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Decision versus compromise for animal groupsin motionNaomi E. Leonarda,1, Tian Shena, Benjamin Nabeta, Luca Scardovib, Iain D. Couzinc, and Simon A. Levinc,1

Departments of aMechanical and Aerospace Engineering and cEcology and Evolutionary Biology, Princeton University, Princeton, NJ 08544;and bDepartment of Electrical and Computer Engineering, University of Toronto, Toronto, ON, Canada M5S 3G4

Contributed by Simon A. Levin, November 8, 2011 (sent for review June 20, 2011)

Previously, we showed using a computational agent-based model

that a group of animals moving together can make a collective de-

cision on direction of motion, even if there is a conict between the

directional preferences of two small subgroups of informed individ-

uals and the remaining uninformedindividuals have no directional

preference. The model requires no explicit signaling or identicationof informed individuals; individuals merely adjust their steering in

response to socially acquired information on relative motionof neigh-

bors. In this paper, we show how the dynamics of this system can be

modeled analytically, and we derive a testable result that adding un-

informed individuals improves stability of collective decision making.Werst present a continuous-time dynamic model and prove a nec-

essary and sufcient condition for stable convergence to a collective

decision in this model.The stabilityof thedecision, which corresponds

to most of the group moving in one of two alternative preferreddirections, depends explicitly on the magnitude of the difference in

preferred directions; for a difference above a threshold the decision is

stable and below that same threshold the decision is unstable. Given

qualitative agreement with the results of the previous simulation

study, we proceed to explore analytically the subtle but important

role of the uninformed individuals in the continuous-time model.

Signicantly, we show that the likelihood of a collective decisionincreases with increasing numbers of uninformed individuals.

collective behavior| Kuramoto| coordinated movement

Explaining the ability of animals that move together in a group

to make collective decisions requires an understanding of themechanisms of information transfer in spatially evolving dis-tributions of individuals with limited sensing capability (16). Ingroups such as sh schools and large insect swarms, it is likelythat individuals can sense only the relative motion of nearneighbors and may not have the capacity to distinguish a well-informed neighbor from the less well informed (2, 3). Further, itis increasingly becoming recognized that the emergent in-telligence of a collective may be more reliable than the in-telligence provided by a few leaders or well-informed individuals(711). This result suggests a subtle but important role in col-lective decision making for those individuals that have no par-ticular information or preference.

In this paper we dene and analyze a continuous-time dynamicalsystem model to examine collective decision making in moving

groups of informed and uninformed individuals that are limited tosensing the relative motion of neighbors and adjusting their steeringin response. Informed individuals have a preference for one of twoalternative directions of motion, whereas uninformed individualshave no preference. The preferences are representative of knowl-edge of the direction to a food source or of a migration route, etc.The model is motivated by the discrete-time model of ref. 1, whichis used to investigate, through computation, mechanisms of de-cision making and leadership in groups moving in the plane; itextends the continuous-time model of ref. 12, which exhibits onlysome of the group behaviors observed in the simulations of ref. 1(compromises but not decisions).

In the discrete-time model of ref. 1 there is no signaling, noidentication of the informed individuals, and no evaluation ofothers information. Nonetheless, it is shown in ref. 1 that the groupcan make a collective decision: With two informed subgroups of

equal population (one subgroup per preference alternative), acollective decision to move in one of the two preferred directions ismade with high probability as long as the magnitude of the pref-erence conict, i.e., the difference in preferred directions, is suf-ciently large. Forsmall conict, thegroup follows the average of thetwo preferred directions. Further, simulations in ref. 13 provideevidence that increasing the population size of uninformed indi-

viduals lowers the threshold on magnitude of conict, making iteasierfor a collective decision to be made.

