Humans can reflect on decisions and report variable levels of confidence. But why maintain an explicit representation of confidence for choices that have already been made and therefore cannot be undone? Here we show that an explicit representation of confidence is harnessed for subsequent changes of mind. Specifically, when confidence is low, participants are more likely to change their minds when the same choice is presented again, an effect that is most pronounced in participants with greater fidelity in their confidence reports. Furthermore, we show that choices reported with high confidence follow a more consistent pattern (fewer transitivity violations). Finally, by tracking participants’ eye movements, we demonstrate that lower-level gaze dynamics can track uncertainty but do not directly impact changes of mind. These results suggest that an explicit and accurate representation of confidence has a positive impact on the quality of future value-based decisions.
As we navigate through life, we are constantly faced with choices that require us to assign and compare the values of different options or actions. Some of these value-based choices seem relatively straightforward (‘what should I eat for lunch?’) and others less so (‘which job offer should I take?’). No matter how simple or complex these choices are, they are often accompanied by a sense of confidence in having made the right choice. Recent work has shown that it is possible to behaviourally and computationally dissociate a value estimate (‘how much do I like something?’) from internal fluctuations in confidence (‘how sure am I?’). For example, at a behavioural level it has been shown that confidence shares only a limited amount of variance with value and instead reflects an assessment of choice accuracy1. This relation between value and confidence is neatly accounted for computationally by assuming that confidence emerges from the dynamics of noisy accumulators in an evidence-accumulation framework1,
According to one view, confidence can be thought of as a by-product of a stochastic accumulation process that is implemented in the ventromedial prefrontal cortex during value comparison. Research indicates the brain constructs an explicit representation of confidence that underpins verbal reports7,8. Studies suggest that the rostrolateral prefrontal cortex represents confidence in both value-based and perceptual decisions1,9,
An alternative view is that explicit representations of confidence are critical for guiding one’s own future behaviour14. Work in perceptual decision-making has revealed commonalities between mechanisms supporting confidence construction and error monitoring15,16, suggesting changes of mind may be informed by confidence4. However, whether confidence is harnessed over a longer timescale to guide future choices is unknown. We aim to test the hypothesis that an explicit (and well-tuned) representation of confidence in a recent choice can guide a decision maker’s choice when faced with the same (or a similar) decision again. To test this hypothesis we presented participants with the same set of choices more than once during the course of two experiments and tested which factors were associated with a change of mind. We then investigated how confidence related to the degree of internal consistency in their patterns of choice. Choice consistency can be quantified by measuring the degree of transitivity across choices. Here we introduced a novel method for tagging choices as conforming to or violating transitivity. Using this method we were able to show that explicit representations of confidence are associated with more consistent patterns of choice as a consequence of changes of mind. Finally, we directly contrasted the effect of explicit confidence reports with lower-level markers of uncertainty that we gathered using eye tracking, revealing that changes of mind were specifically associated with explicit reports of confidence.
We collected data in two experiments in which hungry participants made choices between food items (which they could consume later) while their eye movements were monitored. In the first experiment, the 28 participants were shown high-definition pictures of two snacks and were asked to choose their preferred one (Fig. 1a). In a second experiment, 24 participants chose their preferred snack among three snacks available in each trial (Fig. 1d). After making each choice, participants reported their degree of confidence in having made the ‘correct’ choice, which in this design equates to choosing the higher-valued item. The value for individual items was elicited using a standard incentive-compatible Becker–DeGroot–Marschak (BDM) method17 . The experimental procedure we used was adapted from a task we developed previously1 (see Methods for more details).
Relation between confidence and choice
In line with a wealth of research18,
We used eye tracking to measure the dynamics of eye movements between items during decision-making: both the total amount of time participants spent looking at each item and how frequently gaze shifted back and forth between items (see Supplementary Information). Replicating previous studies23,27 , we found the difference in dwell time (DDT) was a robust predictor of choice in both two-option and three-option experiments (experiment 1: z = 4.95, P < 0.0001; experiment 2: z = 9.81, P < 0.0001; Fig. 1c,f).
