The effects of recursive communication dynamics on belief updating

Abstract

Many social interactions are characterized by dynamic interplay, such that individuals exert reciprocal influence over each other's behaviours and beliefs. The present study investigated how the dynamics of reciprocal influence affect individual beliefs in a social context, over and above the information communicated in an interaction. To this end, we developed a simple social decision-making paradigm in which two people are asked to make perceptual judgments while receiving information about each other's decisions. In a Static condition, information about the partner only conveyed their initial, independent judgment. However, in a Dynamic condition, each individual saw the evolving belief of their partner as they learnt about and responded to the individual's own judgment. The results indicated that in both conditions, the majority of confidence adjustments were characterized by an abrupt change followed by smaller adjustments around an equilibrium, and that participants' confidence was used to arbitrate conflict (although deviating from Bayesian norm). Crucially, recursive interaction had systematic effects on belief change relative to the static baseline, magnifying confidence change when partners agreed and reducing confidence change when they disagreed. These findings indicate that during dynamic interactions—often a characteristic of real-life and online social contexts—information is collectively transformed rather than acted upon by individuals in isolation. Consequently, the output of social events is not only influenced by what the dyad knows but also by predictable recursive and self-reinforcing dynamics.

1. Introduction

People very often rely on others’ judgments to inform their decisions. We ask our family and friends for their opinions on a particular job we are planning to apply for, the colour of a new dress to buy, or the political candidate to favour. When information is costly to acquire, aggregating multiple samples of imprecise information is often more effective than spending time improving a single sample [1]. Aggregating beliefs from diverse and independent sources has long been known to improve judgment reliability [2,3]. How people incorporate others' judgments and advice into their decisions has been of central interest in social and organizational psychology, and more recently in computational social science [4,5]. One particularly fruitful approach to this question is the judge–advisor system paradigm [4], in which participants (the ‘judges’) form independent beliefs, then are asked to revise their initial judgment after seeing the belief of one or more advisors. This approach provides precise control over the information provided by the advice to participants, as well as precisely defined and distinct moments at which an initial decision is formed, advice is presented, and the judge's belief updated. Previous studies have highlighted the factors affecting how personal judgments are influenced by others' beliefs, like confidence [6,7], power [8], expertise [9], agreement among different sources [10], whether the self is involved [11], and how distant is the advice from the judge's own view [12,13].

However, in daily life, advisors' beliefs rarely remain independent from each other or from the advisee's own judgment, as is typically implemented in judge–advisor system studies. Instead, information is collectively transformed and manipulated until it converges to a decision. For example, you might say to the friend recommending the yellow dress that you actually don't like yellow, you like green; to which she replies that she thought green was kind of nice too and when, at that point, a member of staff at the shop interrupts to say that green is going to be next year's fashionable colour, a purchase is made. Here, the processes of decision, advice, and update take place all at the same time, and the line between judge and advisors blurs such that all partners affect each other's beliefs without a clear distinction between cause and effect. Thus, an important class of social interactions involves bidirectional information flow, as in face-to-face conversations and online chats, raising the possibility of recursive dynamics emerging among people's beliefs. Even online, where communication happens in discrete rather than continuous steps (like tweets or replies), interactions allow for a bidirectional back and forth between parties who are thus able to mutually adjust according to each other's positions. We characterize this class of recursive interactions as ‘dynamic’, as opposed to ‘static’ social exposure. While the latter has typically been investigated in micro-scale experimental studies of advice-taking, the former has been investigated within the opinion dynamics research program largely adopting macro-level analytic and simulation-based approaches [14,15].