Simulations of the kind reported in ref. 1 are highly suggestive,but because they contain many degrees of freedom, it is difcult toidentify the inuences of particular mechanisms. In this paper wepresent an approximation to the individual-based model (1) that

allows deeper analysis into the microscopic reasons for the ob-served macroscopic behaviors and a broader exploration of pa-rameter space. The model we propose and study is represented bya system of ordinary differential equations. As in the formulation ofref. 12, each agent is modeled as a particle moving in the plane atconstant speed with steering rate dependent on interparticlemeasurements and, for informed individuals, deviation froma preferred direction. In ref. 12 two timescales, observed in thesimulations of ref. 1, are formally proved to exist for the system ofequations; in the fast timescale, alignment is established withineach subgroup of agents with the same preference (or lack ofpreference), whereas in the slow timescale, the reduced-ordermodel describes the average motion of each of the two informedsubgroups and the uninformed subgroup.

In ref. 12 assumptions are made that simplify the analysis. First,

examination is restricted to the directional dynamics of the par-ticles. Second, each individual is assumed capable of sensing therelative direction of motion of every other individual in the group;i.e., the social information is globally available. Third, the un-informed subgroup is ignored in the analysis of the slow timescaledynamics. A comprehensive bifurcation analysis is presented ofstable and unstable solutions of the reduced-order dynamics; theresults provide insights on stable solutions not explored in thesimulation study, unstable solutions not easily understood throughsimulation, and sensitivity to parameters. However, the simplifyingassumptions yield a model that produces some but not all of thebehavior observed in ref. 1; notably, the group does not select tomove as a whole in one of the preferred directions unless a for-getting feedback is introduced such that informed individualsgradually lose their preference if theynd themselves moving in a

direction far from their preference.The deviation of the results of ref. 12 from those of ref. 1 focusesattention on a small number of assumptions that may be re-sponsible. It is the second and third assumptions of global sensingand neglect of the uninformed individuals that we relax in thispaper. We limit sensing and dene dynamics that represent the

Author contributions: N.E.L. designed research; N.E.L., T.S., B.N., and L.S. performed re-

search; N.E.L., B.N., and L.S. contributed new reagents/analytic tools; N.E.L., T.S., B.N., L.S.,

I.D.C., and S.A.L. analyzed data; and N.E.L. and T.S. wrote the paper.

The authors declare no conict of interest.

1To whom correspondence may be addressed. E-mail: naomi@princeton.eduor slevin@

princeton.edu.

This article contains supporting information online atwww.pnas.org/lookup/suppl/doi:10.

1073/pnas.1118318108/-/DCSupplemental.

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changing sensing neighborhood for each individual. We include theuninformed individuals in our analysis.

With this unique continuous-time model, we show the stabilityof collective decision making without a forgetting factor, and wederive the critical value of magnitude of conict that serves asa threshold for a collective decision. Here a collective decisionrefers to all individuals in one informed subgroup and all un-informed individuals moving together in the informed subgroup spreferred direction; this differs slightly from the denition of

a collective decision in ref. 1, where all individuals achieveconsensus and decide on a preferred direction.Our results agree qualitatively with the results of the study

based on the more complex discrete-time model of ref. 1; ac-cordingly, we use the continuous-time model to explore thesubtle but important role of the uninformed individuals in col-lective decision making. In particular, we derive the sensitivityof the collective decision making to the population size of theuninformed individuals, showing that increasing numbers ofuninformed individuals increases the likelihood that the group

will make a collective decision.

Model

The discrete-time model of ref. 1, like the model of ref. 14,considers a group of individuals, each represented as a self-propelled particle in the plane that adjusts its direction of motionin response to the relative motion of local neighbors and randominuences. In ref. 14, individuals steer to align with the averagedirection of others within a circular neighborhood. In ref. 1,individuals also use circular neighborhoods but make it a priorityto steer away from any others that are too close. If there are nosuch very close neighbors, they steer to align and to attract toneighbors that are not quite so close. Informed individuals sumthe steering term that derives from measurements of neighbors

with a steering term that heads them toward one of two alter-native xed preferred directions. As the individuals move about,relative positions among them can change and thus the localneighborhood of any given individual can change with time.

