Introduction

Disinformation is a pervasive and pernicious feature of the modern world (1). It is linked to negative social impacts that include public-health risks (2–4), political radicalization (5,6), violence (6—8) and adherence to conspiracy theories (8,9). Consequently, there is a growing interest in comprehending how false information propagates across social networks (10–12), including an interest in designing strategies to curb its impact (13–16) albeit with limited success to date (17). However, there is also a considerable knowledge lacuna regarding how individuals learn and update their beliefs when exposed to potential disinformation. Addressing this gap is crucial, as it has been suggested that disinformation propagates by exploiting cognitive biases (18–22). Thus, uncovering whether and how potential disinformation elicits distinct learning biases has the potential to better enable targeted interventions aimed at countering its harmful effects.

We start with an assessment of a prediction that individuals should modulate their learning as a function of the credibility of an information source, and learn more from credible, truthful, information-sources. This prediction is based on Bayesian principles of learning and on previous findings showing that individuals flexibly and adaptively adjust their learning rates in response to key statistical features of the environment. For example, learning is more rapid when observationuncertainty (“noise”) decreases and in volatile, changing, compared to stable environments, particularly following detection of change-points that render re-change knowledge obsolete (23–25). Moreover, human choice is strongly influenced by social information of high (as opposed to low) credibility, such as majority opinions more confident judgments (26) and large group consensus (27). Additionally, people are disposed to follow trustworthy advisors (28), including those who have recommended optimal actions in the past (29,30).

We hypothesised that in a disinformation context individuals would show significant deviations from idealized Bayesian learning, reflecting a diversity of biases. First, filtering non-credible information is likely to be cognitively demanding (31), and this predicts such information would impact belief updating, even if individuals are aware it is untrustworthy. An additional consideration is that humans tend to learn more from positive self-confirming information (32–34), which presents one in a positive light. We conjectured, influenced by ideas from motivated-cognition (35), that low-credibility information provides a pathway for amplification of such a bias, as uncertainty regarding informationveracity might dispose individuals to self-servingly interpret positive information as true and explain-away negative information as false. A final additional consideration is the question of how exposure to potential disinformation impacts on learning from trusted sources. One possibility is that disinformation serves as a background context against which credible information would appear more salient. Alternatively, it might lead individuals to strategically reduce their overall learning in disinformation-rich environments, resulting in diminished learning from credible sources.

To address these questions, we adopt a novel approach within the disinformation literature by exploiting a Reinforcement Learning (RL) experimental framework (36). While RL has guided disinformation research in recent years (37–41), our approach is novel in using one of its most popular tasks: the “bandit task”. This has the advantage that it provides computationally tractable models, that enable estimation of latent aspects of learning processes, such as belief updating. Moreover, our approach also enables an examination of the dynamics of belief updates over short timescales reflecting real-life engagements with disinformation, such as deciding whether to share a post on social media. Moreover, bandit tasks in RL have proven success in characterizing key decision-making biases (e.g., positivity bias (42–44)), albeit in scenarios where learners receive accurate information. . Finally, a previous literature has suggested a role for reinforcement in the dissemination of disinformation, where individuals may receive positive reinforcement (likes, shares) for spreading sensationalized or misleading information on social media platforms, inadvertently reinforcing such behaviours and contributing to a disinformation proliferation (15,40,45).

We developed a novel “disinformation” version of the classical two-armed bandit task to test the effects of potential disinformation on learning. In the traditional two-armed bandit task (36,42,46), participants choose repeatedly between two unfamiliar bandits (i.e., slot machines), that provided rewards with different probabilities, to learn which bandit is more rewarding. Critically, in our disinformation-variant, true choice outcomes (reward or non-reward) were latent, i.e., unobservable. Instead, participants were informed about choice-outcomes by computer-programmed “feedback agents”, who were disposed to occasionally disseminate disinformation by lying (reporting a reward when the true outcome was non-reward or vice versa). As these feedback-agents varied in truthfulness, this allowed us to test the effects of source-credibility on learning. We show across two studies that the extent of belief-updates increases as a function of source-credibility. However, there were striking deviations from ideal-Bayesian learning, where we identify several sources of bias related to processing potential disinformation. In one experiment, individuals learned from noncredible information that should in principle be ignored. Additionally, in both experiments, participants exhibited increased learning from trustworthy information when it was preceded by noncredible information and an amplified normalized positivity bias for non-credible sources, where individuals preferably learn from positive compared to negative feedback (relative to the overall extent of learning).

Results

Disinformation two-armed bandit task

We conducted a discovery (n=104) and main study (n=204). In both studies the learning tasks had the same basic structure but with a few subtle differences between them (see Discovery study and SI Discovery study methods). To anticipate, the results of both studies support mostly similar conclusions, and, in the results section, we focus on the main study, with the final results section detailing similarities and differences in findings across the two studies.

In the main study, participants (n=204) completed the disinformation two-armed bandit task. In the traditional two-armed bandit task (36,42,46), participants choose between two slot-machines (i.e., bandits) differing in their reward probability. Participants are not instructed about bandit rewardprobabilities but instead they are provided with veridical choice feedback (e.g., reward or nonreward), allowing participants to learn which bandit is more rewarding. By contrast, in our disinformation version true choice-outcomes were latent (i.e., unobserved) and participants were informed about these outcomes via three computerized feedback-agents, who had privileged access to the true outcomes.

Before commencing the task, participants were instructed that feedback agents could disseminate disinformation, meaning that they were disposed to lie on a random minority of trials, reporting a reward when the true outcome was a non-reward, or vice versa (Fig. 1a). Participants were explicitly instructed about the credibility of each agent (i.e., based on the proportion of truth-telling trials), indicated by a “star system”: the 3-star agent was always truthful, the 2-star agents told the truth on 75% of the trials while the 1-star agent did so on 50% of the trials (Fig. 1b). Note that while the 1-star agent’s feedback was statistically equivalent to random feedback, participants were not explicitly instructed about this equivalence. Each experimental block encompassed 3 bandit pairs, each presented over 15 trials in a randomly interleaved manner. The agent on each trial was random subject to the constraint that each agent provided feedback for 5 trials for each bandit pair. Thus, in every trial, participants were presented with one of the bandit pairs and the feedback agent associated with that trial. Upon selecting a bandit, they then received feedback from the agent (Fig. 1c). Importantly, at the end of the experiment participants received a performance-based bonus based on true bandit outcomes, which could differ from agent-provided feedback. Within each bandit-pair one bandit provided a (true) reward on 75% of the trials and the other on 25% of trials. Choice accuracy, i.e., the probability of selecting the more rewarding bandit (within each pair), was significantly above chance (mean accuracy = 0.62, t(203) = 19.94, p <.001) and improved as a function of increasing experience with each bandit-pair (average overall improvement over 15 trials = 0.22, t(203)=19.95, p<0.001) (Fig. 1d).