As such, a full picture of social decision-making may require analysis of interaction dynamics, moving beyond the very valuable simplification of a participant working alone as an observer of social information (cf. [16–18]). Consistent with this reasoning, previous work found that individuals performing a task together become more confident and align their linguistic expressions when they are allowed face-to-face verbal interaction, but not when confidence is shared without verbal interaction [19,20]. This finding relates back to earlier work in social psychology on ‘risky shifts’ and group polarization [21–23]. Recently, attention has been devoted to understand belief polarization in more realistic contexts, like networks and social media platforms, due to the far-reaching consequences that these phenomena have in society [24–26]. Bail et al. [27] found that being exposed to disagreeing opinions can polarize individual beliefs even further. This is counterintuitive considering that, from a Bayesian perspective, disagreeing evidence should always decrease the strength of one's own beliefs. This finding may suggest that dynamic interactions among individuals might differ from static social exposure as often studied in the laboratory. Little is known about the direct comparison between uni- versus bidirectional social exchange. To address this issue, in the present study we investigate the effect of recursive social dynamics—typically seen in free social interactions—using a carefully controlled judge–advisor paradigm. We investigate differences between static versus dynamic interactions in social decision-making, specifically in a task in which the information available to participants to make a decision was kept constant across interactions, thus allowing us to characterize the specific impact of interaction dynamics themselves, above and beyond the information brought to the interaction by the individuals involved.

In our study, pairs of participants performed simple perceptual judgements in parallel (figure 1). On each trial, participants first made private, independent judgments (Confidence rating) about which of two boxes contained more dots [28] (Stimulus). They then were asked to continuously monitor and update their judgement, expressed on a semi-continuous confidence scale, based on observing the other participant's belief on the same scale (Social exchange). Each participant was thus both judge and advisor. Importantly, we compared conditions characterized by static information exchange—in which participants only saw their partner's initial judgment—with conditions characterized by dynamic information exchange—in which participants saw their partner's moment-by-moment belief change and thus how they themselves influenced their partner's views. By keeping decision-relevant information constant in the two conditions, we isolate the contrast between static and dynamically evolving social information, and thus characterize the way information is shared and collectively transformed between individuals. Based on previous findings—which show that face-to-face and other dynamic group interactions tend to escalate belief strength [20,22,23,25]—we expected that changing the nature of information exchange (dynamic versus static) would make the final judgments participants converge to after social exchange more extreme, even when matching decision evidence across the two conditions. Thus, contrary to previous literature, we tightly controlled the perceptual evidence accumulated during the private decision phase, which is the only information needed to successfully accomplish our task. Finally, feedback was provided at the end of the trial (Feedback).

Figure 1.

(a) Participants sat on opposite sides of a wooden occluder and used one monitor, one keyboard, and one mouse each, controlled by the same computer. (b) After seeing a perceptual stimulus (stimulus), participants made a judgment independently from one another (confidence rating). During the social exchange part, participants saw on their own scale their partner's initial belief (Static condition) or their partner's evolving belief (Dynamic condition) on alternating blocks. Bidirectional black arrows along the confidence scale represent real-time continuous movement along the scale. The scale used in the actual task had 50 levels per interval. Finally, during the feedback stage, participants received feedback on their binary accuracy (correct versus incorrect) and earnings (not shown) at the end of the trial (feedback). As shown in the figure, the feedback made clear which participant was correct or incorrect on a given trial by using the user-names that participants had to provide at the beginning of the experiment. Notice that although task difficulty was titrated to each participant's performance, correct answers (LEFT/RIGHT) were identical across participants. (Online version in colour.)

Three experiments (N = 72) were performed. Experiment 1 directly compares dynamic and static situations within dyads. We found that when participants are allowed to freely interact in real-time, recursivity in the belief update is observed. Experiments 2 and 3 (reported in electronic supplementary material) largely replicate the findings of Experiment 1, adding controls for potential confounding factors and showing robustness across different instructions and confidence scales. Experiments 2 and 3 tested a simple alternative explanation of the interactive effects reported below for Experiment 1, namely that participants quickly forgot their initial decisions and were only left with their current expressed beliefs as a basis for continued belief updating. We therefore included confidence anchors to remind each participant of the initial, independently expressed confidence in their own judgment (Experiment 2) and of their partner (Experiment 3). We largely replicated results found in Experiment 1, suggesting that this alternative explanation did not account for the data. The study was approved by the University of Oxford ethics board (MS-IDREC-C1-2015-075).

2. Methods

(a). Participants

Twenty-four dyads (N = 48, 37 females, age: M = 23 years, s.d. = 3.2) were recruited in pairs—potential volunteers were invited to bring a friend—for money or course credits. All participants gave informed consent before taking part in the study.

