 | One | Fixed | 48 | 2636 | 5368 |

RL 2 | One | Power | 45 | 2645 | 5381 |

RL 3 | Two | Fixed | 63 | 2569 | 5265 |

RL4 | Two | Power | 72 | 2573 | 5291 |

### Cognitive modeling

To better understand the learning and decision processes underlying value-based decisions, we fit and compared three different classes of models: RL models, the DDM, and RLDDMs, as well as previous attempts of combining RL and the DDM (Pedersen et al., 2017). While RL models can be only fit to choices, the DDM and RLDDM can be simultaneously fit to choices and RTs. However, the DDM does not take trial-by-trial information, such as the reward feedback, into account. In the following section, we report results from the model fitting and model comparison of these models.

*p*

_{WAIC}. Judging by the WAIC, models 2 and 4, having an increasing sensitivity in time, did not outperform models 1 and 3, while the separate learning rates increased fitness of the models. This can be further assessed by looking at the 95% BCI of the posterior predictive distribution of mean accuracy across learning and pairs of options (Fig. 5). All models predicted a nonlinear increase in performance throughout the trials, and a difference between easy (i.e., AC and BD) and difficult (i.e., AB and CD) choices.

*p*

_{WAIC}. The WAIC was lower for model 2 than for model 3, indicating that model 3 could not compensate its higher complexity with a better fit. Checking the posterior predictives in Fig. 6, we can see that: (a) all three versions of the DDM did not predict any learning effect (i.e., both accuracy and RT are stable across trials); (b) only the versions of the DDM with separate thresholds for different option pairs could predict the magnitude effect on RTs, without changing accuracy predictions (i.e., having lower thresholds, responses in higher-valued pairs are not less accurate but only faster); (c) all models could predict difficulty effects on accuracy and RTs; (d) predictions from model 3 were not qualitatively better than predictions from model 2.

Widely applicable information criteria of the diffusion models

Model | | | | −lppd | WAIC |
---|---|---|---|---|---|

DDM 1 | Difficulty | One | 125 | 5194 | 10639 |

DDM 2 | Difficulty | All choice pairs | 183 | 5034 | 10433 |

DDM 3 | All choice pairs | All choice pairs | 232 | 4986 | 10435 |

*p*

_{WAIC}. Having the lowest WAIC suggests that the model’s complexity is compensated by its superior fit to the data. In Fig. 7, posterior predictives reveal how the different models were able to capture the observed patterns in accuracy and RTs. In particular: (a) all models were able to capture learning effects as a decrease in RTs and increase in accuracy over time; (b) all models captured difficulty effects, but only models 5 to 8, by including a non-linear mapping between value differences and drift rate, did not underestimate accuracy for difficult (i.e., AB and CD) decisions; (c) only the models that included a modulating effect of values on the decision threshold could capture the magnitude effect on RTs. While no significant qualitative pattern could be observed for two compared to one learning-rate models, all models with two learning rates had slightly lower WAICs compared to their analogues with only one learning rate. The best RLDDM also outperformed the best DDM, both in terms of WAIC and in terms of posterior predictive checks.

Widely applicable information criteria of the reinforcement learning diffusion decision models

Model | | | | | −lppd | WAIC |
---|---|---|---|---|---|---|

RLDDM 1 | One | Linear | Fixed | 111 | 5129 | 10481 |

RLDDM 2 | Two | Linear | Fixed | 134 | 5051 | 10369 |

RLDDM 3 | One | Linear | Modulated | 145 | 4942 | 10174 |

RLDDM 4 | Two | Linear | Modulated | 159 | 4866 | 10048 |

RLDDM 5 | One | Sigmoid | Fixed | 137 | 4930 | 10135 |

RLDDM 6 | Two | Sigmoid | Fixed | 159 | 4861 | 10039 |

RLDDM 7 | One | Sigmoid | Modulated | 164 | 4672 | 9672 |

RLDDM 8 | Two | Sigmoid | Modulated | 190 | 4613 | 9607 |

Widely applicable information criteria of Pedersen et al., (2017)’s models

Model | | | | | −lppd | WAIC |
---|---|---|---|---|---|---|

Pedersen RLDDM 1 | One | Fixed | Power | 118 | 5100 | 10436 |

Pedersen RLDDM 2 | One | Power | Power | 112 | 5326 | 10875 |

Pedersen RLDDM 3 | Two | Fixed | Power | 141 | 5020 | 10322 |

Pedersen RLDDM 4 | Two | Power | Power | 126 | 5240 | 10732 |

Lastly, to have a measure of uncertainty of the difference in WAIC scores, we calculated the standard error of the difference in predictive accuracy of the best fitting RLDDM with the best fitting DDM, finding a substantial difference between the scores (*e**l**p**d*_{diff} = − 415.4, *S**E* = 38.3), and the best fitting model of Pedersen et al.,, finding a substantial difference between the scores (*e**l**p**d*_{diff} = − 255.1, *S**E* = 33.1).