Figure 1

Credible feedback promotes greater learning

A hallmark of RL value-learning is that participants are more likely to repeat a choice following positive compared to negative reward-feedback (henceforth, “feedback effect on choice repetition”). We tested a hypothesis, based on Bayesian reasoning, that this tendency would increase as a function of agent-credibility (Fig. 3a). Thus, in a binomial mixed-effects model we regressed choice-repetition (i.e., whether participants repeated their choice from the most recent trial featuring the same bandit pair; 0-switch; 1-repeat) on feedback-valence (negative or positive) and agent-credibility (1,2, or 3-star), where these are taken from the last trial featuring the same bandit pair (Methods for modelspecification). Feedback valence exerted a positive effect on choice-repetition (b=0.72, F(1,2436)=1369.6, p<0.001) and interacted with agent-credibility (F(2,2436)=307.11, p<0.001), with a feedback effect being greater for more credible agents (3-star vs. 2-star: b= 0.91, F(1,2436)=351.17; 3-star vs. 1-star: b=1.15, t(2436)=24.02; and 2-star vs. 1-star: b=0.24, t(2436)=5.34, all p’s<0.001). Additionally, we found a positive effect of feedback for the 3-star agent (b=1.41, F(1,2436)=1470.2, p<0.001), and a smaller effect of feedback for the 2-star agent (b=0.49 ,F(1,2436)=230.0, p<0.001). These results support our hypothesis that learning increases as a function of information credibility (note that the feedback effect for the 1-star agent is examined below; see “Non-credible feedback elicits learning”).

To confirm that increased learning based on information credibility is expected under an assumption that subjects adhere to Bayesian reasoning, we formulated two Bayesian models whereby the latent value of each bandit is represented as a distribution over the probability that a bandit is truly rewarding (Fig. 2a, top panel; Fig. S5c for an illustration of the model; for full model descriptions, see Methods). In the instructed-credibility Bayesian model, belief-updates are based on the instructed credibility of feedback-sources. This model is based on an idealized assumptions that during the feedback stage of each trial, the value of the chosen bandit is updated (based on feedback valence and credibility) according to Bayes rule reflecting perfect adherence to the instructed task structure (i.e., how true outcomes and feedback are generated). In contrast, a free-credibility Bayesian model, allows for the possibility that Bayes-rule updates during feedback are based on “distorted probabilities” (47), attributing non-instructed degrees of credibility to sources of false information (despite our explicit instructions on the credibility of different agents). In this variant, we fixed the credibility of the 3-star agent to 1 and estimated the credibility of 2 and 1-star agents as free parameters (which were highly recoverable; see Methods and SI 3.3). Both models additionally assumed uninformative, uniform, priors over reward probabilities of novel bandits and that learning is non-forgetful. Simulations based on both Bayesian models (see Methods) predicted increased learning as a function of feedback credibility (Fig. 3b; top panels; SI 3.1.1.1 Tables S3 and S4 for statistical analysis).

Figure 2

Next, we formulated a family of non-Bayesian computational RL models. Importantly, these models can flexibly express non-Bayesian learning patterns and, as we show in following sections, can serve to identify learning biases deviating from an idealized Bayesian strategy. Here, an assumption is that during feedback, the choice propensity for the chosen bandit (which here is represented by a point estimate, “Q value“, rather than a distribution) either increases or decreases (for positive or negative feedback, respectively) according to a magnitude quantified by the free “Credit-Assignment (CA)” model parameters (48):

where F is the feedback received from the agents (coded as 1 for reward feedback and −1 for nonreward feedback), while f_Q (∈[0,1]) is the free parameter representing the forgetting rate of the Q-value (Fig. 2a, bottom panel; Fig. S5b; see “Methods: RL models”). The probability to choose a bandit (say A over B) in this family of models is a logistic function of the contrast choice-propensities between these two bandits. One interpretation of this model is as a “sophisticated” logistic regression, where the CA parameters take the role of “regression coefficients” corresponding to the change in log odds of repeating the just-taken action in future trials based on the feedback (+/- CA for positive or negative feedback, respectively; the model also includes gradual perseveration which allows for constant logodd changes that are not affected by choice feedback) . The forgetting rate captures the extent to which the effect of each trial on future choices diminishes with time. The Q-values are thus exponentially decaying sums of logistic choice propensities based on the types of feedback a bandit received.

Within this model-family, different model variants varied as to how task-variables influenced CA parameters with the “null” model attributing the same CA to all feedback-agents (regardless of their credibility, i.e., a single free CA-parameter), whereas the “credibility-CA” model availed of three separate CA parameters, one for each feedback agent, thereby allowing us to test how learning was modulated by feedback-credibility. Using a bootstrap generalized-likelihood ratio test for modelcomparison (Methods) we rejected the null model (group level: p<0.001), in favour of the credibility-CA model. Furthermore, model-simulations based on participants best-fitting parameters (Methods) falsified the null model as it failed to predict credibility-modulated learning, showing instead, equal learning from all feedback sources (Fig, 3b; bottom-left panel). In contrast, the credibility-CA model successfully predicted increased learning as a function of credibility (Fig. 3b, bottom-right panel) (see SI 3.1.1.1 Tables S5 and S6).

Figure 3

After confirming CA parameters are highly recoverable (see Methods and SI 3.4), we examined how the Maximum Likelihood (ML) CA parameters from the credibility-CA model differed as a function of feedback credibility (Fig. 3c; see SI 3.3.1 for detailed ML parameter results). Using a mixed effects model (Methods), we regressed the CA parameters on their associated agents, finding that CA differed across the agents (F(2,609)=212.65, p<0.001), increasing as a function of agent-credibility (3-star vs. 2-star: b= 1.02, F(1,609)=253.73 ; 3-star vs. 1-star: b=1.24, t(609)=19.31; and 2-star vs. 1-star: b=0.22, t(609)=3.38, all p’s<0.001).

Substantial deviations from our Bayesian learning models

We next implemented a model comparison between each of our Bayesian models and the credibility-CA model, using a parametric bootstrap cross-fitting method (Methods). We found that the credibility-CA model provided a superior fit for 71% of participants (sign test; p<0.001) when compared to the instructed-credibility Bayesian model, Fig. 2b; and for 53.9% (p=0.29) when compared to the free-credibility Bayesian model, Fig 2c). We considered using AIC and BIC, which apply “off-the shelf” penalties for model-complexity. However, these methods do not adapt to features like finite sample size (relying instead on asymptotic assumption) or temporal dependence (as is common in reinforcement learning experiments). In contrast, the parametric bootstrap cross-fitting method replaces these fixed penalties with empirical, data-driven criteria for model-selection. Indeed, modelrecovery simulations confirmed that whereas AIC and BIC were heavily biased in favour of the Bayesian models, the bootstrap method provided excellent model-recovery (See Fig. S20).