We model the discrete-timedynamics of ref. 1 with a continuous-time model that looks much like a spatial extension of coupledoscillator dynamics (15, 16). That is, an individuals heading angle,

which determines its direction of motion, resembles a phase angle,and the steering laws, which depend on relative headings (andpossibly relative positions) of individuals, serve to dynamicallycouple the phases among the individuals; see refs. 17 and 18. As inref. 12, we include the alignment steering term but neglect the re-pulsion and attraction steering terms of ref. 1; we also includea term that couples the heading angle of each informed individual

with one of the two xed preferred directions. The model is similarto that used in ref. 19 to represent a group of coupled spins in arandom magnetic eld, where each individual oscillator has arandomly assigned pinningangle.

Unlike what is done in any of these continuous-time models,we propose a dynamic model for coupling weights. There issome similarity with coupling weights in the linear consensusdynamic model of ref. 20, which change as a static function of

relative distance, decaying exponentially with distance. Thecoupling weights in our model change as a sigmoidal functionof the integrated relative distance between neighbors; this dy-namic endows individuals with a fading memory of neighbors.The weight dynamics are similar to Hebbian plasticity in neuralcircuits with a saturation; the latter is a reinforcing processthat strengthens effective synapses and weakens ineffective syn-apses (21).

Additionally, we use relative direction of motion rather thanrelative spatial distance as a means of determining neighbors.This is justied by our focus on the decision-making dynamics ofgroups of informed and uninformed individuals that are initiallyclosely aggregated; for an initially aggregated group of individ-uals, those that head in the same direction remain close whereasthose that head in very different directions quickly separate.With our model of neighbors, the steering laws do not depend on

spatial position, and we can analyze the dynamics of the headingdirections independently. The lower dimensionality of theheading plus coupling weight dynamics compared with the di-mensionality of the full spatial dynamics contributes to makingthe analysis tractable.

Our model is deliberately made deterministic so that we caninvestigate mechanisms of collective decision making outside ofstochasticity. The model studied in ref. 12 is also deterministic,and the stability and bifurcation results of ref. 12 were shown to

persist in the presence of randomness in the investigation of ref.22. Simulations of the model presented here with some ran-domness suggest similarly that our results are robust (SI Text).

LetNbe the total number of individuals in a population; eachindividual is modeled as a particle moving in the plane at con-stant speedvc. We denote by angle j(t) the direction of motionof individualj at timet. Then, the planar velocity ofj at time t isvj = (vccos j(t), vcsin j(t)).

We associate every individual with one of three subgroups:The N1 individuals in subgroup 1 have a preference to move inthe direction dened by the angle 1, the N2individuals in sub-group 2 have a preference to move in the direction dened by theangle 2, and the N3 individuals in subgroup 3 have no prefer-ence. We have that N1 + N2 + N3= N.

We de

ne the rate-of-change of direction of motion for eachindividual in subgroup 1 as

dj

dt sin1 jtK1

N

XNl1

ajltsinlt jt

; [1]

in subgroup 2 as

dj

dt sin2 jtK1

N

XNl1

ajltsinlt jt

; [2]

and in subgroup 3 as

dj

dtK1

NX

N

l1ajl

t

sinlt jt: [3]

The constant parameter K1 > 0 weights the attention paid toother individuals versus the attention paid to the preferred di-rection. The dynamic variable 0 ajl(t) 1 denes the weightindividual j puts on the information it gets from individual lattime t. A value ajl= 0 implies that j cannot sense l.

We model the social interaction (coupling) weights ajl(t) asevolving in time according to saturated integrator dynamics thatdepend on how closeindividuals are from one another, wherecloseness is dened in terms of relative heading:

djl

dt K2

jltr

;

ajlt 11 e jlt:[4]

In the model of Eq. 4, jl = lj is an integrated variable, theconstant parameter K2 > 0 quanties the speed at which the

interaction gains evolve, jl jcos1

2j lj gives a measure of

synchrony of direction of motion of l and j, and 0r 1 isa chosen xed threshold representing an individuals sensingrange. It holds that jl 1 ifland j move in the same directionand jl 0 if they move in opposite directions. Ifjl>r, then jandlare close enough to sense each other sojlincreases andajleventually converges to the maximum interaction strength of 1. Ifjl

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dajl

dt K2

1ajlt

ajlt

jltr

: [5]

Equilibrium solutions correspond to ajl(t) = 0 and ajl(t) = 1.The state space for the model of Eqs. 13 and 5 is compactbecause each jis an angle and each ajlis a real number in theinterval [0, 1].

