Quorums enable optimal pooling of independent judgements in biological systems

Effective decision-making is essential in all areas of human society and, more generally, for all organisms. A fundamental question in this context is when a group of decision-makers is superior to individual decision-makers and vice-versa (Galton, 1907; Surowiecki, 2005; Bahrami et al., 2010; Lorenz et al., 2011; Koriat, 2012; Kurvers et al., 2016). Both in human and animal collective decision-making, the Condorcet Jury Theorem is one of the key principles guiding our thinking about this question (List, 2004; Hastie and Kameda, 2005; King and Cowlishaw, 2007; Sumpter et al., 2008; Austen-Smith and Feddersen, 2009; Conradt and List, 2009; Kao and Couzin, 2014a; Marshall et al., 2017). In a nutshell, for pairwise decision problems (e.g. disease-, lie- or predator detection), Condorcet’s Jury Theorem states that a group of decision-makers employing the majority rule is superior to individual decision-makers in contexts where individuals are relatively accurate (i.e. accuracy >50%); conversely, individual decision-makers are superior to majority voting groups in contexts where individuals are relatively inaccurate (i.e. accuracy <50%). Consequently, across diverse fields ranging from organismal behaviour and human psychology to political sciences, two heuristics are commonly employed (List, 2004; Hastie and Kameda, 2005; King and Cowlishaw, 2007; Sumpter et al., 2008; Conradt and List, 2009; Kao and Couzin, 2014a): (i) groups of decision-makers outperform individuals only in contexts where individuals are relatively accurate and (ii) the majority rule is a powerful mechanism to reap the benefits of collective decision-making. We here show that both statements are not true, and in doing so explain the ubiquity of quorum decision rules in the collective behaviour of humans and other social organisms (Seeley and Visscher, 2004; Sumpter and Pratt, 2009; Ward et al., 2012; Pratt et al., 2002; Ross-Gillespie and Kümmerli, 2014; Walker et al., 2017; Bousquet et al., 2011).

Over the past few decades, substantial research effort has focussed on two key explicit assumptions underlying Condorcet’s Jury Theorem, independence (i.e. judgments/votes by different members of the group are assumed to be independent from each other) and homogeneity (i.e. all decision-makers within a group are assumed to be identical, both in competence and in goals) (Kao and Couzin, 2014a; Sumpter and Pratt, 2009; Boland, 1989; Ladha, 1992; Berg, 1993; Marshall et al., 2017). We here focus on a third, implicit, assumption of Condorcet’s Jury Theorem, namely that decision-makers make only one type of error. This assumption stands in contradiction to the well-known fact that, when confronted with a pairwise decision problem like the one studied in Condorcet’s Jury Theorem, two different types of error are possible (i.e. false positive and false negative). Surprisingly, up to now, this basic and well-known feature of pairwise decision problems has not been fully taken into account when investigating Condorcet’s Jury Theorem.

We start by providing a brief summary of the basic model considered in Condorcet’s Jury Theorem. We then introduce an extended model that takes into account the fact that decision-makers can make two types of errors. Based on this extended model, we then show that Condorcet’s Jury Theorem makes several important predictive errors, which apply to the majority of decision scenarios. Moreover, majority voting is frequently suboptimal, whereas quorum-based voting with an appropriate quorum is always optimal, in that it enables groups to simultaneously maximise true positive rate and minimise false positive rate (Wolf et al., 2013). While an analytical solution for the optimal quorum threshold has been derived before (Ben-Yashar and Nitzan, 1997), dependent on true and false positive rates and key ecological characteristics (i.e. classification error cost, prior probabilities), this analysis treated true and false positive rates as independent of these ecological characteristics, whereas in reality the former depend on the latter. In contrast, here we also make use of signal detection theory to optimise individual decision-makers, thereby delineating precisely where non-majority quorums should be used, depending on parameters of the decision ecology. Thus, the simple majority threshold is only a special case of the more general quorum decision mechanism, in which optimal sub-majority or super-majority quorums are the rule rather than the exception.