To further characterise deviations between behaviour and our Bayesian learning models, we used a “cross-fitting” method. Treating CA parameters as data-features of interest (i.e., feedback dependent changes in choice propensity), our goal was to examine if and how empirical features differ from features extracted from simulations of our Bayesian learning models. Towards that goal, we simulated synthetic data based on Bayesian agents (using participants’ best fitting parameters), but fitted these data using the CA-models, obtaining what we term “Bayesian-CA parameters” (Fig. 2d; Methods). A comparison of these Bayesian-CA parameters, with empirical-CA parameters obtained by fitting CA models to empirical data, allowed us to uncover patterns consistent with, or deviating from, ideal-Bayesian value-based inference. Under the sophisticated logistic-regression interpretation of the CA-model family the cross-fitting method comprises a comparison between empirical regression coefficients (i.e., empirical CA parameters) and regression coefficient based on simulations of Bayesian models (Bayesian CA parameters). Using this approach, we found that both the instructed-credibility and free-credibility Bayesian models predicted increased Bayesian-CA parameters as a function of agent credibility (Fig. 3c; see SI 3.1.1.2 Tables S8 and S9). However, an in-depth comparison between Bayesian and empirical CA parameters revealed discrepancies from ideal Bayesian learning, which we describe in the following sections.

Non-credible feedback elicits learning

While our task instructions framed the 1-star agent as highly deceptive, lying 50% of the time, its feedback is statistically equivalent to entirely non-informative i.e., random feedback. Thus, participants should ignore and filter-out such feedback from their belief updates. Indeed, for the 1- star agent, simulations based on the instructed-credibility Bayesian model provided no evidence for either a positive effect of feedback on choice-repetition (mixed effects model described above; b=- 0.01, t(2436)=-0.41, p=0.68; Fig 3b top-left) or a positive Bayesian-CA (b=-0.01, t(609)=-0.31, p=0.76; Fig. 3c). However, contrary to this, we hypothesized that participants would struggle to entirely disregard non-credible feedback. Indeed, we found a positive effect of feedback on choice-repetition for the 1-star agent (mixed effects model, delta(M)=0.049, b=0.25, t(2436)=8.05, p<0.001), indicating participants are more likely to repeat a bandit selection after receiving positive feedback from this agent (Fig. 3a). Similarly, the CA parameter for the 1-star agent in the credibility-CA model was positive (b=0.23, t(609)=4.54, p<0.001) (Fig. 3c). The upshot of this empirical finding is that participants updated their beliefs based on random feedback (see Fig. S7 for analysis showing that this resulted in decreased accuracy rates).

A potential explanation for this finding is that participants do rely on a Bayesian strategy but “distort probabilities”, attributing non-instructed degrees of credibility to lying sources (despite our explicit instructions on the credibility of different agents). Consistent with this, the ML-estimated credibility of the 1-star agent (Fig. 3d) was significantly greater than 0.5 (Wilcoxon signed-rank test, median=0.08, z=5.50, p<0.001), allowing the free-credibility Bayesian model to predict a positive feedback effect on choice-repetition (mixed-effects model: b=0.12, t(2436)=9.48, p<0.001; Fig 3b topright) and a positive Bayesian-CA (b=0.08, t(609)=3.32, p<0.001; Fig. 3c) for the 1-star agent. In our Discussion we elaborate on why it might be difficult to filter out this feedback even if one can explicitly infer its randomness.

Increased learning from fully credible feedback when it follows non-informative feedback

A comparison of empirical and Bayesian credit-assignment parameters revealed a further deviation from ideal Bayesian learning: participants showed an exaggerated credit-assignment for the 3-star agent compared with Bayesian models [Wilcoxon signed-rank test, instructed-credibility Bayesian model (median difference=0.74, z=11.14); free-credibility Bayesian model (median difference=0.62, z=10.71), all p’s<0.001] (Fig. 3a). One explanation for enhanced learning for the 3-star agents is a contrast effect, whereby credible information looms larger against a backdrop of non-credible information. To test this hypothesis, we examined whether the impact of feedback from the 3-star agent is modulated by the credibility of the agent in the trial immediately preceding it. More specifically, we reasoned that the impact of a 3-star agent would be amplified by a “low credibility context” (i.e., when it is preceded by a low credibility trial). In a binomial mixed effects model, we regressed choice-repetition on feedback valence from the last trial featuring the same bandit pair (i.e., the learning trial) and the feedback agent on the trial immediately preceding that last trial (i.e., the contextual credibility; see Methods for model-specification). This analysis included only learning trials featuring the 3-star agent, and context trials featuring the same bandit pair as the learning trial (Fig. 4a). We found that feedback valence interacted with contextual credibility (F(2,2086)=11.47, p<0.001) such that the feedback-effect (from the 3-star agent) decreased as a function of the preceding context-credibility (3-star context vs. 2-star context: b= −0.29, F(1,2086)=4.06, p=0.044; 2-star context vs. 1-star context: b=-0.41, t(2086)=-2.94, p=0.003; and 3-star context vs. 1-star context: b=-0.69, t(2086)=-4.74, p<0.001) (Fig. 4b). This contrast effect was not predicted by simulations of our main models of interest (Fig. 4c). No effect was found when focussing on contextual trials featuring a bandit pair different than the one in the learning trial (see SI 3.5). Thus, these results support an interpretation that credible feedback exerts a greater impact on participants’ learning when it follows non-credible feedback in the same learning context.

Figure 4

Positivity bias in learning and credibility

Previous research has shown that reinforcement learning is characterized by a positivity bias, wherein subjects systematically learn more from positive than from negative feedback (42,44). One account is that this bias might result from motivated cognition influences on learning, whereby participants favour positive feedback that reflects well on their choices. We conjectured that feedback of ambiguous veracity (i.e., from the 1-star and 2-star agents) would promote this bias by allowing participants to explain-away negative feedback as a case of an agent-lying, while choosing to believe positive feedback. Following previous research, we quantified positivity bias in 2 ways: 1) as the absolute difference between credit-assignment based on positive or negative feedback, and 2) as the same difference but relative to the overall extent of learning. We note that the second, relative, definition, is more akin to “percentage change” measurements providing a control for the overall lower levels of credit-assignment for less credible agent. To investigate this bias across different levels of feedback credibility we formulated a more detailed variant of the CA model. To quantify the extent of a chosen-bandit’s value increase or decrease - following positive or negative feedback respectively - the “credibility-valence-CA” variant included separate CA parameters for positive (CA+) and negative (CA-) feedback for each feedback agent. In effect, this model variant enabled us to test whether different levels of feedback credibility elicited a positivity bias (i.e., CA+ > CA-). Using a bootstrap generalized-likelihood ratio test for model comparison (Methods), we rejected, in favour of the valence-credibility-CA model, the null-CA model, the credibility-CA model and a “constant feedbackvalence bias” CA model, which attributed a common valence bias (CA+ minus CA-) to all agents (all group level: all p’s<0.001). This test supported our choice of flexible CA parametrization as a factorial function of agent and feedback-valence.