(b). Procedure

Participants sat on opposite sides of a desk divided by an occluding screen (figure 1a), each with a separate monitor, keyboard, and mouse. The pair performed a series of trials together, each consisting of a private perceptual decision phase followed by social exchange (figure 1b). During the private perceptual decision, participants in parallel performed a dot-count decision task [28]: two boxes containing dots randomly arrayed on a 20 × 20 grid were briefly (160 ms) flashed on each participant's screen to the left and right of a central fixation cross. Participants had to indicate which box contained most dots and their confidence in this decision. Thus beliefs were operationalized as signed confidence on a continuous scale, ranging from −50 (confident Left) to +50 (confident Right). Conversely, confidence was the absolute value of the belief, ranging from 1 to 50. The difficulty of the task was titrated to participants' perceptual threshold, using a staircase procedure that converged at 72.52% ± 0.013 accuracy [29]. As soon as both members confirmed their answer by pressing the space bar, the social exchange stage started, where each dyad member was informed about their partner's belief. At this point, belief updates were recorded continuously. In contrast to the standard judge––advisor system paradigm [4], where belief updates happen in a single step, here we recorded beliefs continuously as they evolved over time: the mouse x-position along the confidence scale was recorded every 200 ms for 5 s (4 s in Experiments 2 and 3). After this social exchange phase, the trial ended with feedback provided to both members of the dyad.

Our manipulation concerned only the social exchange stage. Two conditions alternated across blocks. In the Static condition, the choice and confidence level (the two together representing a ‘belief’) selected by each dyad member in the private phase appeared on their partner's scale as a static coloured cursor. Dyad members were at this point asked to (and were rewarded for) continuously monitor and update their confidence by moving their mouse along the scale. In the Dynamic condition, the social exchange part started exactly as in the static one, with each dyad member's cursor appearing on their partner's scale. However, and for the whole duration of the social part (5 s), if a member updated their belief, this would instantly appear also on their partner's scale and vice-versa. This led to a situation where participants were not only informed of their partner's original beliefs, but also how those beliefs changed in real-time as a function of their own belief changes (figure 1b). In both conditions, participants could update their decision and/or confidence level, thus potentially completely reversing their belief (e.g. switching from ‘certainly left’ to ‘certainly right’). The recursive nature of the Dynamic condition thus captured dynamics typically observed in formal approaches of opinion dynamics [14,15] (see electronic supplementary material for detailed methods).

3. Results

(a). Confidence changes asymmetrically for consensus and condition

During the social stage, the time-series of confidence changes was recorded continuously for 5 s after each participant's decision. The largest changes occurred about 1 s after first exposure to their partner's belief (electronic supplementary material, figure S1). Confidence updates were typically a single, unidirectional transition, towards increased confidence when partners agreed and reduced confidence when they disagreed (figure 2a). The influence that social information has on each dyad member is quantified by the distance δ_C between their post-exchange signed confidence and pre-exchange signed confidence, with larger absolute values corresponding to greater impact of social information. The sign of δ_C is expected to be positive in agreement trials and negative in disagreement trials. The range of possible confidence change however varies in agreement and disagreement: while in agreement the maximum absolute distance between two beliefs is 50 (remember our response scale had 50 points per interval), the maximum absolute distance between two disagreeing beliefs is 100, corresponding to an extreme situation when both participants are maximally confident, but on opposite intervals (e.g. one responding ‘sure LEFT’ the other ‘sure RIGHT’).

Figure 2.

(a) Average confidence change in agreement and disagreement trials, plotted for each condition. (b) Confidence change root density distribution in agreement and disagreement. The corresponding histogram or raw frequencies (no error bars) is shown in electronic supplementary material, figure S5. (c) Toy model emulating how confidence can increase in a Dynamic disagreement trial. y-axis represents belief as signed confidence where the sign represents a decision interval (LEFT/RIGHT) and the absolute value represents confidence. In a Static condition, the agents update their initial belief with their discounted partner's belief (dashed lines). In the dynamic interaction, the same update rule is applied on every iteration until equilibrium is reached (solid lines). (d) Effect of condition on the correlation between absolute confidence changes of the two participants across trials. Average Pearson's correlation coefficient is plotted as a function of consensus and condition. One dyad removed for a missing cell. Error bars represent s.e.m. (Online version in colour.)

