Group-living species frequently pool individual information so as to reach consensus decisions such as when and where to move, or whether a predator is present. Such opinion-pooling has been demonstrated empirically, and theoretical models have been proposed to explain why group decisions are more reliable than individual decisions. Behavioural ecology theory frequently assumes that all individuals have equal decision-making abilities, but decision theory relaxes this assumption and has been tested in human groups. We summarise relevant theory and argue for its applicability to collective animal decisions. We consider selective pressure on confidence-weighting in groups of related and unrelated individuals. We also consider which species and behaviours may provide evidence of confidence-weighting, paying particular attention to the sophisticated vocal communication of cooperative breeders

Andrew N. Radford

Bahrami

Bell

Blumstein

Boland

Bousquet

Conradt

Couzin

Flack

Flower

Freund

Galton

Gavin Brown

Griffin

Grofman

Hamilton

Hare

James A.R. Marshall

Johnstone

Katsikopoulos

Kepecs

Kern

Kiani

King

Koriat

Korman

List

Marshall

Michelena

Milinski

Nitzan

Pettit

Radford

Ramakrishnan

Rands

Ratcliff

Rendall

Schibler

Sorkin

Surowiecki

Thomas

Townsend

Trimmer

Van der Post

Ward

Wheeler

Wolf

Group-living species frequently pool individual information in order to reach consensus decisions, such as when and where to move, or whether a predator is present. Such opinion-pooling has been demonstrated empirically, and theoretical models proposed to explain why group decisions are more reliable than individual ones. In behavioural ecology theory frequently assumes all individuals have equal decision-making abilities, yet decision theory relaxes this assumption and has been tested in human groups. We summarise relevant theory and argue for its applicability to collective animal decisions. We consider selective pressure on confidence-weighting in groups of related and unrelated individuals. We also consider which species and behaviours may provide evidence of confidence-weighting, paying particular attention to the sophisticated vocal communication of cooperative breeders