## Discussion

In the present work, we proposed a new process model for value-based decision-making during learning. To test this model, we collected data from participants performing a multi-armed bandit task, in which both the value difference between options as well as the mean reward of different pairs of options were manipulated. This was done to elicit two value-based behavioral effects known in the literature: the difficulty (e.g., Dutilh and Rieskamp, 2016; Oud et al., 2016; Polania et al., 2014) and the magnitude (e.g., Palminteri et al., 2015; Pirrone et al., 2017; Polania et al., 2014) effects. We first assessed value effects across all trials by fitting regression models on accuracy and RTs. We observed a magnitude effect on RTs only and a difficulty effect on both RTs and choices. To gain insights into the separate learning and value mechanisms, we tested our model against RL models (from the value-based decision-making tradition) and standard DDMs (from the perceptual decision-making tradition). We also compared our model to a previously proposed class of combined RLDDM (Pedersen et al., 2017). Different classes of models were tested, when possible, quantitatively (i.e., whether our model provided a better account of the data using a relative measure) and qualitatively (i.e., whether our model captured the observed patterns that were related to the experimental manipulations).

Notably, our proposed model has stricter constraints than the DDM: When fitting the DDM to behavioral data, all trials are collapsed into two RT distributions for choosing high- and low-value options, meaning that slower decisions will be in the right tail of the distribution, independently of their occurrence at the beginning or at the end of a learning block. In the RLDDM, the trial order is taken into account, as the drift rate and the threshold depend on the learned values in each trial. When fitting the DDM using different parameters per condition, we also have less constraints. By explicitly relating the difference in values to the drift rate, and the mean learned values to the threshold, we propose mapping functions that can accommodate the observed results, providing more mechanistic explanations.

We also compared our best RLDDM to previously proposed RLDDMs. We fit the four best models proposed by Pedersen et al., (2017), and compared them quantitatively (see Table 4) and qualitatively (see Fig. 8) to our models. The quantitative comparison confirmed that our best model had better predictive accuracy than those previous models. The qualitative comparison shows that the models proposed by Pedersen et al., (2017), which assume a linear mapping between value differences and drift rate, largely overestimate the difference in performance between easy and difficult choices. Models that assumed an increasing scaling parameter for the drift rate (i.e., models 2 and 4 in Fig. 8) predicted an almost linear increase in accuracy throughout time, while the data suggest a more asymptotic learning curve. Moreover, all models proposed by Pedersen et al., (2017) predicted that the RTs for difficult (i.e., AB and CD) decisions do not decrease in time as much as it was observed in the data. Because Pedersen et al., (2017) did not show the development of mean RT throughout learning, we cannot assess whether this discrepancy was also present in their data. Finally, since Pedersen et al., (2017) did not manipulate mean reward of pairs of options and did not include a mechanism to account for the magnitude effect, their model is unable to explain this effect.

In this study, by combining DDM and RL models, we aimed at providing a psychological account of the processes underlying behavior in a RL paradigm, where the expected reward and the difficulty varies across trials. As behavioral effects in this task are not necessarily evident in all behavioral measures (in particular, the magnitude effect is only present in RTs), we showed how, by simultaneously fitting choices and RTs and constraining them by feedback data, we can provide a more complete account of the cognitive processes in this task and identify mechanisms that would remain undetected if only choices but not RTs were taken into account.

Future work should test whether the best RLDDM is also successful in describing behavior in different learning paradigms by, for instance, investigating the magnitude effect in the presence of gains and losses or by manipulating the dispersion of the reward distributions. Moreover, RLDDMs could be validated by linking model parameters to neural measures. Trial-by-trial variables such as prediction errors, reward expectation signals, trial-by-trial threshold modulations, and individual parameters such as learning rates for negative and positive prediction errors can be easily estimated using this model. It would be interesting to see whether the model predictions about prediction errors are in line with previous work in RL neuroscience (O’Doherty, Hampton, & Kim, 2007), and whether the trial-by-trial adjustments of decision thresholds are mapped onto the same brain circuitry which has been reported for decision-making paradigms without learning (Gluth & Rieskamp, 2017; Gluth et al., 2012; van Maanen et al., 2011). An obvious limitation of any RLDDM is that it can only be fit to two-alternative forced choice tasks. This is problematic for paradigms such as the Iowa gambling task (Bechara et al., 1994), for example, in which participants choose between four decks in each trial, or for studying context effects in experience-based decisions with more than two options (Spektor, Gluth, Fontanesi, & Rieskamp, in press). A different, multi-alternative SSM that is, for instance, based on the linear ballistic accumulator model (Brown & Heathcote, 2008) could offer an alternative.

In conclusion, by integrating knowledge from two separate research traditions, we showed how an extended computational model can promote our understanding of the cognitive processes underlying value-based decisions during learning.

## Notes

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