For a full list of fitted models and their respective Bayesian information criterion (BIC) scores see the Supplementary Information.
Factors that contribute to confidence
We next investigated which variables contributed to subjective confidence during value-based choices. Our previous work showed an interrelationship between the absolute difference in value (|DV|), response time (RT) and confidence (that is, participants are more confident when |DV| is high and their choices are made more quickly)1 . These findings are in line with the conceptual relation between confidence, strength of evidence (indexed by |DV| in the value-based framework) and decision time3,28 . We observed this same relation in the current study. In both experiments we found that |DV| was a significant predictor of confidence (experiment 1: t = 13.43 P < 0.0001; experiment 2: t = 7.46, P < 0.0001). We also found that RT was a negative predictor of confidence (experiment 1: t = −10.01, P < 0.0001; experiment 2: t = −7.53, P < 0.0001). Additionally, we found that the summed value positively predicted confidence, meaning that participants tended to be more confident when the options were all high in value (experiment 1: t = 3.50, P < 0.005; experiment 2: t = 4.80, P < 0.0001). This finding indicates that overall value might boost confidence, despite paradoxically making choices less accurate. More broadly, these findings highlight how evidence and confidence, although related, play partially independent roles in the decision-making process. Note that all of the predictors analysed in this section were entered into the same hierarchical linear regression; therefore all of the effects hold when controlling for the other variables reported.
We also hypothesized that lower-level features of information sampling may reflect an individual’s explicit confidence reports. To test this idea, we constructed a novel measure that captured uncertainty in information-sampling behaviour. This new measure, which we termed gaze-shift frequency (GSF), indexes how frequently gaze shifted back and forth among the options presented on the screen. This measure is independent of DDT (experiment 1: Pearson’s correlation coefficient (r) = −0.02, experiment 2: r = 0.04): for a constant allocation of time between the options (for example, 3 s for the left-hand option and 5 s for the right-hand option), one may shift fixation only once (switching from left to right after 3 s have elapsed, for example; low GSF) or shift many times between the two options (high GSF). We found that the GSF was a robust negative predictor of confidence in both experiments (experiment 1: t = −3.67, P < 0.005; experiment 2: t = −8.94, P < 0.0001) see Fig. 2a,b. In other words, in trials in which participants shifted their gaze more often between the available options, their confidence was lower, even after accounting for changes in |DV| and RT. The four-way relationship between |DV|, RT, GSF and confidence is plotted in Fig. 2c,d. Correlation tables can be found in the Supplementary Information.
Confidence predicts change of mind
In both experiments, participants saw the same exact choice sets on more than one occasion. In experiment 1 each pair was presented twice; in experiment 2 each triad was presented three times (counterbalancing for different spatial locations). This design allowed us to determine the factors that affect a change of mind when the same choice is encountered again. Note that the way we define change of mind in this study differs from how it is often defined in perceptual decision-making, as a reversal in a continuing motor plan due to further processing of sensory information4,15,29,30 . The hypothesis we sought to test was that an explicit report of confidence in an initial choice at time t would influence behaviour when the same decision was presented again at a future time (t future). In a hierarchical logistic regression, lower confidence at t was indeed associated with increased changes of mind at t future in both experiments (experiment 1: z = −6.70, P < 0.0001; experiment 2: z = –5.71, P < 0.0001). The effect of confidence in predicting change of mind remained robust after controlling for several other factors that might correlate with the stability of a choice such as |DV| and RT. Because |DV| correlated positively with confidence (see the previous section and the Supplementary Information), we checked the covariance matrices and variance inflation factors (VIFs) to ensure that these correlations did not influence the interpretation of our findings. Both the covariances and the VIFs were below standard thresholds, allowing the straightforward interpretation of coefficients (see the Supplementary Information). Furthermore, to rule out the possibility that the effect we observed was driven by the presence of fast motor errors that were later corrected by the participant, we reanalysed the data excluding all trials that were faster than each participant’s mean response time. This analysis produced comparable results (see the Supplementary Information). Notably, GSF (itself a correlate of confidence) did not predict a change of mind when included in the regression analysis (Fig. 3a,b, coefficients in blue), even when excluding reported confidence from the regression analysis (see the Supplementary Information Section 7a). Together, these results suggest that a low-level (and possibly implicit) representation of uncertainty indexed by GSF is insufficient to trigger a future change of mind. On the contrary, individuals may use an explicit representation of uncertainty (expressed through confidence) to reverse their initial decision when the same (or a similar) choice is presented again.