Marshall, James

Brown, Gavin

Radford, Andy

Explore Bristol Research

                          Marshall, J., Brown, G., & Radford, A. (2017). Individual Confidence-Weighting and Group Decision-Making. Trends in Ecology and Evolution,32, 636-645. https://doi.org/10.1016/j.tree.2017.06.004Publisher's PDF, also known as Version of recordLicense (if available):CC BY-NC-NDLink to published version (if available):10.1016/j.tree.2017.06.004Link to publication record in Explore Bristol ResearchPDF-documentThis is the final published version of the article (version of record). It first appeared online via Elsevier (CellPress) at http://www.sciencedirect.com/science/article/pii/S0169534717301520. Please refer to any applicableterms of use of the publisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-termsSUPPLEMENTARY MATERIAL FOR:INDIVIDUAL CONFIDENCE-WEIGHTING AND GROUPDECISION-MAKINGJAMES A. R. MARSHALL, GAVIN BROWN, ANDREW N. RADFORD1. Optimal Decision-MakingIn general to optimise a decision-maker’s expected payoff from decisions, such aswhether or not a predator is present while foraging in an exposed area, we need to takeaccount of the prior probability of states of the world, the payoffs (or losses) from differentdecision outcomes, and finally the accuracy of the decision maker under different statesof the world (e.g. [6, 4]). In decision theory terms for binary classification decisions thesecan be captured in a prior probability vector P, a cost matrix W and a confusion matrixC. These are defined as(1) P =(p (1− p) )where p is the probability that the state of the world is ‘positive’ (e.g. predator present),(2) C =(aP (1− aN )(1− aP ) aN)where aP is the accuracy (probability of true positive) when the state of the world is‘positive’, and aN is the accuracy (probability of true negative) when the state of theworld is ‘negative’, and(3) W =(WTP WFPWFN WTN)where WTP is the cost, or loss, from a true positive, WFP is the loss from a false-positive, WFN the loss from a false negative, and WTN that from a true negative. Hencein matrices C and W columns correspond to the true state of the world, and rowscorrespond to the state the decision maker perceives (‘positive’ then ‘negative’ in bothcases).Given the matrix definitions (1), (2) and (3) the expected loss of a decision-makerclassifying an instance x is defined asE(L(x)) = P(Diag[CTW])(4)where Diag[X] is the leading diagonal of matrix X, which simplifies to give(5) E(L(x)) = p(aPWTP + (1− aP )WFN ) + (1− p)(aNWTN + (1− aN )WFP ).Date: November 18, 2016.12 JAMES A. R. MARSHALL, GAVIN BROWN, ANDREW N. RADFORDThe expected loss is an economic concept that is well known in behavioural biology, forexample in the application of signal detection theory to animal behaviour (e.g. [6, 4]).An optimal decision-maker minimises this loss, everything else being equal.2. Consensus Decision-Making in GroupsWe define the consensus decision of a group, reached by combining the individualdecisions of constituent group members, as(6) H(x) =∑iαihi(x)where hi is the decision of the i-th group member as to whether the state of the worldis ‘positive’ or ‘negative’ (i.e. hi ∈ {−1,+1}), and αi is a weight that individual putson their decision, calculated below. In other words, the group’s decision is a weightedsum of the decisions of its constituent members. Note that ‘positive’ and ‘negative’ aresimply arbitrary labels we assign to the two possible states of the world, such as predatorpresent and predator absent.3. Optimising Consensus DecisionsSince the loss function for the group depends on the discontinuous losses in matrixW, and minimisation over discontinuous functions is inherently difficult, in the followinganalyses we adapt a standard exponential bound on the loss from decisions. This upperbound has been used in, for example, the machine learning literature (e.g. [1]), to ap-proximate losses from decisions. We define the correct classification c(x) ∈ {−1,+1} tobe −1 if the state of the world is ‘negative’ and +1 otherwise, and a group’s hypothesis,H(x) < 0 or H(x) > 0 to indicate their belief about the state of the world (‘negative’or ‘positive’ respectively); thus the group’s decision is given by the sign of equation 6.If we assume for now that the decision-maker incurs a penalty of 1 if they make anincorrect prediction (i.e. the sign of the decision maker’s classification is opposite to thesign of the environment state), and no penalty if they make a correct prediction (i.e. thesign of the decision maker’s classification is opposite to the true sign of the environmentstate), then