After confirming the parameters of this model were highly recoverable (see Methods and SI 3.4), we used a mixed effects model to regress the ML parameters (Fig. 5a; see SI 3.3.1 for detailed ML parameter results) on their associated agent-credibility and valence (see Methods). This revealed participants attributed a greater CA to positive feedback than to negative feedback (b=0.64, F(1,1218)=37.39, p<0.001). Strikingly, for lying agents, participants selectively assigned credit based on positive feedback (1-star: b=0.61, F(1,1218)=22.81, p<0.001; 2-star: b=0.85, F(1,1218)=43.5, p<0.001), with no evidence for significant credit-assignment based on negative feedback (1-star: b=- 0.03, F(1,1218)=0.07, p=0.79; 2-star: b=0.14, F(1,1218)=1.28, p=0.25). Only for the 3-star agent, creditassignment was positive for both positive (b=1.83, F(1,1218)=203.1, p<0.001) and negative (b=1.25, F(1,1218)=95.7, p=<0.001) feedback. We found no significant interaction effect between feedback valence and credibility on CA (F(2,1218)=0.12, p=0.88; Fig. 5a-b). Thus, there was no evidence for our hypothesis when positivity-bias was measured in absolute terms.

Figure 5

However, we found evidence for agent-based modulation of positivity bias when this bias was measured in relative terms. Here we calculated, for each participant and agent, a relative Valence Bias Index (rVBI) as the difference between the Credit Assignment for positive feedback (CA⁺) and negative feedback (CA⁻), relative to the overall magnitude of CA (i.e., |CA⁺| + |CA⁻|) (Fig. 5c). Using a mixed effects model, we regressed rVBIs on their associated credibility (see Methods), revealing a relative positivity bias for all credibility levels [overall rVBI (b=0.32, F(1,609)=68.16), 50% credibility (b=0.39, t(609)=8.00), 75% credibility (b=0.41, F(1,609)=73.48) and 100% credibility (b=0.17, F(1,609)=12.62), all p’s<0.001]. Critically, the rVBI varied depending on the credibility of feedback (F(2,609)=14.83, p<0.001), such that the rVBI for the 3-star agent was lower than that for both the 1-star (b=-0.22, t(609)=-4.41, p<0.001) and 2-start agent (b=-0.24, F(1,609)=24.74, p<0.001). Feedback with 50% and 75% credibility yielded similar rVBI values (b=0.028, t(609)=0.56,p=0.57). Finally, a positivity bias could not stem from a Bayesian strategy as both Bayesian models predicted a negativity bias (Fig. 5b-c; Fig. S8; and SI 3.1.1.3 Table S11-S12, 3.2.1.1, and 3.2.1.2).

Previous research has suggested that positivity bias may spuriously arise from pure choiceperseveration (i.e., a tendency to repeat previous choices regardless of outcome) (49,50). While our models included a perseveration-component, this control may not be preferent. Therefore, in additional control analyses, we generated synthetic datasets using models including choiceperseveration but devoid of feedback-valence bias, and fitted them with our credibility-valence model (see SI 3.6.1). These analyses confirmed that perseveration can masquerade as an apparent positivity bias. Critically, however, these analyses also confirmed that perseveration cannot account for our main finding of increased positivity bias, relative to the overall extent of CA, for low-credibility feedback.

True feedback elicits greater learning

Our findings are consistent with participant modulation of the extent of credit-assignment based solely on cued task-variables, such as feedback-credibility and valence. However, we also considered another possibility: that participants might infer, on a trial-by-trial basis, whether the feedback they received was true or false and adjust their credit assignment based on this inference. For example, for a given feedback-agent, participants might boost the credit assigned to a chosen bandit as a function of the degree to which they believe feedback was true. Notably, Bayesian inference can support a trial-level calculation of a posterior probability that feedback is true based on its credibility, valence and a prior belief (based on experiences in previous trials) regarding the probability that the chosen bandit is truly rewarding (Fig. 6a). The beliefs can partly discriminate between truthful and false feedback. These beliefs can partially discriminate between truthful and false feedback. As proof of this, we calculated a Bayesian posterior feedback-truthfulness belief for each participant and trial featuring the 1- or 2-star agents, (Methods; Recall for the 3-star agent, feedback is always true). On testing whether these posterior-truthfulness beliefs vary as a function of objective feedback truthfulness (true vs. lie), we found beliefs are stronger for truthful trials than for untruthful trials for both agents (1-star agent: mean difference=0.10, t(203)=39.47, p<0.001; 2-star agent: mean difference=0.08 , t(203)=34.43, p<0.001) (Fig. 6b and Fig. S9a). Note that this calculation was feasible because, as experimenters, we had privileged access to the objective truth of the choice-feedback as, when designing the experimental sessions, we generated latent true choice outcomes which could be compared to agent-reported feedback.

Figure 6

To formally address whether feedback truthfulness modulates credit assignment, we fitted a new variant of the CA model (the “Truth-CA” model) to the data. This variant works as our Credibility-CA model, but incorporated a truth-bonus parameter (TB) which increases the degree of credit assignment for feedback as a function of the experimenter-determined likelihood the feedback is true (which is read from the curves in Fig 6a when x is taken to be the true probability the bandit is rewarding). Specifically, after receiving feedback, the Q-value of the chosen option is updated according to the following rule:

where TB is the free parameter representing the truth bonus, and P(truth) is the probability the received feedback being true (from the experimenter’s perspective). We acknowledge that this model falls short of providing a mechanistically plausible description of the credit assignment process, because, participants have no access to the experimenter’s truthfulness likelihoods (as the true bandit reward probabilities are unknown to them). Nonetheless, we use this ‘oracle model’ as a measurement tool to glean rough estimates for the extent to which credit assignment Is boosted as a function of its truthfulness likelihood.

Fitting this Truth-CA model to participants’ behaviour revealed a significant positive truth-bonus (mean=0.21, t(203)=3.12, p=0.002), suggesting that participants indeed assign greater weight to feedback that is likely to be true (Fig. 6c; see SI 3.3.1 for detailed ML parameter results). Notably, simulations using our other models (Methods) consistently predicted smaller truth biases (compared to the empirical bias) (Fig. 6d). Moreover, truth bias was still detected even in a more flexible model that allowed for both a positivity bias and truth-bias (see SI 3.7). The upshot is that participants are biased to assign higher credit based on feedback that is more likely to be true in a manner that is inconsistent with out Bayesian models and above and beyond the previously identified positivity biases.