We next harnessed individual differences in metacognition to provide a more stringent test of this hypothesis. We reasoned that the impact of confidence on changes of mind would be more prominent in participants with enhanced metacognitive skills, that is, those whose explicit confidence ratings more accurately track the level of uncertainty underlying their decision process. To test this hypothesis, we calculated an individual index of metacognitive sensitivity by computing the difference in slope between psychometric functions fitted to high- and low-confidence trials1,31,32 . We then ran a logistic regression to predict changes of mind at t future using confidence measured at t. In line with our initial hypothesis, we were able to show that the impact of confidence on changes of mind (here the negative coefficient of confidence predicting a change of mind) is stronger in subjects with greater metacognitive accuracy (r = –0.35, P = 0.01) (Fig. 3c).
Link between confidence and choice transitivity
In the analyses presented above, we established a link between an explicit representation of confidence and future changes of mind. However, these analyses are agnostic to the quality of the decisions that emerge as a consequence of changes of mind. Not all choices are born equal; some are more consistent than others, which is formally captured by the notion of transitivity. A transitive ranking is characterized by the following structure: if option A is preferred over option B and option B is preferred over option C, then it follows that A should be preferred over C (that is, A ≻ B and B ≻ C then A ≻ C). Transitivity is a normative prescription in utility theory33 ; however, failures of transitivity are commonly observed in human choices and represent a prominent violation of economic rationality and, more generally, of logical consistency34,35 . To test the relation between confidence and transitivity, we found the (idiosyncratic) preference ranking of items that led to the lowest number of transitivity violations for each subject. Finding an optimal ranking of choice sets with more than a handful of items is extremely complex; however, a number of efficient algorithms that approximate a numerical solution have been developed for pairwise comparisons. In our study, we used the minimum violations ranking (MVR) algorithm36 , which minimizes the number of inconsistencies in the ranking of items conditional on each participant’s choices. This method is conceptually similar to other methods that are based on revealed preferences such as Afriat’s efficiency index37,38 . The MVR algorithm provided an optimal ranking of items for each participant so we could tag choices that violate this ranking, hereafter labelled transitivity violations (TVs). Because most of these methods are not suited for ternary choice, the analyses presented in this section were performed only on the data collected for the experiment that used binary choice (experiment 1). An alternative way to assess choice quality is to compute the choice ranking using the BDM method and test whether participants chose the item with the highest ranking. This method gives qualitatively similar results to those reported below (see the Supplementary Information).
After the participants’ choices were ordered according to the MVR algorithm, 4.5% of all of the decisions were classified as TVs. We then split the dataset into trials in which participants reported high confidence and trials in which they reported low confidence (median split). A dramatic reduction in TVs was observed in high confidence trials (16% of TVs) in comparison to low confidence trials (84 % of TVs) (Fig. 4a). Although these results are consistent with previous evidence provided here and elsewhere1 , note that we did not rely on BDM value estimates (collected post-choice), instead we relied only on subjects’ choices to generate the optimal ranking. In other words, the link between confidence and the quality of a value-based decision is robust to the method used to elicit preference. To statistically quantify the relation between confidence and TVs on a trial-by-trial basis (while accounting for other factors that may result in violations of transitivity), we constructed a set of hierarchical logistic regression models. We found that |DV| was a robust negative predictor of TV (z = −6.59, P < 0.0001; Fig. 4b) such that participants were more likely to violate transitivity when items were closer in value. Critically, this same model showed that even when |DV| was accounted for, confidence was a negative predictor of TVs (z = –6.75, P < 0 .0001). In other words, participants were less confident during those trials in which they went against their best-fitting preference order. Finally, both response time (z = 2.55, P = 0.01) and summed value (z = 2.55, P = 0.01) positively predicted TVs, such that trials in which the value of both options was higher and/or in trials in which their responses were slower, participants’ choices were more likely to result in TVs. Similar to change-of-mind analysis, eye-tracking variables did not reliably predict TVs (GSF = −1.74, P = 0.08; |DDT| z = −0.47, P = 0.64) (Fig. 5b). Note that this was still true when reported confidence was excluded from the regression analysis (see the Supplementary Information Section 7b).