Results

The model exhibits fast and slow timescale behavior even formoderate values of gains K1 and K2. LetNk be the subset ofindexes corresponding to individuals j in subgroupk for k = 1, 2,3. For an initially aggregated group, the fast dynamics corre-spond to the individuals in subgroup k (for each k = 1, 2, 3),quickly becoming tightly coupled with one another: The coupling

weights ajl(t) for j Nk and lNk converge to 1, and the di-rection of motion j(t) for each j Nk converges to a commonangle k(t). Also, for each pair of subgroupsm and n wherem n the coupling weights ajl(t) for j Nm and lNn quickly ap-proach a common value of either 0 or 1. Thus, after the fasttransient, individuals in each subgroup move together in thesame direction and the coupling between subgroups becomesconstant; the slow dynamics describe the evolution of the averagedirection of each of the three possibly interacting subgroups.

We can formally derive the fast and slow timescale dynamicsin the case that max1=K1; 1=K2 c,where the critical difference in preference direction cis given by

c cos 1

2r2 1: [7]

On the other hand, manifoldM111 (where all subgroups arecoupled) is stable if 2 < c, i.e., if

Fig. 1. Coupling in manifoldsM010 (Left) andM001 (Right) among sub-groups 1, 2, and 3 as indicated by arrows.

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cos2 > 2r2 1:

The dependency of the stability of the manifolds on the criticalangle c can be interpreted as follows. Given a value of sensingrange parameter r, for sufciently large difference 2 betweenthe two preferred directions, the two informed subgroups will bepulled enough in their preferred directions such that they willlose direct connection with each other. Depending on initialconditions the uninformed subgroup may become connected

with one or the other of the two informed subgroups corre-sponding to the interconnections on M010or M001in Fig. 1. Onthe other hand, for sufciently small difference 2 between thetwo preferred directions, the two informed subgroups can stayconnected with each other and with the uninformed subgroupcorresponding to the fully connected case ofM111.

The stablesolutionof the slow dynamicsof Eq. 6 on the manifoldM010corresponds to all of the informed individuals in subgroup 1and all of the uninformed individuals (subgroup 3) moving steadilyin the preferred direction 1; the informed individuals in subgroup2 are disconnected from the greater aggregation and move offby themselves in their preferred direction 2. We classify this so-lution as (most of) the group making a decision for preference 1.Likewise, the stable solution on the manifoldM001corresponds toall of the informed individuals in subgroup 2 and all of the un-

informed individuals (subgroup 3) moving steadily in the preferreddirection 2 ; the informed individuals in subgroup 1 are discon-nected from the greater aggregation and move off by themselves intheir preferred direction 1. We classify this solution as (most of)the group making a decision for preference 2.

Fig. 2 shows a simulation ofN= 30 individuals obeying thedynamics of Eqs. 14 with N1= N2= 5 and N3= 20. Here r=0.9, which corresponds to c 528. Further, 2 908, which isgreater than c so thatM010andM001are both stable. Indeed,for the initial conditions illustrated in the plot in Fig. 2 (see alsoFig. S1), the solution converges to a group decision for prefer-ence 1 as in the slow dynamics onM010.

Depending on parameters, the slow dynamics of Eq. 6 on themanifoldM111, corresponding to the fully connected case, canhave up to two stable solutions. In the rst stable solution each of

the two informed subgroups compromises between its preferreddirections and the average of the two preferred directions,whereas the uninformed subgroup travels in the average of thetwo preferred directions. Fig. 3 shows a simulation of N = 30individuals obeying the dynamics of Eqs. 14with N1 = N2= 5

andN3= 20. Herer= 0.6, which corresponds to c 1068. As inthe previous example, 2 908, but now this is less than c sothat M010and M001are unstable and M111is stable. Indeed, forthe initial conditions of Fig. 3 (the same as in Fig. 2), the solutionconverges to the compromises as in the rst stable solution of theslow dynamics onM111. IfN3 >2N1, i.e., for a sufciently largepopulation of uninformed individuals,M111 is attractive onlynear the rst stable solution if 2 < c.