We are now ready to formalise when Condorcet gives incorrect predictions, and when simple majority voting is suboptimal. We begin by determining optimal values of a₊ and a_–. Given the prior probability p, the cost and benefits associated with the two states of the world, and the state-dependent characteristics of the cue(s) decision-makers base their decisions on, the optimal realised individual accuracies a₊ and a_– are derived by solving a signal detection problem (Figure 1). Signal detection theory has been applied repeatedly to hypothetical and real world decision problems, for example predator detection by foraging animals (Trimmer et al., 2008; Trimmer et al., 2017); diagnostic decision-making (Kurvers et al., 2016; Wolf et al., 2015); and lie detection (Klein and Epley, 2015). As described in the appendix, for simplicity and tractability our analysis is conducted for the simplest signal detection problem, determining which of two normal distributions a single scalar random variable is drawn from; this could, in the example of a predator detection problem, be the instantaneous volume of a sound heard by a forager. In this case signal detection theory enables us to find a Receiver Operating Characteristic (ROC) curve of optimal a₊, a_– value pairs, dependent on decision difficulty (Figure 1; Green and Swets, 1966; Shettleworth, 2010); the optimal point of this curve gives us a unique pair of a₊ and a_– values dependent on costs and benefits of different decision outcomes, and the prior probability of the + state, p.

We next make the observation that, to simultaneously improve both true positive and false positive rates, a group must choose a quorum q that lies between these two values (Wolf et al., 2013); that is, an optimal group must choose q such that

The intuition behind this result is that group accuracy will converge on the appropriate accuracy for the true state of the world, as group size increases; thus, by setting a quorum between these two accuracies the true state of the world can be determined with high probability for sufficiently large groups. For further details see (Wolf et al., 2013). Since the simple majority quorum q = 1/2, assumed by Condorcet, only satisfies this inequality when both ${\displaystyle a_{+}>1/2}$ and ${\displaystyle a_{-}>1/2}$, when either of these conditions are violated then both simple majority decisions and Condorcet-based reasoning will be deficient. Furthermore, we should never see true positive rate (${\displaystyle a_{+}}$) less than false positive rate (${\displaystyle 1-a_{-}}$) (dark grey region of ROC space in Figure 1), since such a decision-maker could simultaneously improve both their true and false positive rates simply by inverting their decisions and moving above the diagonal. Therefore, the ROC space is divided into two meaningful regions: in the first ${\displaystyle a_{+}>1/2}$ and ${\displaystyle a_{-}<1/2}$ (white region in Figure 1), so simple majority voting is asymptotically-optimal as group size increases, and Condorcet-based predictions are correct. In the second, ${\displaystyle a_{+}<1/2}$ and ${\displaystyle a_{-}>1/2}$, or ${\displaystyle a_{+}>1/2}$ and ${\displaystyle a_{-}<1/2,}$ while ensuring ${\displaystyle a_{+}>1-a_{-}}$ (light grey regions in Figure 1); in these regions simple majority decisions will be sub-optimal, and Condorcet-based reasoning will be erroneous.

We now describe the systematic errors Condorcet leads to when faced with these decision scenarios. These errors can also be described in terms of false positive (frequently referred to as type I) and false negative (type II) errors, in predicting the performance of the Condorcet majority rule, hence we label the errors in such terms.

Error Ia: Condorcet predicts group accuracy approaches 1, but majority groups do not

This error occurs when majority voting is sub-optimal (${\displaystyle a_{+}<1/2}$ and ${\displaystyle a_{-}>1/2}$, or ${\displaystyle a_{+}>1/2}$ and ${\displaystyle a_{-}<1/2}$; light grey regions in Figure 1, as described above), and when expected individual accuracy ${\displaystyle a>1/2}$, since under this last condition individuals on average make more correct decisions than incorrect decisions, and Condorcet thus predicts that group accuracy approaches one as group size increases (King and Cowlishaw, 2007; Boland, 1989). From Equation 1 this requirement is thus that

${\displaystyle p\left(a_{+}-a_{-}\right)+a_{-}>1/2}$

While Condorcet predicts group accuracy (denoted $\stackrel{-}{a}$) approaches 1, that is

(4) $\underset{n\to \infty }{\lim }\stackrel{-}{a}=1$

in fact the majority quorum q = 1/2 is either below 1 – $a_{-}$, or above $a_{+}$, contra inequality 2; thus the group converges to making completely correct choices in one state of the world, and completely incorrect choices in the other state of the world (Wolf et al., 2013). Hence, group accuracy converges to