the loss arising from the decision-maker’s decision is bounded above by anexponential function, as described in appendix A and represented in figure 1.As described at the beginning of the section, in realistic decisions gains or losses fromdecision outcomes need not be all-or-nothing, nor need they be the same under all statesof the world. Rather these gains and losses reflect the costs and benefits of different typesof error and correct classification under different states of the world, as captured in theloss matrix W defined above in equation 3. However the exponential bound presented infigure 1 can be adapted to capture this increased detail and derive an optimal decision-weighting rule for the fully general case presented above, as described in appendix A.The optimal α∗i for this general case, defined in equation A.8 in appendix A, appearsrather complicated but in heuristic terms is relatively simple when visualised (figure 3in appendix A) and, as discussed below, could be replaced by simpler functions that stillprovide an advantage over simple unweighted voting. Simplifying the general analysisto the special case where WFN −WTP = WFP −WTN (so costs for errors are the sameINDIVIDUAL CONFIDENCE-WEIGHTING AND GROUP DECISION-MAKING 3-1.0 -0.5 0.5 1.0 c(x) h(x)0.51.01.52.02.5Figure 1. Realised zero-one loss from a decision (zero loss if correct, oneloss if incorrect; solid lines) is upper-bounded by an exponential function(dashed line) as defined in equation A.1.under positive and negative states of the world), and defining overall expected accuracy(7) ai = paPi + (1− p)aNi ,while noting that if all group members’ weight their vote by the same constant factorthen the consensus decision is unchanged, then as shown in appendix A this reduces to(8) α∗i ∝ ln(ai1− ai),which is the well-known optimal weighting rule from machine learning [1] and decisiontheory [2], presented in the main text. Note, however that the derivation presented inappendix A differs from previous derivations resting on assumed weighting of trainingexamples for classifiers [1] or deriving as a corollary of Bayes’ Theorem [5, 2, 1]. Thisoptimal confidence-weighting is illustrated in Box 2 in the main text.4. Optimal Confidence-Weighting from Optimal IndividualDecision-MakingKiani and Shadlen [3] show how a measure of subjective confidence, the log-oddsratio, can be calculated for a drift-diffusion decision maker (Box 3, main text) facedwith uncertain information quality. Formally, Kiani and Shadlen note that the log odds-ratio of a correct decision(9) logP(〈µ〉 > 0|x(t))P(〈µ〉 < 0|x(t)) = log∑i P(x(t)∣∣µi > 0, |µi|)P(|µi|∣∣µi > 0)∑i P(x(t)∣∣µi < 0, |µi|)P(|µi|∣∣µi < 0) + log P(µ > 0)P(µ < 0) .Assuming that signal magnitude (|µi|) is uncorrelated with signal direction (sign(µi)),and that both states of the world are equally likely, this simplifies to equation 5 in Box3 in the main text.4 JAMES A. R. MARSHALL, GAVIN BROWN, ANDREW N. RADFORDAppendix A. Weighting Opinions by Individual Decision AccuracyReduces Expected Group Decision LossAs described in the main text we assume that the correct classification c(x) ∈ {−1,+1}is −1 if the state of the world is negative and +1 otherwise, and a decision-maker’shypothesis h(x) ∈ {−1,+1} indicates their belief about the state of the world (negativeor positive). We also assume that the decision-maker incurs a penalty of 1 if they makean incorrect prediction, and no penalty if they make a correct prediction, then the lossarising from the decision-maker’s decision is bounded above by the function(A.1) L = e−c(x)h(x),as illustrated in figure 1.Since in general losses from decisions are not all or nothing (eq. 5), and need not bethe same under each state of the world (so the cost for a false positive need not be thesame as the cost for a false negative, for example), expression A.1 can be generalised to(A.2) L ={WTN + (WFP −WTN )eh(x) if c(x) = −1WTP + (WFN −WTP )e−h(x) if c(x) = 1,as illustrated in figure 2.Figure 2. Realised loss from a decision, defined according to a loss ma-trix (blue planes) is upper-bounded by a piecewise exponential function(yellow surfaces) as defined in equation A.2.We now consider how exponential bounds can be used to derive an individual confidence-weighting rule that minimises the group’s expected loss. Since optimal decision rulesshould take account of the prior probability p of the state of the world (5), the upperbound on the expected loss of a group should also take this into account, so substitutingequation 6 into A.2 and weighting by prior probability gives the expected loss