Discovery study

The discovery study (n=104) used a disinformation task structurally similar to that used in our main study, but with three notable differences: 1) it included 4 feedback agents, with credibilities of 50%, 70%, 85% and 100%, represented by 1, 2, 3, and 4 stars, respectively; 2) each experimental block consisted of a single bandit pair, presented over 16 trials (with 4 trials for each feedback agent); and 3) in certain blocks, unbeknownst to participants, the two bandits within a pair were equally rewarding (see SI section 1.1). Overall, this study’s results supported similar conclusions as our main study (see SI section 1.2) with a few differences. We found convergent support for increased learning from more credible sources (SI 1.2.1), superior fit for the CA model over Bayesian models (SI 1.2.2) and increased learning from feedback inferred to be true (SI 1.2.6). Additionally, we found an inflation of positivity bias for low-credibility both when measured relative to the overall level of credit assignment (as in our main study), or in absolute terms (unlike in our main study) (Fig. S3; SI 1.2.5). Moreover, choiceperseveration could not predict an amplification of positivity bias for low-credibility sources (see SI 3.6.2). However, we found no evidence for learning based on 50%-credibility feedback when examining either the feedback effect on choice repetition or CA in the credibility-CA model (SI 1.2.3).

Discussion

Accurate information enables individuals to adapt effectively to their environment (51,52). Indeed, it has been suggested that the importance and utility of information elevate its status to that of a secondary reinforcer, imbuing it with intrinsic value beyond its immediate usefulness (53,54). However, a significant societal challenge arises from the fact that, as social animals, much information we receive is mediated by others, entailing it can be inaccurate, biased or purposefully misleading. Here, using a novel variant of the two-armed bandit task, we asked how we update our beliefs in the presence of potential disinformation, wherein true choice outcomes are latent and feedback is provided by potentially disinformative agents.

We acknowledge that several factors may limit the external validity of our task, including the fact that participants were explicitly instructed about the credibility of information sources. In contrast, in many real-life scenarios, individuals need to learn the credibility of information sources based on their own experience of the world or may even have false beliefs regarding the source-credibility of agents. Moreover, in our task, the experimenter fully controlled the credibility of the information source in every trial, whereas in many real-life situations people can exercise a degree of control over the credibility of information they receive. For example, search engines allow an exercise of choice regarding the credibility of sources. Finally, in our task, feedback agents served as rudimentary representations of social agents, who lied randomly and arbitrarily, in a motivation-free manner. Conversely, in real life, others may strategically attempt to mislead us, and we can exploit knowledge of their motivation to lie, such as when we assume that a used cars seller is more likely to portray a clapped-out car as excellent, rather than state the unfiltered truth. Nevertheless, our results attest to the utility of our task in identifying biased aspects of learning in the face of disinformation, even in a simplified scenario.

Consistent with Bayesian-learning principles, we show that individuals increased their learning as a function of feedback credibility. This aligns with previous studies demonstrating an impressive human ability to flexibly increase learning rates when environmental changes render prior knowledge obsolete (23,55,56), and when there is reduced inherent uncertainty, such as “observation noise” (23,55–57). However, as hypothesized, when facing potential disinformation, we also find that individuals exhibit several important biases i.e., deviations from strictly idealized Bayesian strategies. Future studies should explore if and under what assumptions, about the task’s generative structure and/or learner’s priors and objectives, more complex Bayesian models (e.g., active inference (58)) might account for our empirical findings. In our main study, we show that participants revised their beliefs based on entirely non-credible feedback, whereas an ideal Bayesian strategy dictates such feedback should be ignored. This finding resonates with the “continued-influence effect” whereby misleading information continues to influence an individual’s beliefs even after it has been retracted (59,60). One possible explanation is that some participants failed to infer that feedback from the 1- star agent was statistically void of information content, essentially random (e.g., the group-level credibility of this agent was estimated by our free-credibility Bayesian model as higher than 50%). Participants were instructed that this feedback would be “a lie” 50% of the time but were not explicitly told that this meant it was random and should therefore be disregarded. Notably, however, there was no corresponding evidence random feedback affected behaviour in our discovery study. It is possible that an individual’s ability to filter out random information might have been limited due to a high cognitive load induced by our main study task, which required participants to track the values of three bandit pairs and juggle between three interleaved feedback agents (whereas in our discovery study each experimental block featured a single bandit pair). Future studies should explore more systematically how the ability to filter random feedback depends on cognitive load (61).

Previous reinforcement learning studies, report greater credit-assignment based on positive compared to negative feedback, albeit only in the context of veridical feedback (43,44,62). Here, supporting our a-priori hypothesis we show that this positivity bias is amplified for information of low and intermediate credibility (in absolute terms in the discovery study, and relative to the overall extent of CA in both studies) . Of note, previous literature has interpreted enhanced learning for positive outcomes in reinforcement learning as indicative of a confirmation bias (42,44). For example, positive feedback may confirm, to a greater extent than negative feedback one’s choice as superior (e.g., “I chose the better of the two options”). Leveraging the framework of motivated cognition (35), we posited that feedback of uncertain veracity (e.g., low credibility) amplifies this bias by incentivising individuals to self-servingly accept positive feedback as true (because it confers positive, desirable outcomes), and explain away undesirable, choice-disconfirming, negative feedback as false. This could imply an amplified confirmation bias on social media, where content from sources of uncertain credibility, such as unknown or unverified users, is more easily interpreted in a self-serving manner, disproportionately reinforcing existing beliefs (63). In turn, this could contribute to an exacerbation of the negative social outcomes previously linked to confirmation bias such as polarization (64,65), the formation of ‘echo chambers’ (19), and the persistence of misbelief regarding contemporary issues of importance such as vaccination (66,67) and climate change (68–71). We note however, that further studies are required to determine whether positivity bias in our task is indeed a form of confirmation bias. A striking finding in our study was that for a fully credible feedback agent, credit assignment was exaggerated (i.e., higher than predicted by our Bayesian models). Furthermore, the effect of fully credible feedback on choice was further boosted when it was preceded by a low-credibility context related to current learning. We interpret this in terms of a “contrast effect”, whereby veridical information looms larger against a backdrop of disinformation (21). One upshot is that exaggerated learning might entail a risk of jumping to premature conclusions based on limited credible evidence (e.g., a strong conclusion that a vaccine produces significant side-effect risks based on weak credible information, following non-credible information about the same vaccine). An intriguing possibility, that could be tested in future studies, is that participants strategically amplify the extent of learning from credible feedback to dilute the impact of learning from non-credible feedback. For example, a person scrolling through a social media feed, encountering copious amounts of disinformation, might amplify the weight they assign to credible feedback in order to dilute effects of ‘fake news’. Ironically, these results also suggest that public campaigns might be more effective when embedding their messages in low-credibility contexts, which may boost their impact.