${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\stackrel{-}{a}=\{\begin{array}{cc}p& \text{if }{a}_{+}>\frac{1}{2}\text{and }{a}_{-}<\frac{1}{2}\\ 1-p& \text{if }{a}_{+}<\frac{1}{2}\text{and }{a}_{-}>\frac{1}{2}\end{array}}$

The wide range of decision scenarios in which Condorcet makes this predictive error are illustrated in Figure 2a. An example of this error is presented in Figure 3a; Figure 3b illustrates how the error can be avoided by choosing an appropriate quorum, in this case a super-majority quorum.

Figure 2

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Figure 3

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We have shown that simple majority-based collective decisions are often suboptimal, and that consequently sub- or super-majority quorums should frequently be employed by groups of ‘like-minded’ individuals combining independent decisions. Our results are important for two reasons. First, the majority-based decision rule, and Condorcet-based reasoning, is widespread in several major branches of collective decision theory. In the animal behaviour community the Condorcet prescription on individual accuracy exceeding ½ has been used to recommend when opinions should be pooled (King and Cowlishaw, 2007), and when experts should be favoured over group opinions (Katsikopoulos and King, 2010). Other authors have also invoked Condorcet and majority voting as the gold-standard for collective decision-making (Hastie and Kameda, 2005; Kao and Couzin, 2014a; Kao et al., 2014b; Miller et al., 2013). These results have been useful in highlighting the benefits of information pooling in collective decisions, but such studies implicitly neglect the reality that most decisions have two types of errors. Second, while quorums have been widely studied in collective decision theory, we here present a comprehensive theory that may help explain their prevalence. Sub- and super-majority quorums have been considered theoretically (Sumpter and Pratt, 2009); Sumpter and Pratt implicitly assume only one type of error need be considered, and proceed from that point with their analysis, referring to quorum functions such as managing speed-accuracy trade-offs (Marshall et al., 2009). Ward et al. (2008) consider quorum use for facilitating information transfer in shoaling fish, yet ignore the possibility of different error types despite their great asymmetry in the scenario studied, predator detection. Conradt and Roper (2005) in their review refer to true and false positives, but in explaining quorum usage refer back to earlier analysis as a mechanism to avoid extreme group decisions where individual fitness interests do not completely align (Conradt and Roper, 2003). However List (2004) as well as Conradt and List (2009) noted the effect of cost and prior asymmetry on quorum usage, referring back to earlier political science results (Ben-Yashar and Nitzan, 1997) discussed below. We also note that convergence on an intermediate group accuracy between 0 and 1 has previously been observed, without consideration of signal detection theory; the result presented in Figure 1b of Kao and Couzin (2014a) occurs for a similar reason to our error Ia, in that inappropriate use of a majority decision rule leads increasing group size to result in group accuracy converging on a parameter of the decision problem, in their case the reliability of a low correlation cue.

In contrast to these previous analyses here we have shown the fundamental importance of quorums in one of the simplest possible collective behaviour scenarios, single-shot collective decisions in homogenous groups, where individuals’ interests are aligned, and decision-making abilities do not differ. Thus, we might expect the use of quorums to be widespread in the natural world, even in the simplest of decisions. The widespread use of quorum sensing in bacteria provides evidence of this (Gross, 2017), and may prove a particularly good testbed for our theory given its binary nature and asymmetric state priors and error costs, although evolutionary conflicts of interests within bacterial communities may result in confounds. Moreover, as humans have been shown to employ quorum rules and adaptively adjust the associated quorum thresholds, human decision-making experiments may also provide a powerful approach to test our predictions (Kurvers et al., 2014; Clément et al., 2015).