The second stable solution of Eq. 6 on the manifoldM111 issymmetric to the rst stable solution: The uninformed subgroupmoves in the direction 180 from the average of the two preferreddirections and each informed subgroup compromises between thisdirection and its preferred direction. This is a somewhat patho-logical solution that is very farfrom a group decision. However, thissecond solution does not exist in the presence of a sufciently largepopulation of uninformed individuals, notably in the case that

N3

2N1

2=3> 1

2N1K1

Nsin2=2

!2=3: [8]

Inequality Eq. 8 , which derives from our stability analysis, is al-ways satised for N3 > 2N1 or for sufciently large strength ofsocial interactions given by K1 2. Thus, under the condition

N3>

2N1,M111 is unstable precisely whenM010andM001 arestable. Fig. 4 illustrates stability of decisions (on M010and M001)versus compromise (onM111) as a function of preference dif-ference 2.

Fig. 5,Upper Leftplotsras a function of2given by Eq. 7; thiscurve denes the condition for stability of a collective decisionfor preference 1 as dened by the solution onM010 and forpreference 2 as dened by the solution on M001. The gray regionillustrates the parameter space corresponding to stability ofa collective decision. The decision is unstable in the parameterspace dened by the white region. Given a xed value ofr, thecurve provides a lower bound on the preference difference 2for

which a decision is stable.Now suppose that a number of uninformed individuals are

added to the aggregation; i.e., the density is increased. For any

individual to retain roughly the same number of neighbors afterthe addition of individuals as before, it can decrease its sensingrange. A decrease in sensing range corresponds to an increase inr. As seen in Fig. 5, an increase inrcorresponds to a decrease inthe lower bound c ; i.e., with increased numbers of uninformedindividuals, a collective decision is stable for lower values ofpreference difference 2.

0

Fig. 2. Simulation of dynamics of Eqs. 14withN= 30 individuals, r= 0.9,

and 1 08 and 2 908 as shown with black arrows on the top of the cyl-inder. The solution for each individual is shown evolving on the surface of

the cylinder; the azimuth describes the anglejand the vertical axis describes

time t. For this example, 2 >c 528 and it can be observed that a decision ismade for preference 1.

Fig. 3. Simulation of dynamics of Eqs. 14withN= 30 individuals, r= 0.6,

and 1 08 and 2 908. For this example, 2 < c 1068 and it can be ob-served that no decision is made. Instead, the agents collect in subgroups that

compromise.

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For some range of parameter values for which M010and M001are stable, it is possible that

M000,

M100, and/or

M011 are also

stable. This means that even ifM010 andM001 are stable, forsome initial conditions the solution may converge to the stablesolutions ofM000,M100, and/orM011, none of which corre-sponds to a collective decision for preference 1 or 2. In fact, theonly stable solution on M000corresponds to the three subgroupsmoving apart.M100 can have up to two stable solutions andM011 can have one stable solution; all of these correspond tocompromise solutions. Therefore, we examine the conditionsfor stability ofM000,M100, andM011 to isolate the parameterspace in whichM010 andM001 are the only stable manifoldsamong the eight under investigation.