Our findings show that individuals increase their credit assignment for feedback in proportion to the perceived probability that the feedback is true, even after controlling for source credibility and feedback valence. Strikingly, this learning bias was not predicted by any of our Bayesian or creditassignment (CA) models. Notably, our evidence for this bias is based on a “oracle model” that incorporates the probability of feedback truthfulness from the experimenter’s perspective, rather than the participant’s. This raises an important open question: how do individuals form beliefs about feedback truthfulness, and how do these beliefs influence credit assignment? Future research should address this by eliciting trial-by-trial beliefs about feedback truthfulness. Doing so would also allow for testing the intriguing possibility that an exaggerated positivity bias for non-credible sources reflects, to some extent, a truth-based discounting of negative feedback—i.e., participants may judge such feedback as less likely to be true. However, it is important to note that the positivity bias observed for fully credible sources (here and in other literature) cannot be attributed to a truth bias—unless participants were, against instructions, distrustful of that source.

An important question arises as to the psychological locus of the biases we uncovered. Because we were interested in how individuals process disinformation—deliberately false or misleading information intended to deceive or manipulate—we framed the feedback agents in our study as deceptive, who would occasionally “lie” about the true choice outcome. However, statistically (though not necessarily psychologically), these agents are equivalent to agents who mix truth-telling with random “guessing” or “noise” where inaccuracies may arise from factors such as occasionally lacking access to true outcomes, simple laziness, or mistakes, rather than an intent to deceive. This raises the question of whether the biases we observed are driven by the perception of potential disinformation as deceitful per se or simply as deviating from the truth. Future studies could address this question by directly comparing learning from statistically equivalent sources framed as either lying or noisy. Unlike previous studies wherein participants had to infer source credibility from experience (30,37,72), we took an explicit-instruction approach, allowing us to precisely assess source-credibility impact on learning, without confounding it with errors in learning about the sources themselves. More broadly, our work connects with prior research on observational learning, which examined how individuals learn from the actions or advice of social partners (72–75). This body of work has demonstrated that individuals integrate learning from their private experiences with learning based on others’ actions or advice—whether by inferring the value others attribute to different options or by mimicking their behavior (57,76). However, our task differs significantly from traditional observational learning. Firstly, our feedback agents interpret outcomes rather than demonstrating or recommending actions (30,37,72). Secondly, participants in our study lack private experiences unmediated by feedback sources. Finally, unlike most observational learning paradigms, we systematically address scenarios with deliberately misleading social partners. Future studies could bridge this by incorporating deceptive social partners into observational learning, offering a chance to develop unified models of how individuals integrate social information when credibility is paramount for decision-making.

We conclude by noting previous research has often attributed the negative impacts of disinformation, such as polarization and the formation of echo chambers, to intricate processes facilitated by external or self-selection of information (77–79). These processes include algorithms tailoring information to align with users’ attitudes (80) or individuals consciously opting to engage with like-minded peers (81). However, our study reveals a more profound effect of disinformation, namely that even in minimal conditions, when low credibility information is explicitly identified, disinformation significantly impacts individuals’ beliefs and decision-making processes. This occurs even when the decision at hand entails minimal emotional engagement or pertinence to deep, identity-related, issues. A critical next step is to deepen our understanding of these biases, particularly within complex social environments, not least to enable the development of effective prospective interventions capable of mitigating the potentially pernicious impacts of disinformation.

Materials and methods

Participants

We recruited 246 participants (mean age 39.33± 12.65, 112 female) from the Prolific participant pool (www.prolific.co) who went on to perform the task on the Gorilla platform (82). All participants were fluent English speakers with normal or corrected-to-normal vision and a Prolific approval rate of 95% or higher. UCL Research Ethics Committee approved the study (Project ID 6649/004), and all participants provided prior informed consent.

Experimental protocol

Traditional two-armed bandit task

At the beginning of the experiment participants completed a traditional version of the two-armed bandit task. Participants performed 45 trials, each featuring one of three randomly interleaved bandit pairs (such that each pair was presented on 15 trials). On each trial, participants choose between the bandit-pair, with each bandit being represented by a distinct identicon. Once a bandit was selected it generated a true outcome (converted to bonus monetary compensation) corresponding to either a reward or nothing. Within each bandit-pair, one bandit provided rewards on 75% of trials (with 25% providing no-reward), while the other bandit rewarded on 25% of the trials (75% non-reward trials). Participants were uninformed about the reward probabilities of each bandit and had to learn these based on experience.

At onset of each trial, the two bandits were presented, one on each side of the screen, and participants were asked to indicate their choice within 3 seconds by pressing the left/right arrow-keys. If the 3 seconds elapsed with no choice, participants were shown a “too slow” message and proceeded to the next trial. Following choice, the unselected bandit disappeared, and the participants were presented with the outcome of the selected bandit for 1200ms, followed by a 250 ms ISI before the start of the next trial. Rewards were represented by a green dollar symbol and non-rewards by a red sad face (both in the center of the screen). At the end of the task, participants were informed about the number of rewards they had earned.

Disinformation task

This involved a modified, disinformation version, of the same two-armed bandit task. Participants performed 8 blocks, each consisting of 45 trials. Each block followed the structure of the traditional two-armed bandit task, but with a critical difference: true choice-outcomes were withheld from participants and instead they received reward-feedback from a feedback agent. Participants were instructed prior to the task that feedback agents mostly provide accurate feedback (i.e., the true outcome) but could lie on a random minority of trials by reporting a reward in case of a true nonreward, or vice versa. The task featured three feedback agents varying in their credibility (i.e., probability of truth-telling), as indicated by a “star-rating” system, about which participants were instructed prior to the task. The 3-star agent always told the truth, whereas the other 2 agents were partially credible, reporting the truth on 75% (2-star) or 50% (1-star) of the trials. Feedback agents were randomly interleaved across trials subject to the constraint that each agent appeared on 5-trials for each bandit pair.

At the onset of each trial, participants were presented with the feedback agent for the trial (screen center) and with the two bandits, one on each side of the screen. Participants made a 2-second time limited choice by pressing the left/right arrow-keys. Following choice, the unselected bandit disappeared, and were then presented with the agent feedback for 1200ms (represented by either a rewarding green dollar sign or a non-rewarding red sad face in the center of the screen). All stimuli then disappeared for 250 ms to be followed by the start of the next trial. At the end of each block, participants were informed about the number of true rewards they had earned. They then received a 30-second break before the next block started with new 3 bandit pairs.

General protocol

At the beginning of the experiment, participants were presented with instructions for the traditional two-armed bandit task. The instructions were interleaved with four multiple-choice questions. When participants answered a question incorrectly, they could re-read the instructions and re-attempt. If participants answered a question incorrectly twice, they were compensated for the time but could not continue to the next stage. Upon completing the instructions participants proceeded to the traditional two-armed bandit task.

After the two-armed bandit task, participants were presented with instructions regarding the disinformation task. Again, these were interleaved with six questions wherein participants had two attempts to answer each question correctly. If they answered a question incorrectly twice, they were rejected and received partial participatory compensation. Participants then proceeded to the disinformation task. After completing the disinformation task, participants completed three psychiatric questionnaires (presented in random order): 1) the Obsessional Compulsive Inventory - Revised (OCI-R) (83), assessing symptoms of obsessive-compulsive disorder (OCD); 2) The Revised Green et al. Paranoid Thoughts Scale (R-GPTS) (84), measuring paranoid ideations; and 3) the DOG scale, evaluating dogmatism (85).