for a groupINDIVIDUAL CONFIDENCE-WEIGHTING AND GROUP DECISION-MAKING 5decision asE(L) = p(WTP + (WFN −WTP )e−∑i αihi(x)) +(A.3)(1− p)(WTN + (WFP −WTN )e∑i αihi(x))= p(WTP + (WFN −WTP )∏ie−αihi(x)) +(A.4)(1− p)(WTN + (WFP −WTN )∏ieαihi(x))To add an individual to a group that minimises their contribution to that group’sexpected decision loss, we can see that they thus need to minimiseE(Li) = p(WTP + (WFN −WTP )e−αihi(x)) +(A.5)(1− p)(WTN + (WFP −WTN )eαihi(x))Separating out into expected loss from correct and incorrect decisions, according toaccuracies under positive and negative world states, this becomesE(Li) = p(aPi(WTP + (WFN −WTP )e−αi) +(A.6)(1− aPi)(WTP + (WFN −WTP )eαi)) +(1− p)(aNi(WTN + (WFP −WTN )e−αi)+(1− aNi)(WTN + (WFP −WTN )eαi)),where aPi and aNi refer to decision accuracies for individual i when faced with positiveand negative states of the world respectively.To minimise this contribution we differentiate with respect to individual ‘confidence’αi∂∂αiE(Li) = (p(1− aPi)(WFN −WTP ) + (1− p)(1− aNi)(WFP −WTN ))eαi(A.7)−(paPi((WFN −WTP ) + (1− p)aNi(WFP −WTN ))e−αi .then find the value of αi at which (A.7) is zero, as this minimises E(Li).Rearranging and taking the natural logarithm we find that (A.7) is zero when(A.8) α∗i =12ln(pap(WFN −WTP ) + (1− p)aN (WFP −WTN )p(1− ap)(WFN −WTP ) + (1− p)(1− aN )(WFP −WTN )).Equation A.8 is visualised in figure 3References[1] Yoav Freund and Robert E Schapire. A decision-theoretic generalization of on-line learning and anapplication to boosting. Journal of Computer and System Sciences, 55(1):119–139, 1997.[2] Bernard Grofman, Guillermo Owen, and Scott L Feld. Thirteen theorems in search of the truth.Theory and Decision, 15:261–278, 1983.[3] Roozbeh Kiani and Michael N Shadlen. Representation of confidence associated with a decision byneurons in the parietal cortex. Science (New York, N.Y.), 324(5928):759–64, 2009.[4] James A R Marshall, Pete C. Trimmer, Alasdair I. Houston, and John M. McNamara. On evolution-ary explanations of cognitive biases. Trends in Ecology and Evolution, 28(8):469–473, 2013.[5] Shmuel Nitzan and Jacob Paroush. Optimal decision rules in uncertain dichotomous choice situations.International Economic Review, pages 289–297, 1982.6 JAMES A. R. MARSHALL, GAVIN BROWN, ANDREW N. RADFORDFigure 3. Optimal confidence-weighting parameter α∗ as a functionof individual decision-accuracies under positive (aP ) and negative (aN )states of the world, calculated from equation A.8. Note that the individ-ual’s contribution to the group decision should be negatively-weightedif both its decision accuracies are less than chance (1/2). Note alsothat while standard ROC analysis predicts that the total accuracy ofa decision-maker aP + aN should exceed 1, combinations of accuraciessatisfying this condition that result in a negative optimal weighting canbe found.[6] Pete C Trimmer, Alasdair I Houston, James AR Marshall, Rafal Bogacz, Elizabeth S Paul, Mike TMendl, and John M McNamara. Mammalian choices: combining fast-but-inaccurate and slow-but-accurate decision-making systems. Proceedings of the Royal Society of London B: Biological Sciences,275(1649):2353–2361, 2008.

Individual Confidence-Weighting and Group Decision-Making

Marshall, J.A.R.

Brown, G.

Radford, A.N.

White Rose Research Online

Individual confidence-weighting and group decision-making

Crossref

Trends in Ecology & Evolution

Group-living species frequently pool individual information in order to reach consensus decisions, such as when and where to move, or whether a predator is present. Such opinion-pooling has been demonstrated empirically, and theoretical models proposed to explain why group decisions are more reliable than individual ones. In behavioural ecology theory frequently assumes all individuals have equal decision-making abilities, yet decision theory relaxes this assumption and has been tested in human groups. We summariserelevant theory and argue for its applicability to collective animal decisions. Weconsider selective pressure on confidence-weighting in groups of related and unrelated individuals. We also consider which species and behaviours may provide evidence of confidence-weighting, paying particular attention to the sophisticated vocal communication of cooperative breeders

Marshall, James A R

Radford, Andrew N

The University of Manchester - Institutional Repository

https://research-information.bris.ac.uk/ws/files/125867615/1_s2.0_S0169534717301520_mmc1.pdf

Individual Confidence-Weighting and Group Decision-Making

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