The condition 2 > c is necessary for stability ofM000.However, M000is unstable as long as the initial average headingof the uninformed individuals is greater than 2 and less than

22, i.e., as long as the uninformed individuals are not headed ina direction that is dramatically different from the average of thetwo preferred directions. The latter is not so likely for initially

aggregated individuals. Further, the likelihood ofM000 beingstable shrinks as 2 grows.M100 (coupled informed subgroups) is also unstable if the

initial average heading of the uninformed is not dramaticallydifferent from the average of the two preferred directions.Otherwise, if2 < c, M100is stable about its rst stable solution.The second stable solution ofM100does not exist ifK1 ffiffiffiffiffiffiffiffiffiffiffiffi

1d2p ; d Nsin

2=22N1K1

: [9]

The condition 2 > c is a necessary condition for stability ofM011 (uninformed coupled to uncoupled informed subgroups).However,M011is unstable if either of the following is satised:

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi

1

2 1

2ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2ps

; [10]

where

Nsin2=2

N3K1 Ncos2=2 :

Table 1 summarizes the possible coexistence of stable manifoldsfor different parameter ranges, assuming N3 > 2N1. For theinitial conditions we consider,M000andM100will be unstable,in which case, when M111is stable, it is exclusively stable amongthe eight manifolds. Further, the parameter values that yield theexclusive stability ofM010 andM001 among the eight invariantmanifolds are those that satisfy Eq.10; these values are shown asdark gray regions in the parameter space plots in Fig. 5. In threeplots, the green curve plots ras a function of 2 in the case ofequality in the rst condition of Eq. 10, and the orange curveplots ras a function of 2 in the case of equality in the secondcondition of Eq.10. In each of the plots, N1= N2= 5 andK1=2. The number of uninformed individuals N3 ranges from N3 =11 (Fig. 5, Upper Right) to N3= 50 (Fig. 5, Lower Left) toN3=500 (Fig. 5, Lower Right). The plots show the dark gray region

expanding with increasingN3; i.e., the region of parameter spacethat ensures unique stability of the collective decision for oneor the other preference expands with increasing number of un-informed individuals. An increase in strength of social inter-action K1 also increases this parameter space.

Discussion

The continuous-time, deterministic, dynamical system modelpresented and analyzed in this paper approximates the decisionmaking of a group of informed and uninformed individuals onthe move as studied in ref. 1. In the case that the two informedsubgroups 1 and 2 are equally sized (N1= N2), it is shown in ref.1 that the whole group will decide with high probability to movein one of the two preferred directions, as long as the difference indirections 2 is greater than some critical threshold. Otherwise,

the group will compromise.Our stripped-down model retains dynamically changing, localsocial interactions, but neglects some of the details of the zonal-based interaction rules of ref. 1. Nonetheless, it provides the

Table 1. Possible combinations of stable (S) and unstable (U)

manifolds given N3 >2N1

M101 M110 M000 M010 M001 M100 M011 M111U U S S S U U U

U U S S S U S U

U U S S S S U U

U U S S S S S U

U U U U U S U S

Fig. 4. Stability of decisions (onM010andM001) versus compromise(onM111)illustrated in a plot of direction of uninformed subgroup 3 as a function of

preference difference 2. Herer= 0.707 and so c =2. A solid line denotesa stable solution and a dashed line denotes an unstable solution.

Fig. 5. Curves in the space of parameters2and rthat determine the stability

of manifoldsM010andM001and, thus,the stability of a collectivedecision. Inallplots, K1 = 2 andN1= N2 = 5. (Upper Left) Light gray parameter space corre-

spondsto stability ofM010andM001, independent of N3. (Upper Right) N3 = 11.(Lower Left)N3 = 50. (Lower Right) N3= 500. Dark gray parameter space cor-

responds to M010and M001being the only stable manifolds among the eightinvariant manifolds studied. The dark gray parameter space increases with in-

creasing number of uninformed individualsN3.

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same fundamental result in the case N1= N2 without requiringany additional modeling terms such as a forgetting factor oninformation that is not reinforced (12). Further, simulations ofthe continuous-time model with random terms suggest that theanalysis of the deterministic model is robust to a small level ofuncertainty (SI TextandFig. S2). In the caseN1 N2, our model

yields the same necessary and sufcient conditions for stabilityof a decision (seeSI TextandFig. S3for simulations). In the caseof a decision, simulations show a dominating region of attrac-

tion for the decision to move in the preferred direction of themajority informed subgroup (SI Text and Fig. S3), consistentwith ref. 1.

