Optimally interacting minds

Main Text

To come to an optimal joint decision, individuals must share information with each other, and, importantly, weigh that information by its reliability (1, 2). It is well established that isolated individuals can accurately weigh information when combining different sources of sensory information (3-5). Little is known, however, about how – or even whether – two individuals can accurately combine information that they communicate with each other. To investigate this issue, we examined the behaviour of pairs of individuals in a simple perceptual decision task, and we asked how signals from the same sensory modality (vision) in the brains of two different individuals could be combined through social interaction.

Work on perceptual decision-making has shown that when combining information from different senses, individuals have access not just to magnitudes of sensory signals, but also to their probability distributions, or at least to their means and variances (3-8). This may not, however, be true for interpersonal communication. While probability distributions arising from different sensory modalities are available within an individual’s brain, it is not clear whether such distributions can be passed directly to another person or indeed what types of information can be communicated. To answer this, we considered four models (16), each of which postulated that different types of information could be communicated, and quantitatively compared the predictions of those models to empirical data in a low-level visual decision making task.

The first model proposes that nothing except the decision about the visual stimulus is communicated, and when there is disagreement, the joint decision is no better than a coin flip (CF model). This strategy is expected from previous work on collective decision making without feedback (9). The second model proposes that nothing except the decision is communicated, but that pairs of individuals learn, from trial-to-trial feedback, which of them is more accurate, and so eventually use that individual’s decisions (the behaviour and feedback, or BF, model). This model was motivated by previous work showing that collective decisions are dominated by the most competent group member in situations where clear feedback about “the truth” (in our case, the correct answer) is available (10, 11). The third model, put forward here for the first time, proposes that confidence, which we define as an internal estimate of the probability of being correct (17), is communicated (the weighted confidence sharing, or WCS, model) (16). Finally, the fourth model proposes that the mean and standard deviation of the sensory response to the stimulus about which the decision is made are communicated. This model is used to account for multisensory integration within an individual (3, 4), and also for collective decisions in groups (12). To anticipate our findings, we found that the weighted confidence sharing model was quantitatively consistent with our empirical data, whereas the other three models were not.

Our empirical data were obtained from pairs of participants (dyads) who viewed brief visual displays containing a faint target (contrast oddball; Figure 1A) in either the first or second viewing interval (16). We performed a series of four experiments, each of which followed very similar procedures. Initially, each participant chose the interval they thought contained the target, without consulting the other. Individual decisions were then shared, and if participants disagreed, they discussed the matter until they reached a joint decision. Subsequently both participants were told the correct choice (with the exception of experiment 4 where no feedback was given). Individual and dyad psychometric functions (Figure 1B, left and middle panels) were fit with a cumulative Gaussian function, from which we extracted the slope, denoted s. The slope provided an estimate of sensitivity (the steeper the slope, the higher the sensitivity). More sensitive observers were, by definition, more reliable in their estimates of contrast.

The four models made different predictions for the relationship between the slope of the psychometric function for each individual and the collective dyad, and so by comparing predicted and observed dyad slopes we could distinguish the models. For each of the four models (16) we computed the predicted dyad slopes, $S_{dyad}^{model}$, in terms of the individual slopes, s₁ and s₂, of observers 1 and 2. For the coin flip model, the predicted dyad slope is related to the individual slopes by

for the behaviour and feedback model,

for the weighted confidence sharing model,

and for the direct signal sharing model,

These equations provide upper bounds on performance for each model; e.g., Eq. 3 provides the largest possible dyad slope given that participants share only confidence. If the dyads reach that slope, then they are Bayes optimal given the model assumptions, where by “Bayes optimal” we mean that participants made decisions that maximized their probability of being correct given their model assumptions.

In Figure 1C we plot the predictions (from Eqs. 1-4) for the collective benefit (the ratio s_dyad/s_max) versus relative sensitivity (s_min/s_max), where s_min and s_max are the minimum (i.e. less sensitive) and maximum (i.e. more sensitive) of the individual slopes, respectively. The models clearly make different predictions, but to distinguish them requires experiments with a broad range of s_min/s_max: we would need to investigate dyad members with nearly identical performance (s_min/s_max ~ 1) as well as those with very different performance (s_min/s_max << 1). Experiments 1 and 2 were performed to test the model predictions in different ranges of s_min/s_max.

In Experiment 1, participants viewed identical stimuli, and individual sensitivities of the dyad members were similar (s_min/s_max> 0.5; Figure 2B). The coin flip model (Eq. 1; Figure 1C, dot-dashed line) predicted that dyad sensitivity would never be higher than that of the better participant. The behaviour and feedback model (Eq. 2; Figure 1C, solid black line) predicted that dyad sensitivity would be as good as the better participant. In contrast, the weighted confidence sharing model (Eq. 3; Figure 1C red line) and direct signal sharing model (Eq. 4; Figure 1C green curve) both predicted that, within the relative sensitivity range tested here (i.e. s_min/s_max>0.5), dyad sensitivity would be higher than that of the better participant.