Finally we examined whether intersubject variability in metacognitive ability affected TVs. We reasoned that if a well-calibrated explicit representation of uncertainty plays a role in guiding future decisions, participants with greater metacognitive ability would show a decrease in the number of TVs when the same option was presented a second time. In line with this hypothesis, we observed that greater metacognitive ability was associated with a marked reduction in TVs between the first and second presentation of the same choice (standardized coefficient β = 0.85, s.e.m. = 0.42, z(26) = 2.03, P < 0.05; Fig. 4c). We also confirmed that this effect was not due to a relationship between metacognition and choice instability: the total number of TVs was unrelated to metacognitive accuracy (β = −1.83, s.e.m. = 1.61, z(26) = −1.14, P = 0.25). Together, these analyses show that a more accurate explicit representation of confidence is associated with more optimal choices when participants are given the opportunity to change their minds.
What is the advantage of explicitly representing one’s confidence in value-based decision-making? Most experimental set-ups elicit confidence after a decision has been made and cannot be changed. Our hypothesis was that an explicit representation of confidence might serve an important role in decision-making by signalling the need to explore different alternatives when the same (or a similar) choice is presented again.
Value-based decisions are often perceptually unambiguous (for example, a banana is noticeably different from an apple), and most of the uncertainty is contingent on a number of internal processes such as memories or homeostatic states that are often difficult to manipulate experimentally. For example, a choice between two food items might be affected both by uncertainty about the tastes of the items and by uncertainty about one’s own level of hunger. To take advantage of this information, a decision-maker should be able to correctly monitor uncertainty that arises from the different constitutive computations. A wealth of work has shown that humans can introspect on their choice process and report their level of confidence, an ability that has been associated with the psychological concept of metacognition. However, the functions of these explicit representations of confidence (as opposed to implicit markers of uncertainty such as decision time) have remained unclear. Furthermore, individuals show wide variations in how accurately they can track and report fluctuations in uncertainty (that is, metacognitive accuracy).
In two independent experiments we showed that confidence reports (elicited after a value-based decision) reliably predicted a change of mind when the same choice was presented again. This effect is robust after controlling for other factors associated with the difficulty of a decision, such as difference in value and reaction time. Furthermore, intersubject variability in metacognitive accuracy modulated the degree to which confidence predicted change of mind: confidence was a stronger predictor of change of mind in participants with better metacognitive abilities. Critically, and in contrast to our findings on explicit confidence reports, a lower-level marker of uncertainty (GSF) did not predict subsequent changes of mind, suggesting that an explicit representation of uncertainty expressed through confidence is im portant for guiding future choices. Instead, we suggest that GSF can be considered an ingredient that agents use to construct a subjective sense of certainty, together with decision time and strength of evidence (Fig. 2c,d). An alternative interpretation of our results is that GSF does not contribute directly to subjective confidence but reflects an agent’s attempt to gather more information to adaptively reduce uncertainty (a situation in which confidence would be low and reaction time slow). Future work is required to distinguish between these two hypotheses. A further methodological appeal of GSF as a trial-by-trial measure of uncertainty is that it can be easily gathered in animals. Recent years have seen a resurgence of interest in studying uncertainty and confidence using animal models39 . This promising line of work relies heavily on the development of experimental paradigms (such as opt-out or post-decision wagering) to measure the fluctuation in uncertainty during a decision process. GSF (which can be measured in rodents by tracking head movements) may prove a useful tool to monitor, on a trial-by-trial basis, internal fluctuations in uncertainty and its relation to the neural encoding of decision time and strength of evidence.