The participants took on average 43 minutes to complete the experiment. They received a fixed compensation of 5.48 GBP and variable compensation between 0 and 2 GBP based on their performance on the disinformation task.

Attention checks

The two tasks included randomly interleaved catch trials wherein participants were cued to press a given key within a 3-second limit. None of the participants failed more than one of these attention checks.

Data analysis

Exclusion criteria

Participants were excluded if they: 1) Either repeated or alternated key presses in more than 70% of the trials, and/or 2) their reaction time was lower than 150 ms in more than 5% of the trials. Based on these criteria 42 participants were excluded, while 204 participants were kept for the analyses.

Accuracy

Accuracy rates were calculated as the probability of choosing within a given pair the bandit with a higher reward probability. For figure 1d, we calculated for each participant and for each trial (within a bandit-pair) averaged accuracy across all bandit-pairs. We then averaged accuracy at the trial level across participants. Overall improvement for each participant was calculated as the average accuracy difference between the last and first trials for each of the bandit-pairs.

Computational models

RL Models

We formulated a family of RL models to account for participant choices. In these models, a tendency to choose each bandit is captured by a Q-value. After reward-feedback the Q-value of the chosen bandit was updated conditional on the agent and on whether the feedback was positive or negative according to the following rule:

where CA is a free credit assignment parameter representing the magnitude of the value increase/decrease following feedback receipt F from the agents (coded as 1 for reward feedback and −1 for non-reward feedback), while fQ (∈[0,1]) is the free parameter representing the forgetting rate of the Q-value. Additionally, the value of each of the other bandits (i.e., the unchosen bandit in the presented pair and all the bandits from the other not-shown pairs) were forgotten as per the following:

Alternative model-variants differed based on whether the CA parameter(s) were influenced by agents and/or feedback valence (see Table 1 below), allowing us to test how these variables impacted learning.

1. The “Null” model included a unique CA parameter conveying an assumption that feedback is modulated by neither agent-credibility nor feedback valence.

2. The “Credibility-CA” models included a dedicated CA parameter for each agent allowing for the possibility learning was selectively modulated by agent credibility (but not by feedback valence).

3. The “Credibility-Valence-CA” model included distinct CA parameters for rewarding (CA+) and nonrewarding feedback (CA-) for each agent, allowing CA to be influenced by both feedback valence and credibility.

4. The “constant feedback-valence bias” CA model included separate CA-parameters for each agent, but a single valence bias parameter (VB) common to all agents, such that the CA+ parameter for each agent corresponded to the sum of its CA-parameter and the common VB parameter.

Additionally, we formulated a “Truth-CA” model, which worked as our Credibility-CA model, but incorporated a free truth-bonus parameter (TB). This parameter modulates the extent of credit assignment for each agent based on the posterior probability of feedback being true (given the credibility of the feedback agent, and the true reward probability of the chosen bandit). The chosen bandit was updated as follows:

where P(truth) is the posterior probability of the feedback being true in the current trial (for exact calculation of P(truth) see “Methods: Bayesian estimation of posterior belief that feedback is true”).

Table 1

All models also included gradual perseveration for each bandit. In each trial the perseveration values (P) were updated according to

Where PERS is a free parameter representing the P-value change for the chosen bandit, and f_p (∈ [0,1]) is the free parameter denoting the forgetting rate applied to the P value. Additionally, the P-values of all the non-chosen bandits (i.e., again, the unchosen bandit of the current pair, and all the bandits from the not-shown pairs) were forgotten as follows:

We modelled choices using a softmax decision rule, representing the probability of the participant to choose a given bandit over the alternative:

Bayesian Models

We also formulated a Bayesian model corresponding to an ideal belief updating strategy. In this model, beliefs about each bandit were represented by a density distribution over the probability that a bandit provides a true reward g(p), where p is the probability of a true reward (see full derivation in SI 4.1). During learning, following reward-feedback, the distribution for the chosen bandit was updated based on the agent’s feedback (F) and its associated credibility (C):

At the beginning of each block priors for each bandit were initialized to uniform distributions (g(p)=U[0,1]). In the instructed-credibility Bayesian model, we fixed the credibilities to their true values (i.e., 0.5, 0.75 and 1).

We also formulated a free-credibility Bayesian model, where we only fixed the three-star agent credibility to 1 but estimated the credibility of the two lying agents as free parameters. This model allowed the possibility that participants use distorted instructed-credibilities when following a Bayesian strategy.

For both versions, we modelled choice using a SoftMax function with a free inverse temperature parameter (β):

Where here Q(bandit) is the expected probability, the bandit provides a true reward.

Additionally, we formulated extended Bayesian models to account for choice-perseveration (see SI 3.6.1). These models operate as our instructed-credibility and free-credibility Bayesian models, but also incorporate a perseveration values, updated in each trial as in our CA models (Eqs. 3 and 5). For these extended models, we modelled choices using the following softmax decision rule:

Parameter optimization, model selection and synthetic model simulations

For each participant, we estimated the free parameter values that maximized the summed loglikelihood of the observed choices across all games. Trials where participants showed a response time below 150 ms were excluded from the log-likelihood calculations. To minimise the chances of finding local minima, we ran the fitting procedure 10 times for each participant, using random initializations for the parameters (CA~U[-10,10], PERS~U[-5,5], f_Q~[0,1], fp~[0,1], TB~[-10,10], β~[O,3O], C~U[0,1]).

We performed model comparison between Bayesian and CA models using the parametric bootstrap cross-fitting method (PBCM)(86,87). In brief, this method relies on generating, for each participant, synthetic datasets (we used 201) based on maximal likelihood parameters and each model variant (i.e., the Bayesian model and the CA model), and fitting each dataset with the two models. We then calculated the log likelihood difference between the two fits for each dataset, obtaining two loglikelihood difference distributions, one for each generative model. We determined a loglikelihood difference threshold that leads to best model-classification (i.e., maximizing the proportion of true positives and true negatives). Finally, we fit the empirical data from each participant with the two model variants, calculating an empirical loglikelihood difference. A comparison of this empirical likelihood difference to the classification threshold determines which model provides a better fit for a participant’s data (see Fig. S6 for more information). We used this procedure to compare our Bayesian models (instructed-credibility and free-credibility Bayesian) with a simplified version of the credibility-CA model that did not include perseveration (PERS, fP = 0).

We also performed model-comparisons for nested CA models using generalized-likelihood ratio tests where the null distribution for rejecting a nested model (in favour of a nesting model) was based on a bootstrapping method (BGLRT)(48,88).