We found that the dyad slope was significantly larger than that of the better participant (t(14) = 5.24, p < 10⁻³; paired t-test). These data thus ruled out both the coin flip (Figure 2A, p < 10⁻⁵) and behaviour and feedback (Figure 2A; p < 10⁻³) models, for which the dyad slope can be no larger than that of the better participant, and instead favoured the sharing models (p > 0.1), for which the dyad can outperform the individuals. The sharing models were also able to accurately predict, via Eqs. 3 and 4, the dyad slopes on a case-by-case basis (Supplementary Figure S1). Thus, communication conferred a significant benefit and, at least on this task, two heads did better than one.

Experiment 1 favoured the weighted confidence sharing and direct signal sharing models, but was not able to distinguish between them. For the range of relative sensitivities tested in Experiment 1, the two models made very similar predictions (Figure 2B). To distinguish the models, we now sought to study dyads with very different individual sensitivities (s_min/s_max << 1) for which the weighted confidence sharing model (Figure 1C, red line) made a counterintuitive prediction: if one participant’s sensitivity was no better than ~40% of the other (e.g. s_min/s_max < 2^1/2-1 ≈ 0.4), then two heads should do worse than the better one (s_dyad/s_max < 1) – even when individuals accurately communicated their confidence. In contrast, the direct signal sharing model (Figure 1C, green curve) invariably predicted a benefit for dyads, consistent with the fact that when signals are directly available (as in multisensory integration within a single brain), putting them together is never worse than either one alone (4).

We tested these predictions in Experiment 2. In randomly chosen trials we surreptitiously reduced one or the other (or both) participants’ sensitivity by adding a substantial amount of noise to their stimuli (16) without having told the participants about this manipulation. The four noise regimes were randomized, so on each trial noise was given either to both participants (‘Equal’ condition), to one but not the other (both possibilities combined together as ‘Unequal’ condition), or to neither participant (‘None’ condition). For each participant and dyad, four psychometric functions – corresponding to the four noise regimes – were constructed and the slopes were estimated (Supplementary Figure S2). This showed (Figure 3) that in Equal and None conditions – where participants received identical amounts of noise – robust group benefits were obtained (Figure 3A, for Equal condition, t(10)=2.50, p=0.03; paired t-test; for None condition, t(10)=3.38, p=0.007; paired t-test). This replicated the results of Experiment 1. However, in the Unequal condition dyads did not do better than the better participant, and reliable group benefit was not observed (Figure 3A, t(21)=0.68, p=0.54; paired t-test).

In all three conditions, the results were consistent with the predictions of the weighted confidence sharing model (Figure 3B). Importantly, the majority of the data points for which s_min/s_max < 0.4 fell below the horizontal black line in Figure 3D indicating that, in these cases, two heads did worse than the better one. The direct signal sharing model, on the other hand, was rejected in the Unequal condition (Figure 3C, t(21)=4.52, p= p<10⁻³; paired t-test). Moreover, randomized addition of noise resulted in a wide range of relative sensitivity and a highly significant linear correlation was observed between collective benefit and relative sensitivity (Figure 3D; dotted blue line R² = 0.51; F(42,1) = 43.22; p<10⁻⁷) with slope (0.6±0.09) and intercept (0.74±0.05) that were remarkably close to the slope (1/2^1/2 ≈ 0.71) and intercept (1/2^1/2 ≈ 0.71) predicted by the weighted confidence sharing model.

In these experiments, two aspects of social information contributed to collective decision making: communication and feedback. The experiments, however, could not tell us whether either or indeed both types of information were necessary for collective benefit in sensitivity. To address this issue, we next conducted two more experiments. Experiment 3, tested whether communication was necessary while Experiment 4 tested whether feedback was necessary. We found that communication was necessary but, surprisingly, feedback was not.

It is conceivable that, even if the participants were not able to communicate their confidence on each trial, they would still be able to estimate each other’s average reliability (defined explicitly as the slope of the psychometric curves; see Figure 1B) – not through direct trial-by-trial interaction and confidence sharing, but by accumulating information about one another’s accuracy through feedback. Armed with such an estimate, dyads might conceivably be able to match the performance of those that did communicate, and so match the performance of the weighted confidence sharing model. On theoretical grounds, we did not expect this; instead, we expected performance without communication to match the behaviour and feedback model. We hypothesised that trial by trial communication was necessary and that feedback alone would not be sufficient for achieving collective benefit.

