Swarm Cognition in Honey Bees
Kevin M. Passino
Dept. Electrical and Computer Eng.
Ohio State University
2015 Neil Avenue, Columbus, OH 43210, USA
passino@ece.osu.edu
Thomas D. Seeley
Dept. Neurobiology and Behavior
Cornell University
Ithaca, NY, 14853, USA
tds5@cornell.edu
P. Kirk Visscher
Dept. Entomology
University of California Riverside
Riverside, CA 92521, USA
kirk.visscher@ucr.edu
September 27, 2006
Abstract: We synthesize findings from neuroscience, psychology, and behavioral biology to show
that some key features of cognition in neuron-based brains of vertebrates are also present in beebased swarms of honey bees. We present our ideas in the context of the cognitive task of nest-site
selection by honey bee swarms. After reviewing the mechanisms of distributed evidence gathering
and processing that are the basis of decision-making in bee swarms, we point out numerous similarities in the functional organization of vertebrate brains and honey bee swarms. These include
the existence of interconnected subunits, parallel processing of information, a spatially distributed
memory, layered processing of information, lateral inhibition, and mechanisms of focusing attention
on critical stimuli. We also review the performance of simulated swarms in standard psychological
tests of decision making: tests of discrimination ability and assessments of distractor effects. We
conclude that relating swarm cognition to knowledge from cognitive neuroscience will be fruitful
for broadly understanding the mechanisms of group cognition in social species.
1
Introduction
The study of group decision making sometimes uses a “collective intelligence” or “superorganism”
perspective where the group is viewed as a single decision-maker (Holldobler and Wilson 1990;
Levine et al. 1993; Franks 1989; Seeley 1989 1995; Hinsz et al. 1997; Laughlin 1999; Camazine
1
et al. 2001; Surowiecki 2004; Conradt and Roper 2005). Moreover, it has been recognized for
some time that massively parallel inter-animal communications and actions could form a basis for
complex group cognition processes that are functionally equivalent to ones in neuron-based brains
(Markl 1985; Bonner 1988). The primary goal of this paper is to show explicit examples of how
several of the key elements underlying cognition in neuron-based brains of vertebrates are also
found in bee-based swarms of honey bees.
We focus on swarm cognition in the context of the group decision-making process whereby a
swarm of bees selects its new nest site. Honey bee swarms are formed in the spring through a process
of colony fission wherein the mother queen and about half the workers in a colony leave their nest
and form a cluster on a nearby branch (the biology of swarming is reviewed in Winston 1987). Scout
bees fly from the cluster to search the countryside for potential dwelling places, usually cavities in
trees. Discovered nest sites of sufficient quality are reported on the cluster via the scouts’ waggle
dances, which recruit other bees to evaluate the sites. Higher quality sites evoke stronger dancing
and hence more recruits. When, via recruitment, 10-20 bees are assembled at a candidate nest
site a quorum threshold is reached which triggers choice of the site. The scouts from the chosen
site return to the cluster, initiate lift off, and the swarm flies as a group to the new nest. The
study here of what we call “swarm cognition” during this nest-site selection task uses the model in
(Passino and Seeley 2006) (others are in (Britton et al. 2002; Myerscough 2003)). This model was
validated using the experiments in (Seeley and Buhrman 1999; Camazine et al. 1999; Seeley and
Buhrman 2001; Seeley 2003; Seeley and Visscher 2003 2004b). Overviews of the biology of nest-site
selection are given in (Seeley and Visscher 2004a; Seeley et al. 2006).
Beginning at the “cognition unit-level” (analogous to the cellular neuron-level), we build a
detailed explanation of the nest-site selection process to identify the key elements and functional
organization of swarm cognition. We show that the swarm has identifiable elements that correspond
to neurons, action potentials, inter-neuron communications, lateral inhibition, short-term memory,
neural images, and layers of processing (Kandel et al. 2000). We identify functional similarities to
the networks of neurons that perform certain attention, perception, and choice functions in solitary
animals (Gazzaniga et al. 1998; Kandel et al. 2000). Via simulations we show that the swarm’s
short-term memory (what we call “group memory”) is on average a representation of the relative
quality of the discovered nest sites that leads to good choice performance.
For nest-site selection the swarm benefits by quickly choosing the best discovered nest site. Time
pressure arises due to energy costs and weather risks to the exposed cluster. Also, the quality of
the nest affects hive fitness. Hence, we take the view that the nest-site selection task is a type
of “reaction-time test” studied in psychology (Luce 1986). Such tests have received significant
attention in mathematical and cognitive psychology where diffusion (Ratcliff 1978), accumulator
(Usher and McClelland 2001), and other mathematical models (Busemeyer and Townsend 1993)
have been used for behavioral-level representations of perception-selection reaction-time tests for
humans and animals (see overviews and applications in (Luce 1986; Busemeyer and Townsend 1993;
Ratcliff et al. 1999; Roe et al. 2001; Ratcliff and Smith 2004; Smith 2000)). Such models have been
used for studies of choice speed-accuracy trade-offs, ones studied also for social animals (Franks et
al. 2003; Passino and Seeley 2006; Marshall et al. 2005) and human groups (Karau and Kelly 1992).
Here, we show that our model of swarm nest-site selection (Passino and Seeley 2006) shares important features with the above-listed models for human decision making behavior, but also key
differences. Moreover, we use methods from (Gazzaniga et al. 1998; Treisman and Gelade 1980;
Ratcliff 1978; Roe et al. 2001) to show that for simulated choice tests swarms exhibit several of
the same properties, and choice error characteristics, as humans and animals with neuron-based
2
cognition. Two categories of tests are administered in simulation: discrimination and distraction.
First, we illustrate that the swarm has discrimination abilities that allow it to distinguish between
two relatively close quality nest-sites. This ability is especially strong when site quality differences
are large. Discrimination abilities are amplified for low quality site comparisons and are achieved
in spite of significant assessment noise at the level of individual bees. Second, we show that the
swarm can effectively eliminate from consideration many low quality distractor sites. However, if
the distractors’ qualities are high enough the swarm can make many errors since the best site is
essentially hidden in a field of adequate-quality sites that compete for the swarm’s attention. Supporting evidence for our conclusions on swarm choice behavior is provided in the supplementary
on-line information in the Appendix. Moreover, we discuss implications for group decision making
in other species.
2
Process Dynamics of Nest-site Selection
The model from (Passino and Seeley 2006) is summarized via the flow-diagram in Figure 1. Consider
a sequence of “expeditions” indexed by k = 0, 1, 2... by B “scout” bees that take on different roles
in the nest-site selection process (we use B = 100). When an “explorer” bee finds a candidate
nest site it evaluates its attributes in order to form a quality assessment. We denote the quality of
nest j as N j ∈ [0, 1] with “1” representing a perfect site. We let the position of bee i be θ i , and
J(θ) denote the “landscape” of nest quality, with θ = [0, 0]⊤ the position of the cluster. We have
J(θ i ) = N j if bee i is at site j, but bee i has assessment noise wi (k), and a quality threshold ǫt = 0.2
below which it will ignore a site. Hence, bee i’s assessment of a site is S i (k) = J(θ i (k)) + wi (k),
if J(θ i (k)) + wi (k) > ǫt , and zero otherwise. Here, wi (k) is uniformly distributed on (−0.1, 0.1)
to represent up to a ±10% error in scouts’ nest-site quality assessment. Any bee that finds an
above-threshold nest site dances for it, and hence becomes “committed” to that site. Bees die with
a small probability pd = 0.0016 on each expedition so that less than 10% die over the whole process.