A 2 × 2 ANOVA on confidence change with factors of consensus with respect to the initial, private decisions (disagreement versus agreement) and condition (static versus dynamic) revealed significant main effects of both consensus (F_1,47 = 150.26, p < 0.001, ${\eta }_G^2=0.7$) and condition (F_1,47 = 9.40, p < 0.01, ${\eta }_G^2=0.005$), but no significant interaction (F < 1). As expected, average δ_C was negative in disagreement and positive in agreement (figure 2b). The main effect of condition indicated that participants' final level of confidence was greater in the Dynamic condition than in the Static condition, with separate paired t-tests confirming this effect held regardless of consensus: participants increased their confidence more when in agreement (t₄₇ = 2.69, p = 0.009) and decreased their confidence less when in disagreement (t₄₇ = 2.08, p = 0.04), in dynamic compared to static blocks. The results indicate that belief change is dominated by the information content conveyed (i.e. agreement versus disagreement), with the nature of interaction (static versus dynamic) modulating rather than fundamentally altering this pattern. Nevertheless, the observed increase in confidence in the dynamic case is non-trivial when considered in some parts of the belief space (figure 3) and when considering that it represents an average including trials when participants did not change their confidence at all (figure 2b). Similar results were obtained when using a trial-level mixed regression approach (electronic supplementary material, table S1).

Figure 3.

Confidence change observed for every social situation. On a given trial, the x-axis represents the initial confidence of the trial-dominant member, and the y-axis the initial confidence of the trial-dominated member in relation to the former (+50 indicates confident agreement, −50 indicates confident disagreement). Horizontal and vertical coordinates thus represent the pre-exchange dyad state, while the colour (z-axis), represents confidence change. Panels a–c represent confidence change in dominant trials in Static condition, Dynamic condition, and their contrast, respectively. Panels d–f represent confidence change in dominated trials in Static condition, Dynamic condition, and their contrast, respectively. Panels c and f represent contrasts between the two conditions (contrasts (b,a) and (e,d), respectively). Regions of the space labelled ‘x’ and ‘y’ correspond to regions of the belief space where the effect of condition is strongest as described in the main text. Trials within each condition and dyad were linearly interpolated, due to data sparseness. Each panel contains 175 data points. (Online version in colour.)

Figure 2b shows the density distribution of confidence changes broken down by consensus (agreement versus disagreement). Both agreement and disagreement distributions peaked around zero, which was by far the most common change (notice that the y-axis in panel b uses a square root scale), indicating that very often participants ignored social information. Also of note is that on some disagreement trials participants actually increased their confidence, and agreement trials they decreased it. This is a surprising result if we consider that, from a normative (e.g. Bayesian) perspective, disagreement with an independent observer should always lead to a reduction of confidence and agreement should always lead to an increase of confidence (if we assume that the participant believes that their partner performs better than chance).

To explore this surprising pattern of confidence change more formally, a two-way repeated measures ANOVA was computed on the probability of an irrational confidence change, defined as confidence decreases in cases of agreement and confidence increases in cases of disagreement, again with factors of consensus and condition. To avoid including trials where increases/decreases in confidence were simply due to involuntary cursor movements (a ‘trembling hand’), we defined a confidence ‘change’ as a shift larger than 5 confidence points in the unexpected direction. The findings were, however, consistent across cut-offs greater than zero. Results show a significant effect of consensus (F_1,47 = 7.88, p = 0.007, ${\eta }_G^2=0.07$) but not of condition (F_1,47 = 2.56, p = 0.11, ${\eta }_G^2=0.001$) and a significant interaction between the two (F_1,47 = 9.90, p = 0.002, ${\eta }_G^2=0.005$), indicating that irrational confidence changes were 4.5 times more frequent in disagreement than agreement (0.018 versus 0.004 of trials) and that the Dynamic condition was 33% more likely to produce irrational confidence increases than the Static condition (0.020 versus 0.015 of trials, t₄₇ = 2.59, p = 0.01), as well as (numerically) 25% less likely to produce irrational confidence decreases (0.003 versus 0.005 of trials, t₄₇ = 1.98, p = 0.05). The interaction between condition and consensus was however not replicated in Experiment 2 (electronic supplementary material), indicating that the result was not robust to changes in the use of the confidence scale.