It is surprising that signal detection theory has seen relatively scant application to collective decision-making. Wolf et al. (2013) noted the potential for different error types in identifying how quorums can improve collective decision accuracy; while motivated by signal detection theory they did not directly apply it to optimise the individual decision-makers and relate this back to group behaviour. Kirstein and von Wangenheim, 2010 also noted the possibility for independent error types, again with reference to signal detection theory; they noted the potential for Condorcet to make the same qualitatively incorrect predictions that we note here, however they did not apply the relevant theory to delineate the situations under which Condorcet reasoning is incorrect, nor did they consider the possibility for quorum-based decision rules to rescue these situations. Sorkin et al. (2001) applied signal detection theory to Condorcet-like models with varying super-majority quorums, but did not find the mechanism by which group decisions can be optimised (Wolf et al., 2013). Laan et al. (2017) noted that Condorcet and majority voting can be suboptimal, and considered ways in which voting can be improved; while they discussed signal detection theory and voting mechanisms they neither explicitly considered error types, nor quorum thresholds, focussing mainly on correlations between decision-makers and the impact of the cost function used, as well as suggesting a data-driven machine learning approach to improving collective decision-making rules. The results that most closely anticipate ours are those of Ben-Yashar and Nitzan (1997), who analytically solved for the general optimal decision rule by recognising both error cost and prior asymmetry, as well as simultaneously considering the case of variable individual decision-ability (e.g. Marshall et al., 2017). These results describe the relationship between prior asymmetry and optimal quorum threshold, and error cost and optimal quorum threshold. However, because they did not apply signal detection theory to optimise individual agents’ decisions (i.e. they treated true and false positive rates of individuals as independent from the ecological parameters error cost and prior asymmetry), they were unable quantitatively to uncover the complex nonlinear relationship between these three quantities (Figure 2).

In contrast to earlier work, by applying signal detection theory we have simultaneously shown here both how fragile Condorcet and majority-voting are, and how the use of sub- or super-majority quorums should relate to decision ecology. Although simple collective behaviour models have been well studied and highly influential, our results, and others relaxing other assumptions of such models (Marshall et al., 2017), indicate the subtlety that may be revealed in collective decision-making by a richer consideration of individual decision theory. Other approaches to such problems should be pursued in the future. For example, it is possible to optimise individual quorums (Ben-Yashar and Nitzan, 1997) rather than simply set them within a suitable interval as we do here, thereby giving greater benefits to smaller groups. Similarly, rather than apply signal detection theory one could apply Bayesian decision-theory (e.g. Pérez-Escudero and de Polavieja, 2011; Arganda et al., 2012; Pérez-Escudero and de Polavieja, 2017), thereby attempting to deal with further complexities such as non-independence of individual decisions. In many scenarios errors and correct decisions may be correlated although even when multiple individuals observe the same stimulus their information can be considered independent due to sensory noise (Marshall et al., 2017). We believe that our simple approach has, however, the benefit of tractability while still revealing the complexity of collective decision-making even in the simplified scenario considered.

Our work takes inspiration from political science and decision theory to address questions in behavioural ecology, but may additionally have the potential to inform work in the design of artificial decision-making systems, machine learning and robotics. For example, in the field of ensemble learning, in which predictions from multiple weak classifiers such as neural networks are combined to improve decision accuracy, variable quorums, referred to as ‘threshold shift’, are used (e.g. Dmochowski et al., 2010). However majority voting is still among the simplest and most ubiquitous vote fusion rules discussed (Sagi and Rokach, 2018; Krawczyk et al., 2017). Hence, we suggest that the simple perspective on how to combine votes presented here may also yield technological insight.

All data generated or analysed during this study are included in the manuscript and supporting files. Source data files have been provided for Figures 1, 2 and 3.

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Author details

1. James AR Marshall

   Department of Computer Science, University of Sheffield, Sheffield, United Kingdom

   Contribution

   Conceptualization, Formal analysis, Writing—original draft, Writing—review and editing

   For correspondence

   james.marshall@sheffield.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-1506-167X

2. Ralf HJM Kurvers

   Centre for Adaptive Rationality, Max Planck Institute for Human Development, Berlin, Germany

   Contribution

   Conceptualization, Writing—review and editing

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-3460-0392

3. Jens Krause

   Department of Fish Behavior and Ecology, Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany

   Contribution

   Conceptualization, Writing—review and editing

   Competing interests

   No competing interests declared

4. Max Wolf

   Department of Fish Behavior and Ecology, Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany

   Contribution

   Conceptualization, Formal analysis, Writing—original draft, Writing—review and editing

   For correspondence

   m.wolf@igb-berlin.de

   Competing interests

   No competing interests declared

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