An unsuccessful explorer returns to the cluster and seeks to observe a dance. The expedition
that bee i first discovers site j is kji and if the sensed quality of the site is above the quality
threshold, this is a successful explorer and it returns to the cluster and dances with a strength
Lij (kji ) = γS i (kji ) waggle runs where γ = 150. After dancing, this committed bee returns to the
site, and then back to the cluster, possibly several times; however, each time it returns it dances
ǫs = 15 fewer waggle runs than the initial time. The sequence of waggle runs produced by bee i
over the whole process is Li and the total number of waggle runs produced on the cluster for all
sites at step k is Lt (k). We call a sequence of dances by one bee for one site, from the time of the
initial dance to when the dance strength goes to zero, a “dance decay series.” After a committed
scout’s dance strength has decayed to zero it rests, and rejoins the process (by seeking to observe a
dance) at each step with a probability
pm = 0.25. Bees that seek to observe will end up exploring
2
1 Lt (k)
where σ = 4000, representing that when there is not much
with probability pe (k) = exp − 2 σ2
dancing on the cluster (small Lt (k)), then there will be more exploring, and vice versa. There are
Br (k) resters, Bo (k) bees that seek to observe, Bu (k) = Bo (k) + Br (k) uncommitted bees, and
Bc (k) committed bees. With probability 1 − pe (k) observer bees will observe dances, and with
i (k)
probability pi (k) = PBL
will be recruited by the ith committed dancing bee. Bees recruited
c (k) i
i=1
L (k)
to site j visit and dance for it according to their own assessment as described above.
3
Swarm cluster
B is total number of scouts
Bcis number of committed scouts
pe probability
will explore
when seek
to observe
Probability
recruited by
bee i, p i
Tree
Recruited bees (visit sites)
Nest site 1
Committed bees (after dance, revisit)
Dance
j
decay γ S
series
i
L (k)
i
εs
k, expedition
Observe
dances
Candidate nest sites
Explorers (Betotal)
Waggle dances
Go
da
Seek to
observe dances
Bo observers
nc
eo
nc
lus
ter
pm probability rester will
seek to observe
Br resters
i
L (k) is dance strength of bee i
ij
L (k), for site j, Lt(k) is total
γ is initial dance strength for
time bee i discovers j (k ji )
ε s dance decay rate
Committed scout
Explorer (successful)
Bu= Bo+Br is number of
uncommitted scouts
Committed scout (that finished dancing for site)
Explorer (unsuccessful)
θ
J(θ i )
Nj
Sj
wi
εt
bee position
nest-site quality landscape
quality of nest j
bee’s assessment of j
assessment noise
quality threshold
J(θ)
N
j
θ1
θ2
Bd bees die at k, Bdt total
pd is probability of death
εq
Nest site j
quorum
threshold
Agreement time Ta,
piping, heating, lift-off
Figure 1: Nest-site selection process.
Even though at any particular time the swarm of bees is spatially distributed across the environment (explorers), candidate nest sites, and the cluster, it is best to think of the process dynamics
in terms of the simultaneous activities at all of these places. So, a key part of the process occurs
simultaneously with the activities on the cluster via the nest-site quality assessments and quorum
sensing at each candidate nest site where the bees sense the number of other bees at the site they
are visiting. When the quorum threshold ǫq is reached, bees return to the cluster, they produce
piping signals that elicit heating, and then the entire swarm lifts off flies to the chosen site. Each
expedition is assumed to take 30 min, and the maximum amount of time to decide is set at 32 hrs,
so there are up to 64 expeditions. Due to the time pressure to decide, the possibility of close-quality
alternatives, and randomness, there can be simultaneous quorum achievement at two or more sites
resulting in a “split decision.” In this case, the process is restarted by having the swarm reform the
cluster. Also, the process can fail to come to agreement before 64 expeditions are completed; this is
called a “no-decision failure.” These failures can arise if a site of sufficient quality is not discovered
early enough, one that will generate a recruitment rate that will assemble the required ǫq bees at
a nest site.
2.1
Distributed Evidence Gathering and Feedback
Within the decision-making process, the evidence of the quality of a site takes two forms: dances on
the cluster and bees at the sites. First, bees that are committed to a site perform a series of dances
for it on the cluster to advertise it to the observers. The number of bees recruited to each site
occurs in proportion to the number of waggle runs for each site at each step. If a recruit assesses
4
the site to be of similar quality to what its recruiter found, it will perform a similar dance decay
series. A positive feedback is created since the number of recruiters and their recruited bees will
grow exponentially (assuming an infinite pool of recruits). There are, however, also two types of
negative feedback, one induced by dance decay rate ǫs and the other by scout deaths as defined by
pd . Since the probability of death on an expedition is low, its impact on the dynamics is small.
However, the impact of the dance decay rate impact is large. Since decay rate is the same for
different quality sites, the dancing for poor sites will quickly fade; hence there are few recruits to
poor sites. This ensures that not too much time or too many assessors will be dedicated to poor
sites.
The second form of evidence of a site’s quality is the number of bees assembled at the site.
Assuming that the site is not independently discovered by many bees, if several bees are at a site
it is due to recruitment to the site which only occurs if the site is of adequate quality. And, if
there are many bees at the site, this could only occur if all those bees had judged it to be of good
quality. The numbers of bees visiting each site changes dynamically as new sites are discovered and
evidence is gathered by bees. Recruitment at the cluster is driven by the number of bees visiting
the sites, yet the recruitment also impacts the number of bees visiting the sites. This tight coupling
between activities at the cluster and candidate sites demands a broad perspective on the spatially
distributed dynamics of the process, one where equal attention is given to all locations where the
bees congregate, not just the cluster. Indeed, concurrent quorum sensing activities occur at all the
candidate nest sites and when one or more quorum thresholds are reached, this signals that the
process is complete; without the quorum sensing the process will not terminate.
2.2
Mechanisms of Selection
The ability of the swarm to discriminate between sites of different quality depends on an interplay
between positive and negative feedback. We illustrate this via a simple example. Assuming no
assessment noise, if two bees are dancing on the cluster for two different above-threshold sites j
′
and j ′ with qualities N j > N j , then the difference in the number of recruits for the two cases (and
′
hence strength of positive feedback) is proportional in a nonlinear way to the difference N j − N j .
′
′
As an example, first consider two low quality sites N j = 0.4 and N j = 0.2 so N j − N j = 0.2. A
′
bee that dances for site j ′ does so with γN j = 150(0.2) = 30 waggle runs in the first visit back to
the site, and then 15 fewer, or 15 waggle runs on the second visit. Then, the dance decay series
ends. The total number of runs in this case is 45. In contrast, a bee that dances for site j has
a dance strength sequence of 60, 45, 30, 15, 0 for a total of 150 waggle runs in its dance decay
series. The percentage increase when quality increases from 0.2 to 0.4 is 105
45 × 100 = 233%. Hence,
there are many more recruits for the better site compared to the inferior site. Now, if there are two
′
′
relatively high quality sites N j = 1 and N j = 0.8, again with N j − N j = 0.2, the total numbers
of waggle runs per dance decay series for qualities of 0.8 and 1 are 540 and 825 respectively. The
percentage increase when quality increases from 0.8 to 1 is 285
540 × 100 = 53%, much smaller than
the above case. Hence, when site qualities are relatively low, a small difference in quality leads to
a big percentage difference in the number of dances and the positive feedback gain on recruitment
rate to that site. So, the swarm has a a very good ability to discriminate between nest-site qualities
when they are relatively low quality sites When site qualities are higher, the same size difference in
quality leads to a lower percentage difference in the number of dances and hence recruits. Hence,
the swarm’s site discrimination abilities for high quality sites is less than for low quality sites.
At the same time, it is important to highlight that the evidence gathering dynamics have three
5
features that make it difficult for a swarm to always succeed at picking the best nest site. First,
each bee has a noisy assessment of the quality of any site, hence all evidence is noisy. The noise is
amplified by the nonlinear relationship between differential site quality and the number of recruits,
which helps discriminate between sites that differ in quality. However, the quality assessment noise
of each individual bee is filtered out at the group level to a significant extent by:
1. Averaging the effects of multiple dancing bees on the cluster: Some bees will assess quality
low, and some high, but the relative mean number of dances per site will closely represent
the relative site quality.
2. Quorum threshold effect: The proportioning of dancing on the cluster results in a proportioning of bees assessing each site in accordance with its relative quality. Since the number
allocated to a site must be above the quorum threshold before it is chosen this ensures that
averages are taken over a sufficient number of error-prone bees so that the swarm does not
make mistakes.