Although not normatively prescribed (e.g. in a Bayesian framework), belief aggregation strategies described in the literature [3,30,31] can explain irrational decreases in confidence in cases of agreement. For example, averaging of confidence would lead to this outcome when a partner agrees but is much less confident than the participant, such that the participant concludes that they should not have been so confident in the first place. Of more interest, therefore, are irrational increases in confidence after disagreement, which occurred more frequently than irrational decreases but which are difficult to reconcile with any obvious confidence-update strategy. We notice that this irrationality could occur through recursive dynamics introduced by real-time interaction. Consider an example trial in which a participant starts off on a confidence level of 6 while their partner weakly disagrees with a confidence level of -4 (the negative sign indicates disagreement). Suppose next that both participants use a simple update strategy, namely summing their own initial signed confidence with their partner's weighted signed confidence (here: weight = 0.80) (cf. [31]). In a situation without recursive interaction, participants can only use their partner's initial belief to update their beliefs (dashed lines). If recursive dynamics are introduced, each participant can use his/her partner's current belief to update their own. Figure 2c shows that this simple strategy leads to an oscillatory update (solid lines) that stabilizes for the more confident participant (in red) on a higher confidence (distance from 0) than initially held. The effect reflects the fact that if the low-confident partner (in blue) crosses the decision boundary, disagreement turns into agreement—thus supporting one's initial belief—instead of providing contradictory evidence, and therefore leads to an increase in confidence (electronic supplementary material, figure S9).

To test for recursive dynamics in our behavioural data, we counted, for each condition, the average number of vacillations in a trial, namely the number of times the direction of the update (i.e. stationary/increase/decrease) changed in the update window (see electronic supplementary material). Supporting our intuitions, we found that both the average number of vacillations in a trial and the total number of irrational increases were significantly more frequent in the dynamic than Static condition. Thus, irrational increases in confidence could have arisen because participants treated the observed updates of their partner (influenced by their own judgment) as if they were independent evidence. Figures S6–S9 show individual trial trajectories in belief space, including agreement and disagreement, vacillations, and irrational confidence increases.

(b). Dyadic interactions in belief space

The preceding analyses explore belief change when the data are aggregated across broad categories (e.g. agreement versus disagreement trials). To explore more nuanced behaviour as a function of participants' absolute and relative levels of confidence, we explored our data as a function of a two-dimensional ‘belief space’ [31] as shown in figure 3. The figure plots confidence change following interaction within this space. Here, the x-axis value indicates the confidence of whichever participant in the dyad is the more confident on any given trial in the private judgment phase, henceforth the ‘dominant’ member on the trial, thus ranging from 1 (minimum confidence) to 50 (maximum confidence). The y-axis value gives the confidence of the less confident, or ‘dominated’, member of the dyad in their initial judgment on the trial, on a scale ranging from −50 (disagreement with maximum confidence) to 50 (agreement with maximum confidence). This plot creates a grid of possible social situations in which the dyad's state—both members' choices and their confidence—is fully represented by a pair of coordinates, while collapsing across the particular side of participants’ choices (left versus right box). In figure 3, pixel colour indicates the median change in confidence from pre- to post-exchange of the dominant (upper panels) and dominated (lower panels) member of the dyad on each trial. The trial-dominant and dominated participants' confidence change can be combined into a single vector field (electronic supplementary material, figure S2) visualizing dyadic transitions in state space [31]. Dynamic animations of confidence transition in this space for each condition are provided via an Open Science Framework (OSF). Figure 3 shows once again the overall increase in confidence seen in dynamic versus static interactions, with dynamic interaction characterized by greater increases in confidence when partners agreed (cf. larger red area in the upper half of figure 3b than 3a) and smaller decreases when they disagreed (cf. smaller blue area in lower half of figure 3b than 3a). The contrast plots (figure 3c,f) more precisely identify the locus of these between-condition differences, while also highlighting the magnitude of the effects in certain conditions. Thus, dynamic interaction leads to particularly marked confidence increases when partners began the social stage in agreement but with low confidence (points marked ‘x’ in figure 3). Confidence change in the Dynamic condition (panel b,e) in these conditions of uncertain agreement is 20–30 confidence points, and approximately 15 points greater than in static interaction blocks. Thus, when interacting dynamically, but not statically, two uncertain partners tended to reinforce each other's belief so that together they converged on the maximum possible confidence level. The other key point of interest in the contrast plots lies in the disagreement half of the belief space, specifically at the points labelled ‘y’ in figure 3c,f. These were trials in which the dominant member was highly confident and the dominated member weakly disagreed, a situation described in our simple simulation above. The warmer colour at ‘y’ in figure 3c indicates that disagreement had markedly less impact on the dominant partner's confidence when the dyad interacted dynamically rather than statically. The corresponding point in figure 3f indicates that, similarly, larger shifts in the trial-dominated participant's confidence toward the trial-dominant individual's position were observed in dynamic compared to static blocks. Overall, therefore, this belief space analysis identifies the conditions under which dynamic interaction has its largest impact—when partners agree with symmetric low levels of confidence, and when they disagree with asymmetric levels of confidence—and shows that this impact is substantial in these conditions.