Tracking the level of decision uncertainty is helpful in guiding behaviour in a number of contexts; for example, in guiding learning40
, in deciding whether to explore a new alternative or stick with the current one41,42
or in evaluating an alternative course of action18
. At the neural level, the rostrolateral prefrontal cortex and frontopolar cortex have been shown to play key roles in tracking trial-by-trial evolution of uncertainty43,
Another question we sought to address was whether changes of mind are associated with more optimal decisions. In value-based decisions the difference between a correct decision and an incorrect one is often murky because value is a subjective construct. However, when people make a series of value-based choices across a set of options, their pattern of decisions is characterized by a variable degree of internal consistency. In experiment 1 we used a recently developed algorithm to find an optimal ranking of items that produced the lowest number of TVs for each individual. In this way we identified when participants’ decisions were inconsistent with their overall (idiosyncratic) pattern of decisions. TVs are a paradigmatic example of irrationality in economic choice as they are easy to exploit. For example, when individual preferences are not transitive, it is possible to construct a choice set in which each decision appears fair on its own but when combined guarantees a loss (a phenomenon known as a Dutch book or arbitrage in finance)49 . We showed that choices made with high confidence are overall more transitive and therefore more optimal according to the normative prescriptions of utility theory. Noticeably, this effect is robust after controlling for the absolute difference in value and reaction time. This finding suggests that individuals can monitor and report that a given decision was noisier and therefore more likely to result in a decision that is inconsistent with their overall preference patterns, establishing confidence as a correlate of choice accuracy without relying on the BDM procedure to derive independent estimates of subjective utility. This result also resonates with the well-established finding in perceptual decision-making that people are able to detect and signal errors as soon as they respond16,50 and with the proposal that confidence can facilitate cognitive control51 . We suggest a similar process might operate in value-based decisions, in which errors can be thought of as choices that are at odds with one’s overall preferences. Consistent with this proposal, we found that individuals who have a more accurate representation of confidence (greater metacognitive ability) were more likely to move towards a more internally consistent decision-making pattern over time. Our work sheds light on the reasons for an explicit representation of confidence in human decision-making. It explores value-based choices (aka economic choices) by borrowing methods and concepts from perceptual decision-making52 . Similar to perceptual decision-making, we found that the same ‘strength of evidence’ in value (that is, |DV|) is accompanied by a variable level of uncertainty that is represented explicitly as confidence. We suggest these representations play a functional role not only in allowing confidence to be shared with others but also in guiding our own future choices. Taken together, our results show that an explicit and accurate representation of confidence can have a positive impact on the quality of future value-based decisions.
Participants were required to make binary choices between 16 common snack items. Participants were asked to choose between each combination of the items (N = 120) twice, counterbalanced across the left–right spatial configurations (total number of choices = 240). After each choice, participants indicated their confidence in their decision on a continuous rating scale. Neither choices nor confidence ratings were time-constrained. The trial order was randomized with the only constraint being that the same pair was never repeated in subsequent trials. Participants’ eye movements were recorded throughout this task.
At the end of the experiment, one choice from this phase was played out and the subject had the opportunity to buy the chosen item by means of an auction administered according to the BDM procedure: the experimenter randomly extracted a price from a uniform distribution (£0–3)—the ‘market price’ of that item. If the participant’s bidding price (willingness to pay) was below the market price, no transaction occurred. The computer-generated value was drawn to a precision greater than two decimals to avoid the possibility of a tie but was rounded to pennies in the event of a transaction. If the subject’s bidding price was above the market price, the participant bought the snack item at the market price17 . At the end of the experiment, participants had to remain in the laboratory for an additional hour. During this hour, the only food they were allowed to eat was the item purchased in the auction, if any. At the end of the waiting period, participants were debriefed and thanked for their participation. Participants were paid £25 for their time, less the cost of the food item, if they bought any. Both tasks were programmed using MATLAB 8.0 (MathWorks) running the Psychophysics toolbox (http://psychtoolbox.org) as well as the Eyelink toolbox extensions53,54 . The procedure of this experiment was approved by the UCL Research Ethics Committee (project ID: 3736/004).