The second feature that conspires against the swarm making the best choice is the simple
observation that evidence of the quality of a site, in the form of returning bees that dance, arrives
at the cluster asynchronously. Evidence arrives randomly in time depending on the time a site is
discovered and the bee returns to dance for it, and the times of occurrence of subsequent dances by
recruits. Even high quality discoveries near the very end of the process will be unlikely to impact
the swarm’s choice since many of the scouts are apt to be committed to other sites and hence not
available for recruitment. It is possible that a relatively low quality site discovered early in the
process will be chosen, especially if no high quality site is found, since then there is enough time
so that enough bees can be assembled at the poor site to achieve the quorum threshold.
The third feature that negatively impacts accurate choice is the presence of “distractor” nest
sites, ones of sufficient quality to be evaluated, but that should not be chosen since they are of
inferior quality. Strong negative feedback due to dance decay generally enables the swarm to
consider a distractor site only briefly. However, if there are many distractors the strength of the
positive feedback for the best site is attenuated, perhaps even low enough that a distractor is
chosen. Essentially, the swarm is too busy paying attention to many inferior sites to allow it to
focus on the assessment and choice of the best site. Noise can amplify this distraction effect, but
the averaging discussed above will reduce its effect at the group level.
2.3
Search-Select Phases and Dynamic Internal Coupling
Nest-site selection has a search phase and a selection phase. Even though these two phases are
always interleaved and simultaneous, and hence blend together, it is useful to characterize their
general features. The search phase is characterized by a high value of Be (k), relatively low value
of Bc (k), and a small value of Lt (k) since there is not much dancing on the cluster since not many
bees are visiting sites. The select phase is characterized by the exact opposite of this situation so
that Lt (k) is higher, pe (k) is lower, not as many bees explore, and this raises the positive feedback
on recruitment so that a quorum threshold ǫq can be quickly reached and a crescendo can occur
resulting in agreement at time Ta .
The feedback processes discussed above are modified by internal coupling (cross-inhibition)
between variables associated with different candidate nest sites. Recruitment to one site means
6
that the recruits will not be recruited to other sites until they possibly dance for the site they
are recruited to, rest, and re-enter the process as a new recruit rather than an explorer. Hence,
if the total strength of dancing on the cluster and number of visitors to one site goes up, bees
are inhibited from visiting other sites, and the number of visitors at other sites may even decrease
(since if bees finish dancing for a poor site they will be likely to be recruited to the better one).
The size of the pool of uncommitted scouts and the phase of the process (search vs. select)
both impact the type of internal coupling. If there is a large amount of dancing on the dance
floor, then independent of the number of bees producing the dances, the probability of exploring
will go down, the probability to follow dances will go up, which then strengthens the potential
for positive feedback due to recruitment. If there are many distractors, then there are generally
fewer uncommitted scouts, but still a relatively low amount of total dancing, so that exploration
can stay high when not enough good sites of sufficient quality have been found. If a large number
of uncommitted bees is available, then recruiters can achieve their maximum recruitment rate
which will tend to focus the swarm on a single great site and inhibit consideration of relatively
low quality sites. Hence, in the beginning of the process (during the search phase) there is a type
of flexibility in that many alternatives are often considered, but their consideration is coupled in
complex ways depending on discovery times and qualities. The nature of the coupling changes as
the phase switches from search to selection. Near the end of the process, the coupling is such that
there is a strong tendency of the swarm not to consider new evidence. Normally, the relatively low
quality alternatives have become significantly less visited due to dance decay, and the best one gets
many more recruits and the crescendo occurs. Hence, the cross-inhibition helps to avoid having
oscillations between different alternative choices, at least when there are clear quality differences.
2.4
Dynamics Create a Speed-Accuracy Trade-off
There is a trade-off between speed and accuracy of choice (Passino and Seeley 2006) that is a direct
consequence of the dynamics. Generally, error rate reduction typically costs more time Ta or more
dances, or both. The time when agreement is reached, Ta , is a random variable that is affected by
the pattern of times of discovery of sites, their qualities, and indeed all aspects of the dynamics
P
described above. Let
Lt denote the total amount of dances that have been performed by time
Ta . The agreement time is in a sense controlled by the swarm: the swarm will reach agreement
when it has evaluated enough alternatives of sufficient quality to make it likely that it picks the
P
best one. Natural selection will favor reducing Lt so that the swarm invests less time and energy
in scouting and dancing to achieve a decision. Also, it will favor reducing Ta so that the swarm can
lift off, fly to the new nest, and quickly establish its new home. Generally, choice errors are made
P
in trying to minimize Ta and
Lt . The main mechanism that tries to make the decision occur
quickly is the positive feedback aided by cross-inhibition that forces discrimination and a quick
transition from the search to select phases. Correspondingly, the negative feedback tends to slow
the process. The balance between the two, mediated by coupling and randomness in assessments
and discovery times, leads to the agreement time.
Specific scenarios serve to highlight how the dynamics affect the speed-accuracy trade-off. First,
if there are no sites discovered for a period of time, or only low quality sites are discovered, then
the positive feedback is not strong enough to completely switch from a search to a select phase,
which is good since the swarm needs more time to find a sufficiently high quality site. There can
also be an increase in Ta due to the presence of similar quality sites that require more “deliberation
time” (coupled dynamic interactions that inhibit each other from achieving a sufficient recruitment
7
P
rate and hence a quorum), which normally costs more dances (i.e., an increased
Lt ). Such
extra deliberation time can even result from the need to distinguish between inferior distractors of
P
different quality. When luckily a great nest site is found quickly, both Ta and Lt can be relatively
low and the error rate can be low.
3
Features of Swarm Cognition
If N j ∈ [0, 1] is the quality of nest site j, its relative quality is N j / j N j . For nest site j, let dj
be the distance from the hive to site j and let φj be the angle to site j for a coordinate system
with origin at the cluster and the x axis pointing to an appropriate reference point, the one used
P
in bee dance communication on the cluster. Let i Lij (k) denote the total number of waggle runs
P
for site j at step k; hence “ i ” denotes the sum over all bees dancing with (dj , φj ). The total
P P
number of waggle runs for site j up to Ta is denoted by k i Lij (k). Its mean over many nestP P ij
site selection processes is E[ k i L (k)]. This expectation is a measure of the total amount of
signaling (advertisement) for site j. For plotting purposes we will consider the relative mean total
P P
P
P P
amount of dancing for site j, which is E[ k i Lij (k)]/( j E[ k i Lij (k)]). Let B(j, k) be the
number of bees that visit site j at step k. Then, maxk B(j, k) denotes the maximum number of bees
that visited site j up to Ta . Its mean over many nest-site selection processes is E[maxk B(j, k)].
This expectation is a measure of the maximum number of bees that a site can “muster” before
the agreement is reached when its value reaches quorum for some j (the chosen site). For plotting
P
purposes, we use the relative mean E[maxk B(j, k)]/( j E[maxk B(j, k)]) for each site j.
P
3.1
Interconnected Cognition Units
An individual scout bee is the fundamental unit of cognition (analogous to a neuron). The unit can
change its role and location. It can sense, act, and communicate with other units. The swarm’s
sensory process is the sum of the sensory processes of all the scout bees. The swarm’s spatial
“field of view” (Kandel et al. 2000) is dynamically modulated by scout bee choices (e.g., during
exploration or site assessment) to appropriately span many square kilometers and possibly many
candidate nest sites.
Dances by bee i of strength Lij for site j are analogous to action potentials (Kandel et al. 2000).
For this, there is an assessment of quality relative to threshold ǫt before such dancing is activated
(analogous to activation thresholds (Kandel et al. 2000) in neurons). The dances play a special
role in that they act as signals between different units. If we view bees as nodes and bee-to-bee
dance communications as directed arcs, a time-varying “random graph” can describe the dynamic
interconnection topology of the inter-bee communications on the swarm cluster. The form of
the graph is driven by where bees dance on the cluster and which bees happen to be present and
observe this dancing. If another graph is used to represent the union of all communications until the
agreement time, there would be arcs clustered around bees that discovered relatively high quality
sites, and arcs pointing from the bee that first discovered each site on a path (of recruiting recruits)
to the last bee recruited for that site. Hence, while it is not a fixed network of communicating units
as is often the case for neural networks, the interconnection topology for bee cognition units has a
structure with some predictable properties.