(c). Coupling of confidence changes during interaction

The analyses above consider belief change for each dyad member separately. To investigate how the magnitude of partners' belief changes co-varied across trials, we calculated across-trial correlations of absolute confidence change between initial and final confidence (δ_C) between the two members of each dyad. Pearson's r coefficients were calculated for each dyad as a function of dyadic consensus (agree versus disagree) and interaction condition (dynamic versus static). A 2 × 2 ANOVA on the resulting values across dyads (figure 2d) revealed a main effect of consensus (F_1,22 = 20.93, p < 0.001) but not of condition (F_1,22 = 1.71, p = 0.20), and a significant interaction between the two (F_1,22 = 38.39, p < 0.001). When dyad members did not see each other's confidence changes in the social exchange stage (Static condition), confidence changes did not correlate significantly between partners (h₁ = r > 0, t₂₂ < 1.28, p > 0.2, d < 0.26). This finding is not unexpected, but nor is it trivial—for example, a positive correlation might be expected in agreement trials as a consequence of a boost to both participants' confidence when they agreed but were initially uncertain, such that agreement led to increased confidence for both. This did not seem to occur. By contrast, in the Dynamic condition, partners’ confidence changes became coupled: in agreement trials, the correlation was positive, indicating that the more one member increased their confidence, the more their partner also increased their confidence. In disagreement, the correlation was negative, indicating that the more one member decreased their confidence in their initial decision, the less the partner decreased theirs. Pairwise contrasts showed that, compared to the Static condition, in the Dynamic condition correlation coefficients were significantly greater for agreement (t₂₂ = 4.89, p < 0.001, d = 1.20) and somewhat smaller for disagreement (t₂₂ = 2.02, p = 0.05, d = 0.52). The negative correlation found in disagreement dynamic trials, was replicated in Experiment 3, but not in Experiment 2 (electronic supplementary material). Further analyses (electronic supplementary material) showed that the effect could not be explained by participants using their partner's reaction times (i.e. ‘unwillingness to move’) in the dynamic but not in the Static condition. Coupling of confidence changes in interaction suggests that participants were influenced by their partner's confidence change when updating their own belief, rather than basing their change solely on their partner's initial (independent) judgment as normatively prescribed.