A decision in the continuous-time model corresponds to oneinformed subgroup and the uninformed individuals choosing tomove together in the same preferred direction. This decision differsslightly from the decision in ref. 1 where all individuals move to-gether in the preferred direction. However, the result is qualita-tively the same, and the continuous-time model has the advantageof analytical tractability. Indeed, the critical threshold c isexplicitlydened in Eq. 7 (and illustrated in Fig. 5, Upper Left). Thisthreshold provides a sharp condition for stability of the two sym-metric collective decision solutions versus stability of a compromisesolution. Further, the decision in ref. 1 can be recovered with thecontinuous-time model by the addition of a mechanism inspired bythe repulsive term in the dynamics of ref. 1.

The analytical tractability of the continuous-time model allowsformal investigation into the sensitivity of the decision-makingresults to model parameters. In particular, our analysis permitsa formal examination of the role of the uninformed population sizein the group decision-making dynamics. Our results provide formalevidence that an increase in uninformed population size N3 canimprove decision making for a group in motion by increasing thelikelihood that the group will make a decision rather than com-promise. A rst supporting result concerns the second stablecompromise solution on M111. This solution is worse for decisionmaking than the rst stable compromise solution because not onlydoes the group not make a decision, but also it moves in the di-rection opposite the average of the two preferred directions. Thepresence of a sufcient number of uninformed individuals preventssuch a solution, throwing off the delicate balance that is required

for its existence. Further, a largeenoughN3 limits the attractivenessof the rst stable compromise solution, making the sufcient con-dition for stability ofM111also a necessary condition.

A second supporting piece of evidence derives from the resultthat the minimum difference in preference direction required

for a group decision decreases with a decreasing sensing range(equivalently, an increasing thresholdron synchrony of directionssensed) (Eq. 7). This result suggests that the more local thesensing is, the better the sensitivity to the conict in preference;

when individuals sense too much of the group, the result is a l-tering of the local inuences and an averaged (compromised)collective response. By increasing the density of the group, evenby adding uninformed individuals, an individual can reduce itssensing range and keep track of the same number of neighbors; in

such a way an increase in population size of uninformed individ-uals lowers the critical difference in preference direction, makinga group decision more likely.

The third supporting result, illustrated in three plots in Fig. 5,shows that an increasing uninformed population size N3 increasesthe region of parameter space for which a decision solution is ex-clusively stable among the eight solutions studied. A sufcientlylarge number of uninformed individuals throws off the delicatebalance for the uninformed individuals to be connected to both in-formed subgroups without the two informed subgroups connecting

with each other (M011). The uninformed individuals provide a kindof glue; indeed, the larger N3 provides the same effect as in-creasing the social interaction strength K1. Overall, the result showsthat with larger numbers of uninformed individuals, a collectivedecision is more likely and more robust to variations in parameters

r(sensing range) and

2 (difference in preferred directions).The improvements we have shown in decision making withincreased uninformed population size are striking and providea testable result. Adding individuals that do not invest directly inan external preference provides a low-cost way in which groupscan enhance decision making. Our analysis addresses the sym-metric case of a group in motion in which there are two equallysized informed subgroups, each preferring to move in one of twoalternative directions. Our results on stability of decision versuscompromise persist in the case of unequal sized informed sub-groups. In related work (24), we study the inuence of un-informed individuals in the case that there is heterogeneityamong informed individuals in the strength of their response topreference relative to social interactions.

ACKNOWLEDGMENTS. This research was supported in part by Ofce ofNaval Research Grant N00014-09-1-1074 (to N.E.L., T.S., L.S., and I.D.C.), AirForce Ofce of Scientic Research Grant FA9550-07-1-0-0528 (to N.E.L., B.N.,and L.S.), Defense Advanced Research Planning Agency Grant HR0011-09-1-0055 (to S.A.L., N.E.L., and I.D.C.), and Army Research Ofce Grant W911NG-11-1-0385 (to N.E.L., I.D.C., and S.A.L.).

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