8
3.2
Group Memory
The group of bees committed to nest site j is a “population” of units (analogous to a population
of neurons (Kandel et al. 2000)) that can accumulate quality evidence for the site. The set of
such groups for each discovered site serves as a short-term “group memory” for the swarm. The
group memory is an internal model of the pattern of nest-site qualities currently in the swarm’s
field of view (analogous to a “neural image” (Kandel et al. 2000)). The group memory is spatially
distributed across the cluster and the candidate sites with the distribution defined by the current
locations of the committed scouts in the populations for each site j:
1. Group memory at the cluster: Group memory on the cluster can be decoded from the cluster
surface where dances are performed. For each dancing bee, the value of (dj , φj ) can be found.
Then the populations of bees can be formed for each nest site j. The dancing on the cluster
at any step k < Ta is a representation of the cluster’s current estimate of the nest-site quality
P P
landscape. Mathematically, when j i Lij (k) > 0
P ij
i L (k)
P P
ij
iL
j
(k)
is the relative total amount of waggle runs for site j at step k, the cluster’s internal model of
P
N j / j N j at time k. If a site has not been found it will not be in the representation. The
representation becomes more accurate as more bees visit the site (the averaging filters the
noise due to individual assessment errors).
2. Group memory at the candidate nest sites: The component of group memory distributed
across the nest-sites is composed of the number of bees at each candidate site B(j, k). We
call B(j, k) the current “swarm preference” level for each nest site j. When swarm preference
for site j reaches the quorum threshold, site j is chosen.
For examples of the dynamical time evolution of the cluster internal model estimate and swarm
preferences see (Passino and Seeley 2006) or for corresponding experimental data see (Seeley et
al. 2006).
The swarm’s distributed group memory is distinct from the internal neural-based memory of
each individual bee. What is known by the swarm is actually far more than the sum of what is
known by the individual bees since the swarm’s knowledge includes what is in the bees’ brains and
information coded in the locations of the bees and their actions. No bee can know all the locations
and activities for all other bees. But this information is coded at the swarm level and, as discussed
below, is explicitly used in swarm decision making.
3.3
Layered Early/Late Processing at the Cluster and Nest Sites
Individual bees can sense group level variables and memory, but with noise, and this provides for
“layered” processing of information via “parallel and converging paths” as found in neural systems
(Kandel et al. 2000; Gazzaniga et al. 1998). There are three forms of sampling of group memory
variables:
1. Allocation to exploration vs. recruitment: By sensing delays in finding dancers, bees can
estimate the total number of dancing bees, i.e., a current group memory value. They can
9
use such a value to make decisions about whether to explore or get recruited. The individual
decisions can be error-prone (e.g., by being unlucky and not seeing any dancers even though
there are many), but on average the group of bees that use this cue can make a correct decision.
Indeed, as the sizes of groups of committed scouts increase, as is the case for relatively high
quality sites, the variance on the groups’ sample will decrease. Hence, group memory is in a
sense more accurate for better sites.
2. Proportionate recruitment: Even though a bee cannot measure the sizes of the individual
groups dancing for the candidate sites, they can be influenced directly by these values in that
they are proportionally recruited according to the proportion of dancing for each site. Hence
cluster-based group memory is used by the swarm to allocate the cognition units to nest sites.
Again, the key is that while the individual cannot estimate the proportions of bees dancing
for each site, the group of observers can. Moreover, they will do it more accurately for higher
quality sites.
3. Nest-site bee assembly: At each nest site the assembly of bees provides a group memory of
the number of bees that have assessed (or is assessing) its quality. Via the simultaneous and
self-referential process of quorum sensing, the bees take actions based on this group-level
memory by deciding when the process should be terminated.
Thus there is a layering of processing in the swarm, with bees playing roles in the early processing
of sensory signals and dancing, and roles in late processing where the overall level of dance activity,
proportions of committed bees, or numbers of bees assembled at sites are used.
3.4
Functional Relationships to Attention, Perception, and Choice
Key functionalities of neuron-based attention, perception, and choice processes (Gazzaniga et
al. 1998) are found in swarm decision making. For attentional systems, the group memory sets
the overall relative level of attention paid to site j at step k. More attention will be paid to higher
quality sites when they are found. Distractor nest sites will enter into swarm’s field of view, but
they will be purged from the memory by low average quality assessments and dance decay. There
is a coupling between late and early attentional processes where via recruitment, swarm resources
are directed to the more interesting (higher quality) points in the spatial field of view of the swarm.
Distractors require swarm resources and hence can degrade the quality of the focusing process.
Focusing dynamics are slowed by a cluttered field of close quality sites. Cross-inhibition between
site variables affects focusing dynamics.
Relative to neural perception and choice systems, there are receptive fields (Kandel et al. 2000)
set by individual bees. “Lateral inhibition” (Kandel et al. 2000) is closely related to what we
call cross-inhibition above and it defines coupling in the process due to: (i) individual bees not
comparing dances, (ii) abandonment via dance decay of relatively poor sites, (iii) the finite size of
the pool of potential recruits, and (iv) since sites winning recruits inhibit other sites from getting
recruits. Cross inhibition is driven by the distance between alternatives as in neural systems
(Kandel et al. 2000), but “distance” in this case is not, for instance, physical distances between
nest sites, but is defined via (i) relative site quality, (ii) closeness in times of discovery, and (iii) the
random interconnectedness of all site variables via the dance floor. There is a type of “two-point
discrimination” process (see above discussion) that is due to “on-center, off-surround” feature found
in neural perceptual systems (Kandel et al. 2000; Gazzaniga et al. 1998). There is a process of
10
“feature detection” (Kandel et al. 2000) that occurs in what can be viewed as the later processing by
the swarm. The “feature” that is detected in this late processing is the “best-of-N ” discovered sites.
There is a type of “winner-take-all” process that emerges via the quorum threshold achievement,
agreement, and lift-off and it is analogous to ones in neural systems (Gazzaniga et al. 1998).
3.5
Simulation: Group Memory Quality and Impact on Choice Performance
For the model in (Passino and Seeley 2006) we use 1000 simulation runs where sites 1-6 have
qualities of 0.5, 0.6, 0.7, 0.8, 0.9, 1, respectively (“case 2” from (Passino and Seeley 2006)). This is
a representative, yet challenging quality landscape since, for instance, the sites of low quality will
distract the swarm, and it will be difficult to discriminate between the sites of close quality.
Consider the top-right plot in Figure 2. This shows that group memory as measured by the
relative mean total amount of dancing for site j (cluster-portion, white bars) and relative mean
maximum number of bees visit site j (nest-site portion, light gray bars) is skewed per the relative
site quality (black bars). The skewing is seen as you scan from site 1 (low quality), to site 6 (high
quality), where the white and light gray bars increase more than relative nest-site quality. The
swarm’s group memory (both at the cluster and nests) views relatively low quality sites as worse
than they are (see site 1). The swarm’s group memory views relatively high site qualities (site 6)
as better than they are. The swarm’s group memory is used to make choices. The skew represents
that the swarm forms good memory for the task at hand; the skew helps the swarm discriminate
between sites. This shows that on average the swarm develops a useful group memory of the quality
landscape.
Next, consider the top- and bottom-left plots. The probability distribution of Ta shows that it
never decides too fast, or too slow (there were zero no-decision failures). The distribution is “heavytailed” which means that long agreement times are not highly unlikely. Early decisions are reached
via quick discovery of a great site, and few or no other good discoveries, so that a maximum level of
positive feedback is achieved and the quorum is reached quickly. Longer Ta times result from slow
discoveries and long deliberation times in trying to discriminate between sites. The distribution for
P
Lt has similar characteristics and is also somewhat heavy-tailed. Finally, the bottom-right plot
shows that almost no bees dance for two or more sites which illustrates that the swarm choice has
little dependence on bees switching from dancing for one site to another (i.e., choice is a group-level
phenomenon).
4
Swarm Choice Tests
Viewing the swarm as a superorganism, in this section we evaluate its choice performance. We
use the values of the behavioral parameters validated in (Passino and Seeley 2006), and consider
performance for several types of nest-site quality landscapes. For each landscape quality pattern
we use 100 nest-site selection processes that terminate with a single site chosen.
4.1
Discrimination
To test discrimination, let all sites have zero quality, except sites 5 and 6, which both start out at a
quality of N 6 = N 5 = 0.75 and differentially move to 0.5 and 1. The results are shown in Figure 3.