4. Discussion

The present study compared social exchange involving static, one-step communication with exchange characterized by dynamic and recursive interaction. We hypothesized that real-time recursive dynamics, which characterize many daily-life interactions of social influence [18], would have systematic impact on decisions made in a social context, over and above the effects of the information brought by each individual to the interaction, as studied in traditional judge–advisor systems [4,32]. Across conditions with equal information available—because in both the dynamic and static conditions, dyad members viewed perceptual evidence separately and for 160 ms only—we observed different belief aggregation strategies according to the nature of communication between partners. Specifically, dynamic interaction produced higher confidence changes in agreement and smaller confidence changes in disagreement by breaking the independence of dyad members' beliefs: confidence changes of the two participants became correlated during dynamic interaction compared to a static baseline so that, in agreement, greater increases in confidence for one member were associated with greater confidence increases for their partner, leading to belief escalation. In disagreement, greater changes in confidence for one member were associated with smaller changes in confidence for their partner, reducing the impact of disagreement on belief updating. These combined results can be understood in terms of individuals making use of their partner's change in confidence to update their belief, without taking into account that this change was biased (and indeed generated) by exposure to their own judgment. People are known to quickly reach decisions even when information is scarce by taking into account a host of circumstantial variables that are known to co-vary with problem-specific evidence, but that are not themselves strictly task-relevant [33]. Interpreting someone's changes of mind as another cue for confidence is sensible in many daily-life situations. Indeed, if somebody's beliefs are fickle, we have reasons to believe s/he must be uncertain. In the case of social interaction, however, this heuristic leads to potentially sub-optimal self-reinforcing dynamics, when a person uses the impact of his/her own belief on the other person as evidence for the belief itself. The impact of this circular reasoning was particularly marked for low confidence agreement. In these cases, dyads often escalated together towards maximal confidence in their beliefs.

This micro-level effect may provide some insights into poorly understood group phenomena of certainty escalation and ‘confidence effects’. Confidence increases are observed when individuals are allowed to freely communicate in groups and the more people are exposed to social information, with no improvements or even damaging effects on accuracy [20,34–36]. The effect of group polarization has long been studied, and has gained renewed attention and meaning in the context of online interactions [23,24,27]. The present study adds to this growing body of evidence by showing that a potential cause of belief escalation is recursive interaction. When the interaction allows for recursive dynamics, the use of redundant task-irrelevant information becomes more likely. People should use only each other's independent beliefs to arrive at a final decision, because this is the only information that carries task-relevant value. However, they also (incorrectly) use how much their own belief is affecting their partners. This strategy creates dependencies that can potentially create escalation dynamics. Our dynamic model of belief update, even though based on dyadic interaction and highly simplified (figure 2c), shares important features with models of opinion dynamics—including the operationalization of belief and belief update as signed one-dimensional continuous values—which use formal models drawn from engineering and physics to study the nonlinear properties of a network of individual nodes holding beliefs. The study of these systems, though less complex than real societies, has nevertheless proven valuable for social scientists interested in emerging macroscopic influence dynamics [14,15,37]. We add to this literature by showing how recursive dynamics can be empirically captured in simple experimental paradigms and cognitive models of belief update.

Beyond belief escalation as described above, a more subtle but nevertheless consistent effect of bidirectional dyadic influence in our data was that participants occasionally increased their confidence despite initially learning that their partner disagreed with them. This surprising irrationality occurred more frequently than the mirror effect of decreasing confidence when a partner agreed. Similar phenomena have been observed in the context of political beliefs in online environments [27]. Our results show that the effect was more common in dynamic rather than static trials, and suggest a possible low-level mechanism for it. This phenomenon cannot easily be explained by static aggregation rules like averaging, summing, or Bayesian integration [3,30,31], but is predicted by a recursive update model. Importantly, rather than providing an exact description of participants' behaviour, our model aimed to broadly capture the intuition that others’ changes of mind can sometimes be perceived as supporting evidence for one's views. When a partner initially disagrees, but loses confidence in this view (or even reverses it) on subsequently learning the participant's view, the participant can take this as evidence in favour of their initial position, and therefore increase their confidence.

Why do people make this mistake? Our interpretation implies that people seem to have limited ability to assess the independence of evidence, a conclusion that converges with previous findings using the judge–advisor system paradigm [38]. Another possibility is that when making joint decisions, we reduce our individual responsibility [39]. Feeling less responsible, people might afford to be more confident. A related social context explanation might apply to our observation that confidence actually increases on some disagreement trials: this phenomenon might occur because making a good decision is not the only goal of the decision-maker. Another aim might be winning an argument [40]. Once the player knows that their ‘rival’ is less confident, they will be encouraged to prevail by increasing further their confidence. However, due to the non-verbal perceptual features of the task, we expect argumentative reasons to have less weight here. Future studies should investigate whether irrational confidence increases are also observed in disagreement when logical and linguistic arguments are exchanged.