Participants gave their willingness to pay for 72 common snack food items on a scale ranging from £0 to £3 in a BDM procedure17 that is similar to that in experiment 1. They then completed a choice task: in each trial, they had to pick their favourite item out of three options. The triplets presented in the choice task were tailored for each participant from their willingness-to-pay ratings. The items were divided into high-value and low-value sets by a median split. The 36 high-value items were randomly combined into 12 high-value triplets; this procedure was mirrored to generate 12 low-value triplets. The high-value and low-value items were then mixed to generate medium-value triplets, with 12 triplets consisting of two high-value items and one low-value item and 12 triplets with the reverse ratio. This resulted in 48 unique triplets, with counterbalanced spatial configurations (total trials = 144) split into three blocks. Each triplet was shown once in each block; the presentation order inside blocks was randomized with the constraint that the triplet that ended one block was never shown first in the next block.
In the subsequent choice task, the triplets were presented inside three squares in an equidistant 2 × 2 grid (one randomly determined position on the grid was left empty). We used a gaze-contingent paradigm in which the items were only visible when the participant fixated inside one of the squares, so that the participant could only see one item at a time. They had unlimited time to make up their mind and could make as many fixations as they wished. After each choice, participants indicated their confidence in their decision on a visual analogue rating scale without any time constraints. Participants’ eye movements were recorded throughout the choice task. Both the choice task and the willingness-to-pay procedure were programmed in Experiment Builder version 1.10.1640 (SR-Research).
Following the choice task, an auction based on the BDM ratings was held (see experiment 1). After the auction, participants had to remain in the laboratory for an additional hour, as in experiment 1. At the end of the waiting period, participants were debriefed and thanked for their participation. Participants were paid £15 for their time, less the cost of the food item, if they bought any. The procedure of this experiment was approved by the University of Cambridge Psychology Research Ethics Committee (application no. Pre2014.113).
Because the aim of the experiment was to explore the relationship between confidence and value, it was essential that we had enough measurement sensitivity in both the confidence scale and in the value scale (the BDM ratings) and that participants’ choices reflected their stated preferences. We therefore excluded participants if any of the following criteria were met:
Participants used less than 25% of the BDM scale.
Participants gave exactly the same BDM rating for more than 25% of the items.
Participants used less than 25% of the confidence scale.
Participants gave exactly the same confidence rating for more than 25% of their choices.
Participant choices did not correspond to their BDM ratings (when predicting choices from differences in value, the DV coefficient deviated by more than 2 standard deviations (s.d.) from the experiment mean).
A total of 30 participants took part in the study. One participant did not complete the task and one participant was excluded because the BDM estimates were poor predictors of his choice (failed criterion 5). Thus 28 participants were included in the analysis (13 females, aged: 19–73). All participants were required to fast for 4 hours before taking part in the experiment. Blood glucose levels were taken to test their adherence to this criterion (mean glucose level = 83.57 mg dl−1, s.d. = 10.90 mg dl−1; by comparison, the mean fasting blood glucose levels for adults is 86.4 mg dl−1)55 . All participants gave informed consent before participating in this experiment.
Of the 30 participants who completed the study, three were excluded due to a limited range in their BDM ratings (failed criterion 2). An additional three participants were excluded for a limited range in their use of the confidence scale (failed criterion 4). In total, 24 participants were included in the main analyses (17 females, aged: 21–38). All participants were required to fast for 4 hours before doing the experiment. All participants gave informed consent before participating in this experiment.