11
RelQual(blk);Cluster(wht)
Sites(ltgray);RelChoice(dkgray)
250
200
150
100
50
0
Number of times per Σ Lt bin
#splits=286; #no dec=0
300
0
5
0.2
0.1
0
10
Ta, hrs.
3
4
Nest site
Mean=gray dotted vert line
At T :MeanB =51.808;MeanB =18.634;Mean p =0.36807
200
150
100
50
1
1.5
2
ΣL
t
2.5
1
a
250
0
0.3
#Bees2.5,25,50,75,97.5%quant
Number of times per Ta bin
Vert lines=2.5,25,50,75,97.5% quantiles
3
4
x 10
c
2
5
6
e
e
80
60
40
20
0
0
2
4
#Nest sites visited
6
Figure 2: Simulation results, case 2 quality landscape. Top-left: Number of times terminated at Ta
with vertical lines the indicated quantiles (e.g., 2.5% of the cases terminated with a Ta less than
the left-most vertical bar) with the mean indicated by the gray-dotted line. Bottom-left: Similar to
P
top-left, but number of times terminated at Ta with Lt . Top-right: horizontal is site no., vertical:
P
black (relative site quality, N j / j N j ), white (relative mean total amount of dancing for site j,
P P
P
P P
E[ k i Lij (k)]/( j E[ k i Lij (k)])), light gray (relative mean maximum number of bees visit
P
site j, E[maxk B(j, k)]/( j E[maxk B(j, k)])), dark gray (proportion times chosen). Bottom-right:
quantile plot of number of bees that dance for zero, one, two, etc. sites.
Before interpreting the results, note that besides the horizontal axes, labeling of all plots in the
remainder of this paper is the same.
To interpret the results in Figure 3 note that ideally, once the two sites have different qualities
the swarm should always choose the best one. However, the results show that many errors are
committed by the swarm, especially for low values of differential quality. The swarm can, however,
amplify the quality differences (at a rate of increase higher than the relative quality of the two
sites) and often make the correct choice, with choice performance increasing as the differential site
quality increases. The slope of the curves in the top-right plot are positively correlated with the
level of discrimination ability. The slope of the choice percentage curve for the best site (site 6)
is about (1-0.5)/0.4=5/4; we will compare other values to this one (e.g., in the Appendix). When
12
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
1
15
10
5
0
0.1
4
0.4
0.4
0.2
0
0.5
0
0.1
0.2
0.3
0.4
0.5
1.6048
11.2
44
42
1.2036
8.4
33
31
0.8024
5.6
22
21
0.4012
2.8
11
10
0
0.1
0.2
0.3
0.4
Differential quality
0.5
0
e
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
55
52
0
Split(x),no dec(o)(right)
Mean#Bees visit
t
0.3
0.6
14
x 10
Mean (.)/std(+)Σ L
0.2
0.8
0
0
0.1
0.2
0.3
0.4
Differential quality
0.5
c
20
a
T , hrs
25
MeanB (x)MeanB (+)
%choice(blk);%qual(gray)
30
0
Figure 3: Discrimination effect, linear differential quality. Top-left: middle line in each box is
the median value of Ta , boxes with notches that do not overlap represent that the medians of
the two groups differ at the 5% significance level, whiskers (dashed lines) represent 1.5 times the
interquartile range, and outliers are designated with a “+”. Top-right: percentage of times each
nest site is chosen (black lines), with site 1 designated by ⊲ (“tR” represents “triangle right”, a
right-pointing triangle), site 2 by △ (“tU” represents “triangle up”), site 3 by 3 (“dia”), site 4 by
2 (“sq”), site 5 by ◦, and site 6 by ∗. The gray lines with markers show the relative site qualities,
P
P
Lt (solid line,
N j / j N j . Bottom-left: left-vertical axis and the black lines show the mean
dots), and its standard deviation (dash-dot line, + marker), while gray lines and right-vertical axis
show the number of split decision (×) and no-decision (◦) cases that occur for the 100 nest site
selection processes that terminate with a single choice. Bottom-right: left-vertical axis and the
black lines show the mean number of bees out of the 100 total that visit 0 sites (designated with
⊲), 1 site (△), 2 sites (3), 3 sites (2), 4 sites (◦), 5 sites (∗), and 6 sites (×), and right-vertical
axis shows via the gray lines the mean number Bc of committed scouts (×) and mean number of
explorers Be (+) at the agreement time Ta .
the differential site quality is close to zero the swarm “generalizes” and treats the two sites as
having the same quality. Note that there is uniformly distributed noise in individual bee quality
assessments on (−0.1, 0.1), yet the swarm can often discriminate (due to filtering with multiple bee
assessments) two sites with less then a 0.1 quality difference. There is little change in the median
13
P
Ta or mean Lt for a change in differential quality. For low values of differential quality there are
many split decision cases due to the build-up of bees at each of the sites being similar due to the
close quality sites. The number of no-decision cases is higher for lower differential quality simply
because in that case there are no obviously superior sites for the swarm to pick. The number of
bees that visit no sites is around half the bees; these bees explore or rest. Of the bees that visit
sites, most visit only one. Of the bees that do visit sites, the relative mean number of bees visiting 2
sites is about 6/40=15% for all cases, which demonstrates that there is some coupling between site
variables via dance decrease to zero and then reentry to the process via recruitment to a different
site. The mean number of bees that are committed to sites and explore is relatively constant.
4.2
Distraction
To test distraction, let N 6 = 1, N 1 = 0, a distractor quality variable D = N 5 = N 4 = N 3 = N 2 ,
and consider D ∈ [0, 1]. See Figure 4. The key feature is that the swarm his very effective at
evaluating then discarding from consideration distractors so long as they are of a quality D < 0.4
since for this range the swarm always makes the correct choice. As the distractor quality D gets
closer to one, more errors are made. In fact, at about D = 0.55 the percentage of times the correct
site is chosen is about 80% as in (Seeley and Buhrman 2001). The percent of correct choices quickly
decreases for D > 0.55 where by about D = 0.8 the percentage of times the swarm picks the best
site is close to that for the distractors, even though the best site is markedly better. The median
value of Ta decreases significantly representing that the errors are made fast, yet the value of the
P
mean of Lt increases since the swarm tries to discriminate between many distractors and the best
site and this requires many dancers. For high distractor quality there are many split decisions and
the relatively high coupling in the process decreases and this is why the bees lock in on incorrect
decisions. The bees are busy discriminating between sites, but this reduces the coupling and hence
the number of correct decisions in this case. Indeed, the number of explorers decreases significantly
for high distractor quality since bees quickly find and recruit to all the sites and this leads to a
reduction in cross-inhibition between site variables so that there is no clear winner.
In the Appendix we study the impact of the number of distractors via a Treisman-type test
(Treisman and Gelade 1980), several interactions between discrimination and distraction effects,
context-dependent effects, the effect of individual bee assessment noise magnitude, and cognition
mechanism adaptation.
5
Discussion
This synthesis paper provides evidence that bee-based swarms have a cognition process that shares
key features with neuron-based brains, both at the level of the neuron/bee and at the level of the
brain/swarm.
5.1
Relations to Neuroscience and Psychology
In Section 3, we noted numerous similarities between swarms and brains in their functional organization for information processing. This entailed showing close relationships to key ideas from
neuroscience (Kandel et al. 2000; Gazzaniga et al. 1998), including analogs to: early sensory processing (field of view, receptive fields), neurons (bees), activation levels (quality thresholds), action
14
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
1
%choice(blk);%qual(gray)
30
20
a
15
10
5
0.2
0.8
Split(x),no dec(o)(right)
0
0.2
0.4
0.6
0.8
1
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
57
45.6
1.3076
34.2
0.8717
22.8
0.4359
11.4
0
0.2
0
1
1.7435
0
0.4
0.2
0.4
0.6
0.8
Distractor quality
1
0
47
42
35
31
23
21
12
10
e
t
Mean (.)/std(+)Σ L
x 10
0.6
0.6
0
0
0.2
0.4
0.6
0.8
Distractor quality
1
c
4
2.1793
0.4
0.8
MeanB (x)MeanB (+)
0
Mean#Bees visit
T , hrs
25
0
Figure 4: Distractor effect, four distractors (see Figure 3 caption for axes explanation).
potentials (dances), neuron populations (groups of dancing bees), neural network structure and
communications (bee-to-bee communications on a random topology), neural image and short-term
memory (spatially distributed internal model, group memory), lateral inhibition (cross-inhibition),
late processing based on group memory (for explore/get recruited allocation, proportional allocation of recruits, and quorum sensing), and parallel and converging paths (simultaneous assessment
of multiple sites, yet late processing for agreement on the best-of-N ). The study of group memory
accuracy in Section 3.5 showed that group memory on average provides the swarm with a representation of the relative nest-site qualities under consideration that enhances its choice performance.