Importantly, notwithstanding these differences in confidence updating across static versus dynamic interactions, participants' overall accuracy showed a consistent benefit from social information exchange. A significant effect of decision stage (pre- versus post-interaction) was found on both accuracy (F_1,47 = 47.00, p < 0.001, ${\eta }_G^2=0.16$) and confidence calibration (F_1,47 = 89.58, p < 0.001, ${\eta }_G^2=0.25$) (see electronic supplementary material information for details). The benefits of social exchange were of similar magnitude across static and dynamic interaction conditions, whether this benefit was measured in terms of overall accuracy, confidence in the correct answer, or the calibration between confidence and objective performance (see electronic supplementary material). This parity was observed despite different interaction dynamics across conditions, at least in part because these dynamics led to opposing effects on accuracy across trials. A standing question is whether interaction might amplify errors in the presence of systematic biases across members (e.g. [31]), due to belief escalation in incorrect answers [41,42].

In the electronic supplementary material, we compare empirically observed behaviour with an optimal model grounded in a probabilistic interpretation of confidence [43], and show participants clearly departed from the Bayesian strategy commonly explored in the belief aggregation literature [30,44], in which beliefs are combined in relation to expressed confidence. We used inverse Bayes theorem to compare the objective information conveyed from social partners with the information ‘perceived’ by a participant. Results evidenced that social information was distorted by the receiver in a self-serving manner and asymmetrically for agreement and disagreement (electronic supplementary material, figure S3), in line with the advice-taking literature suggesting the presence of egocentric and confirmation biases [45]. More specifically, the weighting of partner's information (i.e. use of social information) seemed to follow a bimodal distribution, with peaks around 50%, corresponding to uninformative social evidence, and 100%, corresponding to maximally supporting social evidence. These results are consistent with the hypothesis that people are solving a categorical inference problem (cf. [46]). Instead of using continuous social information as it is provided by their social partners, participants seem to classify each trial as ‘partner is wrong’ versus ‘partner is correct’ and, once this categorization is performed, use social information accordingly to update their views. Accordingly, participants would try to minimize situations of uncertainty (e.g. 0.25 or 0.75 evidence), thus maximizing the impact of social information on final confidence. Notice that this is in stark opposition with traditional opinion dynamics models of social contagion, where belief updates are modelled as a linear combination of self and others' beliefs [14,47].

Collectively, these findings show that social influence depends not only on private beliefs—here, the only task-relevant information—but also on the modality in which information is shared and transformed across individuals. In the aggregate, the impact of recursive dynamics is subtle but consistent, evident as a general increase in confidence in decisions made. However, the impact is very marked in specific situations, notably when shared but uncertain beliefs become mutually reinforced to a state of near certainty, and when a decision-maker interprets vacillation in a partner's weak disagreement as positive evidence for their views. The relevance of these basic dynamics might extend beyond human groups, to include other social animals, for example, in the case of collective motion direction [48]. Our findings contribute to the debate on group polarization in online and physical environments by providing a fine-grained description of within-subject belief dynamics in recursive and static social exchanges. Real-time interaction in many daily social situations is recursive in nature. Effective interventions aimed at reducing belief escalation online and offline will require a cognitive-level understanding of these dynamics.

Ethics

The study was approved by the University of Oxford's ethics committee with Ethics Approval Ref: MS-IDREC-C1-2015-075.

Data accessibility

Data available from osf.io/7b6py [49].

Authors' contributions

N.P. and N.Y. designed the study and drafted the manuscript. N.P. collected the data, performed the data analysis, and curated data availability and reproducibility. N.Y. supervised the overall data collection and analysis.

Competing interests

We declare we have no competing interests.

Funding

This work was completed with the support of the Clarendon Fund, Christ Church College and the Department of Experimental Psychology of the University of Oxford.