Sample size was determined a priori. A power estimation was based on previously published work that used a similar experimental set-up26 . We implemented a fixed sample stopping rule set a priori (N = 30). Statistical inferences were conducted only after allof the data were collected. A participant who did not fulfil one of the exclusion criteria (decided before data collection) would have been excluded from the analysis without replacement.
For experiment 1, eye gaze was sampled at 250 Hz with a head-mounted SR Research Eyelink II eye-tracker (SR-Research). For experiment 2, eye movements were recorded at 1,000 Hz with an EyeLink 1000 Plus eye-tracker (SR-Research).
Preparation of the eye-tracking data
Areas of interest (AI) were defined by splitting the screen in half to create two equal-sized areas. Fixations in the left AI were assumed to be directed towards the left snack item and vice versa. We constructed two variables from the eye-tracking data: the DDT between the two AIs and GSF. DDT was calculated by subtracting the total dwell time on the left side from the total dwell time on the right side. GSF was calculated using the number of times that participants shifted their gaze from one AI to the other during each trial.
AIs were pre-defined by the three squares that participants had to fixate on to view the items (given the gaze-contingent design). We derived two variables from the eye-tracking data: the total dwell time in each AI for a given trial and GSF. Following experiment 1, GSF measured the number of fixations in one AI immediately followed by a fixation in another AI. To ensure that participants paid attention, we excluded trials where participants had not fixated on every option available at least once. Of the 3,457 trials, 13 were excluded from the analysis for this reason.
All of the hierarchical analyses reported in the results section were conducted using the lme4 package56 (version 1.1-7) in R. For the linear models, degrees of freedom and P values were obtained using the Kenward–Roger approximation, as implemented in the pbkrtest package57 . For the choice models (Fig. 1c,f), we ran two hierarchical logistic regressions: in experiment 1 we predicted the log odds ratio of picking the right-hand option on a given trial; for experiment 2 we predicted the log odds ratio of picking the reference item. The reference item was determined as the first item encountered according to reading order in Latin languages (that is, the upper-left item for the trials when an item was presented in that position and the upper right item for the remaining trials). Fixed-effect confidence intervals were estimated by multiplying the standard errors by 1.96 (ref. 58 ). Because these confidence intervals are estimates that do not take the covariance between parameters into account59 , they should not be interpreted too strictly, but rather serve to give the reader a sense of the precision of the fixed-effect coefficients. Note that all predictors reported are z-scored on the participant level and that all models allow for random slopes at the participant level. For completeness, we report coefficients from the full model, while noting that this model is not always the most parsimonious. For a comprehensive list of models tested and a formal model comparison using BIC scores see the Supplementary Information Section 3.
Note that the regression models for confidence in experiment 1 had issues converging. We addressed these issues by square-root-transforming the |DV| predictor. Notably, for the individual difference analyses that investigated the change of mind and transitivity, we did not implement hierarchical models but instead unpooled (individual-level) models. The rationale behind this choice was that for both analyses we were interested in studying between-subject variation (Fig. 3c and Fig. 4c) that could be potentially affected by the shrinkage of parameters towards the group mean that is characteristic of hierarchical models60.
The code for the analyses presented in this article can be found at the BDM Lab GitHub page: https://github.com/BDMLab.
How to cite this article: Folke, T., Jacobsen, C., Fleming, S. M. & De Martino, B. Explicit representation of confidence informs future value-based decisions. Nat. Hum. Behav. 1, 0002 (2016).
This work was supported by the Wellcome Trust and Royal Society (Henry Dale Fellowship no. 102612/Z/13/Z to B.D.M.) and the Economics and Social Research Council (PhD scholarship for T.F.). The funders had no role in the study design, the data collection and analysis, the decision to publish, or the preparation of the manuscript. We would like to thank Y. Yamamoto for sharing the methods he developed to rank choice in experiment 1 and C. Street and S. Bobadilla Suarez for help in collecting the data and pre-processing the eye-tracking raw data used in experiment 1. We also thank C. Ruff for suggesting an appropriate name for the GSF variable.
Supplementary Figures 1–7, Supplementary Tables 1–16, Supplementary Methods and Supplementary Results