Identification and evaluation of these cognition elements allowed us to relate swarm cognition to
well-studied attention-perception-choice tasks from cognitive neuroscience (Gazzaniga et al. 1998).
We explained at a conceptual level the surprising similarities to attentional systems with a view of
the swarm as eliminating distractors from consideration, and simultaneously focusing on the best
sites. Similarly, for perception-choice tasks we explained how the swarm considers its field of view,
develops a representation of its problem domain, and then uses that to choose.
The series of behavioral tests we administer in simulation for the swarm borrowed from ideas
in neuroscience and psychology (Gazzaniga et al. 1998; Treisman and Gelade 1980; Ratcliff 1978;
Roe et al. 2001). First, we performed discrimination tests and showed that in spite of significant
15
individual assessment noise the swarm was able to discriminate between close quality sites. In the
Appendix we show that the ability to discriminate depends not only on the differential quality,
but the absolute quality. In particular, discrimination performance improves for lower quality
sites. Next, we studied the impact of distractor sites. First, we showed that multiple distractors
can be effectively eliminated from consideration provided their quality is relatively low. However,
when distractors are of higher quality, fast but erroneous choices are made. It is as if the best
site is “hidden” by the set of distractors. In the Appendix we administer a version of a Treisman
feature search test (Treisman and Gelade 1980). This shows that choice performance only degrades
slightly if relatively low quality distractors are added to the task of finding the best quality site.
The original idea that this provides evidence of parallel processing in the brain (Treisman and
Gelade 1980) clearly also holds for the swarm. Moreover, we show that if the high quality site
is removed, the swarm takes much longer to decide which site to choose. It was proposed that
this was due to a switch in humans to a sequential search mode (see discussion in (Treisman and
Gelade 1980; Gazzaniga et al. 1998)). Here, however, the delay is clearly induced by the dynamics
of the process that leads to deliberation. This provides an alternative way to interpret the delays
that occur in reaction time tests and this may be useful to infer mental dynamics and structure in
other species.
To gain more insights into discrimination and distraction effects we report additional studies in
the Appendix. First, we show that distractors can attenuate the ability of the swarm to discriminate. Second, we show how discrimination mechanisms try to overcome the negative impacts of
distractors. Third, we study whether swarms exhibit “irrational” choice behavior commonly found
in human decision making (Luce and Suppes 1965; Huber et al. 1982; Tversky 1972; Simonson 1989;
Simonson and Tversky 1992) in the presence of “context-dependent effects” (i.e., certain patterns
of choice alternatives that can conspire to mislead the decision-maker). Context-dependence has
also been studied for human group decision making (Steiner 1966; Laughlin and Ellis 1986; Laughlin 1999; Kerr and Tindale 2004; Hastie and Kameda 2005) as summarized in (Hinsz et al. 1997).
We use an approach like in (Ratcliff et al. 1999; Ratcliff and Smith 2004; Roe et al. 2001; Busemeyer
and Townsend 1993) where humans are subjected to reaction-time tests. For a very wide variety of
nest-site quality patterns, our simulations show that the swarm cannot be tricked into misordering
its choice percentages in relation to the nest-site quality pattern. This implies that violations of
strong stochastic transitivity (Luce and Suppes 1965), the similarity effect (Tversky 1972), and
the comparison effect (Simonson 1989; Simonson and Tversky 1992) will not occur. Finally, we
show that for a special nest-site quality pattern (where a discrimination-distraction interaction is
induced) the attraction effect (Huber et al. 1982) can occur. However, this leads to improved choice
performance.
Diffusion, accumulator, and other models of reaction-time tests (Ratcliff 1978; Luce 1986; Ratcliff et al. 1999; Usher and McClelland 2001; Ratcliff and Smith 2004; Roe et al. 2001; Busemeyer
and Townsend 1993; Smith 2000) have a number of similarities to our model of nest-site selection
(Passino and Seeley 2006). For instance, they have representations of the time evolution of preference based on probabilistic arrival of evidence regarding multiple alternatives. There is internal
coupling between the dynamical evolution of the preferences and preference thresholds that trigger
choice. Moreover, a range of speed-accuracy trade-offs have been discovered with these models as
also found by Passino and Seeley (2006). In spite of these broad similarities, direct use of these
models to represent the nest-site selection process is not possible since: (i) unique nonlinearities
exist in the swarm (e.g., due to dance decay and explorer allocation); (ii) swarm cognition is a distributed process which leads to a significantly different “information structure” (i.e., when and how
16
variables interact with each other); (iii) cross-inhibition is present in the swarm but primarily via
the indirect path of dance decay and re-entry of bees to the recruitment process; and (iv) there are
spatially-distributed feedback paths in the swarm that couple the process of evidence gathering on
the cluster with preference dynamics at the discovered nests (e.g., as preference for a site increases
the swarm gathers more evidence on that site via dispatching more assessors).
Recent work from the diffusion model literature, however, does resonate with our approach.
Smith and Ratcliff (2004) describe their initiatives to use the diffusion models from psychology
to model neural-level signals in a reaction test for monkeys. This innovative work shows that the
well-studied behavioral-level diffusion models can actually model the system at the neural level.
Their descriptions of the dynamics of decision making for monkeys are strikingly similar to our
description of the nest-site selection dynamics in (Passino and Seeley 2006) and Section 2. The
model in (Passino and Seeley 2006) essentially starts at the other end of the spectrum, the “neurallevel” for the swarm, to create a model. This is possible due the unique feature that honey bee
swarms can be “dissected” and studied while functioning largely intact (Seeley 1995). This paper
expands on the work in (Passino and Seeley 2006) by identifying the elements of cognition and
adding a detailed behavioral-level assessment of swarm choice performance. So, while we have
proceeded in the bottom-up direction, and the work in (Smith and Ratcliff 2004) rests on a topdown method, we have met in the middle with models of reaction time tests. It is an important
direction to determine if the paths, functions, and dynamics in swarm cognition could give useful
hints for the search for neural circuit structures and dynamics.
5.2
Relations to Behavioral Ecology
The speed-accuracy trade-off and irrationality have also been studied in the field of behavioral
ecology. For instance, in hoarding gray jays simultaneous choice errors decrease as the rate of availability of choices decreases, since then choice errors are costly (Waite 2001; Waite and Field 2000;
Waite 2002). Honey bees and gray jays have been shown to exhibit context-dependent decision
making (Shafir et al. 2002). In these studies errors (“irrationality”) seem to arise due to sensory
noise, cognitive processing limitations, and physical constraints (which all cause choice errors in
nest-site selection also). Consideration of the work in (Shafir et al. 2002) on honey bees also naturally leads us to wonder if, for the same task and species, context-dependent effects are amplified
or attenuated when comparing cases of social and solitary choice.
Our analysis of the adaptive “tuning” of the processes underlying swarm cognition in the Appendix allows us to study the cognitive ecology (Dukas 1998) of the honey bee swarm. We showed
that several behavioral parameters underlying swarm cognition (e.g., ǫq , γ as studied in the Appendix, and ǫs in (Passino and Seeley 2006)) have values evidently shaped by natural selection to
balance speed and accuracy of choice. Moreover, we showed that the accuracy of the group memory is insensitive to bee-level assessment errors, and is also the result of natural selection balancing
choice speed and accuracy. The swarm exploits the information from multiple assessor bees, averages this information, and holds it in a group-level memory. Simultaneously, in late processing it
uses this information to efficiently allocate assessors and terminate the process via quorum sensing.
Our results demonstrate that the key component of swarm cognition, group memory, is a robust
component that mediates early and late processing.