To assess the mechanistic predictions of each model, we generated synthetic simulations based on the ML parameters of participants. Unless stated otherwise, we generated 5 simulations for each participant (1020 total simulations) with a new sequence of trials generated as in the actual data. We analysed these data in the same way as we analysed empirical data, after pooling together the 5 simulated data set per participant.

Parameter recovery

For each model of interest, we generated 201 synthetic simulations based on parameters sampled from uniform distributions (CA~U[-10,10], PERS~U[-5,5], f_Q~U[0,1], f_p~U[0,1], β~U[0,30], C~U[0,1]). We fitted each simulated dataset with its generative model and calculated the Spearman’s correlation between the generative and fitted parameters.

Mixed effects models

Model-agnostic analysis of agent-credibility effects on choice-repetition

We used a mixed-effects binomial regression model to assess whether, and how, value-learning was modulated by agent-credibility, with participants serving as random effects. The regressed variable REPEAT indicated whether the current trial repeated the choice from the previous trial featuring the same bandit-pair (repeated choice=1, non-repeated choice=0) and was regressed on the following regressors: FEEDBACK coded whether feedback received in the previous trial with the same bandit pair was positive or negative (coded as 0.5, −0.5, respectively), BETTER coded whether the bandit chosen in that previous trial was the better -mostly rewarding- or the worse -mostly unrewarding-bandit within the pair, coded as 0.5 and −0.5 respectively, AGENT_2-star indicated whether feedback received in the previous trial (featuring the same bandit pair) came from the 2-star agent (previous feedback from 2-star agent=1, otherwise=0) and, AGENT_3-star indicated whether the feedback in the previous trial came from the 3-star agent. The model in Wilkinson’s notation was:

In figure 2a and 2b, we plot the choice-repeat probability based on feedback-valence and agentcredibility from the preceding trial with the same bandit pair. We independently calculated the repeat probability for the better (mostly rewarding) and worse (mostly non-rewarding) bandits and averaged across them. This calculation was done at the participants level, and finally averaged across participants.

Model-agnostic analysis of contextual credibility effects on choice-repetition

We used a different mixed-effects binomial regression model to test whether value learning from the 3-star agent was modulated by contextual credibility. We focused this analysis on instances where the previous trial with the same bandit pair featured the 3-star agent. We regressed the variable REPEAT, which indicated whether the current trial repeated the choice from the previous trial featuring the same bandit-pair (repeated choice=1, non-repeated choice=0). We included the following regressors: FEEDBACK coding the valence of feedback in the previous trial with the same bandit pair (positive=0.5, negative=-0.5), CONTEXT_2-star indicating whether the trial immediately preceding the previous trial with the same bandit pair (context trial) featured the 2-star agent (feedback from 2-star agent=1, otherwise=0), and CONTEXT_3-star indicating whether the trial immediately preceding the previous trial with the same bandit pair featured the 3-star agent. We also included a regressor (BETTER) coding whether the bandit chosen in the learning trial was the better -mostly rewarding- or the worse -mostly unrewarding- bandit within the pair. We included in this analysis only current trials where the context trial featured the same bandit pair. The model in Wilkinson’s notation was:

In figure 4c, we independently calculate the repeat probability difference for the better (mostly rewarding) and worse (mostly non-rewarding) bandits and averaged across them. This calculation was done at the participants level, and finally averaged across participants.

Effects of agent-credibility on CA parameters from credibility-CA model

We used a mixed-effects linear regression model to assess whether, and how, credit assignment was modulated by feedback-agent, with participants serving as random effects (data from Fig. 2c). We regressed the maximal likelihood CA parameters from the credibility-CA model. The regressors AGENT_2-star and AGENT_3-star indicated, respectively, whether the CA parameter was attributed to the 2- star or the 3-star agent. The model’s Wilkinson’s notation was:

Effects of agent-credibility and feedback valence on CA parameters from credibility valence-CA model

We used a second mixed-effects linear regression model to test for a valence bias in learning, and how such bias was modulated by feedback credibility, with participants serving again as random effects (data from Fig. 3a). The maximal likelihood CA parameters from the credibility-valence-CA model served as the regressed variable, which was regressed on: AGENT_2-star and AGENT_3-star (defined in the same way as the previous model), and VALENCE coding whether the CA parameter was attributed to positive (coded as 0.5) or negative (coded as −0.5) feedback. The Wilkinson’s notation of the model was:

We used a separate mixed-effects linear regression model to test how relative valence bias was modulated by feedback credibility. We first computed the relative valence bias index (rVBI) for each credibility level, and we then regressed these values on AGENT_2-star and AGENT_3-star (defined in the same way as the previous models).

Bayesian estimation of posterior belief that feedback is true

We calculated the Bayesian posterior conditional probability of feedback truthfulness (Fig. 4a and 4b) follows. First, we calculated the probability of each true outcome, r (0-non-reward; 1-reward) conditional on the feedback, f (0: non-reward, 1: reward), the credibility of the agent reporting the feedback (C) and the history of experiences from past trials (H):

Where proportionality omits terms independent of r, is the expected probability of the chosen bandit is rewarding (conditional on past-trial history), and g(p|H) is the density over the probability (the chosen bandit) is rewarded (conditional on the history of previous trials).

Next, we normalized the two terms (for r=0,1) to sum to 1 (to correct for the proportionality in (14)). Finally, the posterior belief in truthfulness was taken as P(r=f|f,C,H).

In Fig. 4b, we calculated for each participant the mean posterior belief of truthfulness separately for trials where each agents told the truth or lied, and we compared these mean beliefs between the two kinds of trials using a paired t-tests (one test per agent).

Code and Data Availability

All code and data used to generate the results and figures in this paper will be made available on GitHub upon publication.

Acknowledgements

We thank Bastien Blain, Lucie Charles and Stephano Palminteri for helpful discussions. We thank Nira Liberman, Keiji Ota, Nitzan Shahar, Konstantinos Tsetsos and Tali Sharot for providing feedback on earlier versions of the manuscript. We additionally thank the members of the Max Planck UCL Centre for Computational Psychiatry and Ageing Research for insightful discussions. The Max Planck UCL Centre is a joint initiative supported by UCL and the Max Planck Society.

J.V.P. is a pre-doctoral fellow of the International Max Planck Research School on Computational Methods in Psychiatry and Ageing Research (IMPRS COMP2PSYCH). We acknowledge funding from the Max Planck research school to J.V.P. (577749-D-CON 186534), and funding from the Max Planck Society to R.J.D. (549771-D.CON 177814). The project that gave rise to these results received the support of a fellowship from “la Caixa” Foundation (ID 100010434), with the fellowship code LCF/BQ/EU21/11890109.

J.V.P. contributed to the study design, data collection, data coding, data analyses, and writing of the manuscript. R.M. contributed to the study design, data analyses, and writing of the manuscript. R.J.D. contributed to the writing of the manuscript.

Additional files

Supplementary information