Experiment 3 tested this prediction, using the same paradigm as Experiment 1 but now modified so that participants were now not allowed to communicate anything but their choice. Whenever the participants disagreed in their decision, one of the two participants (chosen randomly by the computer) made a decision individually by arbitrating between their own choice and that of the other participant. Feedback about correct choice was then given to both participants. (16). The results were unequivocal. Contrary to Experiment 1, dyad sensitivity now did not exceed that of the more sensitive observer (Figure 4A, red bar; t(13)=0.18; p=0.85, paired t-test) as predicted by the behaviour and feedback model. More importantly, dyad sensitivity was significantly lower than the upper bound predicted by the weighted confidence sharing model (Figure 4B, red bar; t(13)=5.91; p<10⁻⁴, paired t-test), demonstrating that knowledge of current choice and previous outcomes was not adequate for the dyads to reach the level of performance observed in Experiment 1, expected from the weighted confidence sharing model.

Experiment 3 showed that communication was necessary and that feedback alone was not sufficient for dyads to achieve a collaboration benefit. However, the results do not address the question whether communication alone, without feedback, is sufficient for achieving collaboration benefit. Could dyads achieve any group benefit at all without ever receiving any objective feedback about the accuracy of their decisions? This is an important question, since feedback is not formally incorporated in the confidence-sharing model (16). Taking this model seriously at face value, one may draw the extremely counterintuitive prediction that as long as accurate communication of confidence is ensured, dyad benefit can still be achieved without any feedback i.e. without any definitive knowledge of decision outcomes.

To test this, in Experiment 4 we now removed the feedback stage of the task to test this prediction (16); i.e., after the joint decision was made (either automatically in the agreement trials or after interaction in the disagreement trials), the participants were not told the correct answer. All other aspects of the experiment were identical to Experiment 1. Consistent with our prediction, even without feedback, the dyads nevertheless achieved a significant collaboration benefit (Figure 4A, blue bar; t(10)=2.68; p=0.022, paired t-test) and dyad sensitivity was statistically indistinguishable from the prediction of the confidence sharing model (Figure 4B, blue bar; t(10)=1.16; p=0.27, paired t-test). This indicates that, indeed, objective feedback was not necessary and communication alone was sufficient for achieving collective benefit.

Our results show that interactive decision making between two individuals can significantly improve perceptual sensitivity – but, importantly, only for similarly sensitive observers. Moreover, such joint behaviour is Bayes optimal under the assumption that participants accurately communicate their internal estimate that they are correct (where, as mentioned above, “Bayes optimal” means the participants make decisions that maximize the probability of a correct choice under their model). Our findings show that human-to-human interpersonal communication is adequately rich to permit sharing of subjective estimates of confidence, and humans are adequately perceptive to make optimal use of this information. Moreover, communication of trial by trial confidence is necessary for collective benefit, but, somewhat surprisingly, feedback about decision outcomes is not.

Quantitatively, we tested four models, and only one – the weighted confidence sharing model, in which participants communicated only an internal estimate of their reliability on each trial – was consistent with the data. Of the three models that were not consistent with the data, one, the direct signal sharing model, posited that participants communicated both the perceived contrast and their estimate of its reliability on each trial. That model was rejected because it outperformed the dyads in Experiment 2. This leaves open the possibility that the participants did communicate contrast and reliability, but used that information suboptimally. This seems unlikely as we never observed any dyads explicitly communicating contrast and reliability separately. However, our data cannot rule this out and further research is needed to distinguish between optimal use of weighted confidence sharing versus suboptimal direct signal sharing.

The general consensus from extensive earlier work on collective decision making is that groups rarely outperform their best members (10, 13). Even in one of the rare cases where consistent collaborative benefit was established, group performance failed to reach the bound predicted by the proposed ideal combination of individual decisions (12). That study, however, employed the direct signal sharing model (see Eq. 4) to estimate the ideal, expected group sensitivity. As shown in Experiments 1 and 2, however, the predictions of that model deviate significantly from empirical data if individuals’ sensitivities differ markedly. In particular, experiment 2 demonstrated the detrimental side-effect of collective decision making based on Bayesian combination of confidence: individuals with very different sensitivities are best advised to avoid collaboration, and instead should rely entirely on the more sensitive individual. In fact, the weighted confidence sharing model and the results of Experiment 2 (Figure 3D) set a quantitative limit on the usefulness of cooperation that, to our knowledge, is not predicted by current economic and social theories of collective decision making (13). An important question for future research is to test the generality of this limit in other types of dyadic interactions.

These findings have direct bearing on studies in social psychology that have discovered numerous situations where groups fail to do better than their individuals. Many explanations for such ‘process loss’ have been suggested, such as reduced effort in the presence of others (e.g. ‘social loafing’ (14)), interpersonal competition (10) and groupthink (15). Our results raise the rather different possibility that, when the communicated evidence (i.e. perceived contrast) cannot be separated from its reliability (i.e. slope), then such failures of collective decision-making may be the natural consequence of a perfectly reasonable strategy (e.g. weighted confidence sharing). Indeed, we know all too well about the catastrophic consequences of consulting “evidence” of unknown reliability on problems as diverse as the existence of weapons of mass destruction and the possibility of risk free investments.

Supplementary Material