17
5.3
Implications for Other Species?
It seems likely that elements and functionalities of cognition can be identified in other social animal
species and human groups. For instance, ants performing nest-site selection (Marshall et al. 2005;
Mallon et al. 2001; Pratt et al. 2002; Franks et al. 2002 2003; Pratt 2005; Pratt et al. 2005) use
quorum sensing in a way similar to honey bees. Hence, the number of tandem runs and quorum
levels for these ants may also correspond to types of group memory. Similarly, consensus decision
making in other species is likely to contain group memory (Conradt and Roper 2005). In this
paper, the physical elements of group memory were built from the biotic environment (groups of
bees performing activities at locations). In other species or decision-making processes the abiotic
environment could also be exploited for group memory. For instance, it is plausible to view the
network of pheromone trails laid by several species of ants as a type of abiotic group memory. Trails
are built in a distributed fashion by individuals, used to guide foraging activities of the group, and
forgetting occurs via trail evaporation (Holldobler and Wilson 1990). Other forms of “stigmergy”
(Camazine et al. 2001) may be related to group memory.
With respect to human group decision making, note that dynamics and evolutionary adaptation
are key current directions identified in (Kerr and Tindale 2004). To address these challenges, we
suggest that experiments be designed where human groups perform a time-constrained choice task
(that perhaps also involves searching for choice alternatives). Then, data representing time-histories
of individual decision making and individual-to-individual communications should be gathered and
represented with a mathematical model (unlike in (Kerr 1982; Abelson 1964) and perhaps via
extending the models in (Ratcliff 1978; Usher and McClelland 2001; Roe et al. 2001)). Simulations
can then be used to evaluate implied group choice dynamics, performance, and adaptation in the
context of speed-accuracy trade-offs. This could lead to the identification of features of human
group cognition mechanisms (e.g., group memory). Such discoveries could significantly deepen our
understanding of human collective intelligence and enable us to more effectively structure groups
to enhance their performance in business, economics, law, and politics (Steiner 1966; Laughlin and
Ellis 1986; Laughlin 1999; Kerr and Tindale 2004; Hastie and Kameda 2005; Surowiecki 2004).
Acknowledgements: The authors would like to thank B. Moore, P. Laughlin, R. Ratclliff, and
T.A. Waite for their inputs. The research reported here was supported in part by the U.S. National
Science Foundation (grant no. IBN02-10541 to TDS).
References
Abelson, R. P. (1964). Mathematical models of the distribution of attitudes under controversy.
In N. Frederiksen and H. Gulliksen (Eds.), Contributions to mathematical psychology (pp.
142–160). New York: Holt, Rinehart, and Winston.
Bonner, J. T. (1988). The evolution of complexity by means of natural selection. Princeton, NJ:
Princeton University Press.
Britton, N. F., Franks, N. R., Pratt, S. C., and Seeley, T. D. (2002). Deciding on a new home:
how do honey bees agree? Proc R Soc Lond B, 269, 1383–1388.
Busemeyer, J. R. and Townsend, J. T. (1993). Decision field theory: A dynamic-cognitive approach
to decision making in an uncertain environment. Psych Rev, 100 (3), 432–459.
Camazine, S., Deneubourg, J.-L., Franks, N. R., Sneyd, J., Theraulaz, G., and Bonabeau, E. (2001).
Self-organization in biological systems. NJ: Princeton Univ. Press.
18
Camazine, S., Visscher, P. K., Finley, J., and Vetter, R. S. (1999). House-hunting by honey bee
swarms: collective decisions and individual behaviors. Insectes Soc, 46, 348-360.
Conradt, L. and Roper, T. J. (2005). Consensus decision making in animals. Trends in Ecology
and Evolution, 20 (8), 449–456.
Dukas, R. (Ed.). (1998). Cognitive ecology. Chicago, IL: Univ. Chicago Press.
Franks, N. R. (1989). Army ants: A collective intelligence. American Scientist, 77, 138-145.
Franks, N. R., Dornhaus, A., Fitzsimmons, J. P., and Stevens, M. (2003). Speed versus accuracy
in collective decision making. Proc R Soc Lond B, 270, 2457–2463.
Franks, N. R., Pratt, S. C., Mallon, E. B., Britton, N. F., and Sumpter, D. J. T. (2002). Information
flow, opinion polling, and collective intelligence in house-hunting social insects. Proc R Soc
Lond B, 357, 1567–1583.
Gazzaniga, M. S., Ivry, R. B., and Mangun, G. R. (1998). Cognitive neuroscience: The biology of
the mind. NY: W. W. Norton and Co.
Hastie, R. and Kameda, T. (2005). The robust beauty of majority rules in group decisions. Psychol.
Rev., 112 (2), 494–508.
Hinsz, V. B., Tindale, R. S., and Vollrath, D. A. (1997). The emerging conceptualization of groups
as information processors. Psychol. Bulletin, 121 (1), 43–64.
Holldobler, B. and Wilson, E. (1990). The ants. Cambridge, MA: Belknap Press of Harvard Univ.
Press.
Huber, J., Payne, J. W., and Puto, C. (1982). Adding asymmetrically dominated alternatives:
Violations of regularity and the similarity hypothesis. J. Consumer Research, 9, 90–98.
Kandel, E., Schwartz, J., and Jessell, T. (Eds.). (2000). Principles of neural science. NY: McGrawHill.
Karau, S. J. and Kelly, J. R. (1992). The effects of time scarcity and time abundance on group
performance quality and interaction process. J. Experimental and Social Psychol., 28, 542–
571.
Kerr, N. L. (1982). Social transition schemes: Model, method, and applications. In H. Brandstetter,
J. H. Davis, and G. Stocker-Kreichgauer (Eds.), Group decision making. London: Academic
Press.
Kerr, N. L. and Tindale, R. S. (2004). Group performance and decision making. Annu. Rev.
Psychol., 55, 623–655.
Laughlin, P. R. (1999). Collective induction: Twelve postulates. Organizational Behavior and
Human Decision Processes, 80, 50–69.
Laughlin, P. R. and Ellis, A. L. (1986). Demonstratability and social combination processes on
mathematical intellective tasks. J. Experimental and Social Psychol., 22, 177–189.
Levine, J. M., Resnik, L. B., and Higgins, E. T. (1993). Social foundations of cognition. Annu.
Rev. Psychol., 44, 585–612.
Luce, R. D. (1986). Response times: Their role in inferring elementary mental organization.
Oxford: Oxford Univ. Press.
Luce, R. D. and Suppes, P. (1965). Preference, utility, and subjective probability. In R. D.
Luce, R. R. Bush, and E. Galanter (Eds.), Handbook of mathematical psychology (Vol. 3, p.
249-410). New York: Wiley.
Mallon, E. B., Pratt, S. C., and Franks, N. R. (2001). Individual and collective decision-making
during nest site selection by the ant Leptothorax albipennis. Behav Ecol Sociobiol, 50, 352–359.
Markl, H. (1985). Manipulation, modulation, information, cognition: some of the riddles of
communication. In B. Holldobler and M. Lindauer (Eds.), Experimental behavioral ecology
and sociobiology (p. 163-194). New York: Sinauer Assoc.
19
Marshall, J. A. R., Dornhaus, A., Franks, N. R., and Kovacs, T. (2005). Noise, cost and speedaccuracy trade-offs: decision-making in a decentralized system. Proc R Soc Lond B, ?, ??–??
Myerscough, M. (2003). Dancing for a decision: a matrix model for nest-site choice by honey bees.
Proc R Soc Lond B, 270, 577–582.
Passino, K. M. and Seeley, T. D. (2006). Modeling and analysis of nest-site selection by honey bee
swarms: The speed and accuracy trade-off. Behav Ecol Sociobiol, 59 (3), 427–442.
Pratt, S. C. (2005). Quoroum sensing by encounter rates in the ant Temnothorax albipennis. Behav
Ecol, 16, 488–496.
Pratt, S. C., Mallon, E. B., Sumpter, D. J. T., and Franks, N. R. (2002). Quorum sensing,
recruitment, and collective decision-making during colony emigration by the ant Leptothorax
albipennis. Behav Ecol Sociobiol, 52, 117–127.
Pratt, S. C., Sumpter, D. J. T., Mallon, E. B., and Franks, N. R. (2005). An agent-based model
of collective nest choice by the ant Temnothorax albipennis. Anim Behav, ?, ??–??
Ratcliff, R. (1978). A theory of memory retrieval. Psych Rev, 85, 59–108.
Ratcliff, R. and Smith, P. L. (2004). A comparison of sequential sampling models for two-choice
reaction time. Psych Rev, 111 (2), 333–367.
Ratcliff, R., VanZandt, T., and McKoon, G. (1999). Connectionist and diffusion models of reaction
time. Psych Rev, 106 (2), 231–300.
Roe, R. M., Busemeyer, J. R., and Townsend, J. T. (2001). Multialternative decision field theory:
A dynamic connectionist model of decision making. Psych Rev, 108 (2), 370–392.
Seeley, T. D. (1989). The honeybee as a superorganism. American Scientist, 77, 546-553.
Seeley, T. D. (1995). The wisdom of the hive. Cambridge, Mass: Harvard University Press.
Seeley, T. D. (2003). Consensus building during nest-site selection in honey bee swarms: the
expiration of dissent. Behav Ecol Sociobiol, 53, 417–424.
Seeley, T. D. and Buhrman, S. C. (1999). Group decision making in swarms of honey bees. Behav
Ecol Sociobiol, 45, 19-31.
Seeley, T. D. and Buhrman, S. C. (2001). Nest-site selection in honey bees: how well do swarms
implement the “best-of-n” decision rule? Behav Ecol Sociobiol, 49, 416–427.
Seeley, T. D. and Visscher, P. K. (2003). Choosing a home: how the scouts in a honey bee swarm
perceive the completion of their group decision making. Behav Ecol Sociobiol, 54, 511–520.
Seeley, T. D. and Visscher, P. K. (2004a). Group decision making in nest-site selection by honey
bees. Apidologie, 35, 1–16.
Seeley, T. D. and Visscher, P. K. (2004b). Quorum sensing during nest-site selection by honey bee
swarms. Behav Ecol Sociobiol, 56, 594–601.
Seeley, T. D., Visscher, P. K., and Passino, K. M. (2006, May/June). Group decision making in
honey bee swarms. American Scientist, 94 (3), 220–229.
Shafir, S., Waite, T. A., and Smith, B. (2002). Context-dependent violations of rational choice in
honeybees (apis mellifera) and gray jays (perisoreus canadensis). Behav Ecol Sociobiol, 51,
180–187.
Simonson, I. (1989). Choice based on reasons: The case of attraction and compromise effects. J.
Consumer Research, 16, 158–174.
Simonson, I. and Tversky, A. (1992). Choice in context: Tradeoff contrast and extremeness aversion.
J. Marketing Research, 24, 281–295.
Smith, P. L. (2000). Stochastic dynamic models of response time and accuracy: A foundational
primer. Math Psych, 44, 408–463.
Smith, P. L. and Ratcliff, R. (2004). Psychology and neurobiology of simple decisions. Trends in
Neurosciences, 27 (3), 161–168.
20
Steiner, I. D. (1966). Models for inferring relationships between group size and potential group
productivity. Behavioral Science, 11, 273–283.
Surowiecki, J. (2004). The wisdom of crowds. NY: Doubleday.
Treisman, A. and Gelade, G. (1980). A feature-integration theory of attention. Cog Psychol, 12,
97–136.
Tversky, A. (1972). Elimination by aspects: A theory of choice. Psych Rev, 79, 281–289.
Usher, M. and McClelland, J. L. (2001). The time course of perceptual choice: The leaky, competing
accumulator model. Psych Rev, 108, 550–592.
Waite, T. A. (2001). Background context and decision making in hoarding gray jays. Behav Ecol,
12, 318-324.
Waite, T. A. (2002). Interruptions improve choice performance in gray jays: prolonged information
processing versus minimization of costly errors. Anim Cog, 5, 209–214.
Waite, T. A. and Field, K. L. (2000). Erroneous choice and foregone gains in hoarding gray jays
(Perisoreus candensis). Anim Cog, 3, 127-134.
Winston, M. L. (1987). The biology of the honey bee. Cambriage: Harvard University Press.
21
Appendix: Supplementary On-Line Information
Alternative Quantifications of Group Memory
In this section we reconsider how to quantify group memory. Let L̄j (k) be the mean number of
dances for site j at time k (mean taken across all bees that are dancing with (dj , φj ), and zero if
no bees are dancing for it). Then,
L̄∗j = max L̄j (k) : 0 ≤ k ≤ Ta
is the maximum mean number of dances for site j before Ta . Let E[L̄∗j ] be the mean taken over
P
many nest-site selection processes, and then the relative mean for site j is E[L̄∗j ]/( j E[L̄∗j ]). This
relative mean is an alternative measure of group memory at the cluster. It represents the maximum
number of waggle runs that a site can muster.
P
P
Let k B(j, k) be the total number of bees that visit site j up to Ta and E[ k B(j, k)] is
the mean total number of bees that visit site j over many nest-site selection processes. It is
an alternative measure of group memory at the nest sites. It represents the total amount of
bees that visit a nest, and hence is a measure of the amount of signaling via quorum sensing at the site (notice the parallel with the cluster measure used earlier). The relative mean is
P
P
P
E[ k B(j, k)]/( j E[ k B(j, k)]) for each j.
Consider the top-right plot in Figure 5. This shows that group memory as measured by the
relative mean maximum amount of dancing for site j (cluster-portion, white bars) and relative mean
total number of bees visit site j (nest-site portion, light gray bars) closely matches the relative site
quality (black bars) (but there is no skewing as found in Section 3.5). These measures are not
the ones that are the basis for choice (right bars); if they were then poorer choice performance
would result. However, they do demonstrate that the swarm holds valid group memory when it is
measured via alternative metrics.
Discrimination Amplification
Let all sites have zero quality, except sites 5 and 6, which both start out at a quality of N 6 = N 5 =
0.65 and differentially move to 0.4 and 0.9. The results in Figure 6 show that discrimination ability
goes up relative to Figure 3, indicated by increased slope on the percentage of times the correct site
(site 6) is chosen as they move away from each other (in the initial part, the slope is approximately
(1-0.5)/0.3=5/3, compared to 5/4 when the sites start at 0.75). Hence, discrimination is better
when site qualities are lower since mistakes are more costly. There is a slight decrease in the median
P
value of Ta and mean value of Lt since in this case the quality of site 5 is much lower so that it
is not nearly as viable of a candidate so it is easier to choose the best site (i.e., without as much
deliberation). Notice that compared to Figure 3 there is more coupling in the process here due to
more bees abandoning the low quality site and switching to the high quality site (here, for some
values of differential quality about 11/40=27.5% dance for two sites).
Effect of Number of Distractors: Treisman Test
Let N 6 = 1 and N 5 = 0.55, then successively add sites 1, 2, 3, and 4 as additional distractors of
quality 0.55. So, there are a total of 2, 3, 4, and 5 distractors. This is a Treisman feature search
22
RelQual(blk);Cluster(wht)
Sites(ltgray);RelChoice(dkgray)
250
200
150
100
50
0
Number of times per Σ Lt bin
#splits=286; #no dec=0
300
0
5
0.2
0.1
0
10
Ta, hrs.
3
4
Nest site
Mean=gray dotted vert line
At T :MeanB =51.808;MeanB =18.634;Mean p =0.36807
200
150
100
50
1
1.5
2
ΣL
t
2.5
1
a
250
0
0.3
#Bees2.5,25,50,75,97.5%quant
Number of times per Ta bin
Vert lines=2.5,25,50,75,97.5% quantiles
3
4
x 10
c
2
5
6
e
e
80
60
40
20
0
0
2
4
#Nest sites visited
6
Figure 5: Simulation results, 1000 runs, case 2 quality landscape. Top-left: Number of times
terminated at Ta with vertical lines the indicated quantiles (e.g., 2.5% of the cases terminated with
a Ta less than the left-most vertical bar) with the mean indicated by the gray-dotted line. BottomP
left: Similar to top-left, but number of times terminated at Ta with Lt . Top-right: horizontal is
P
site no., vertical: black (relative site quality, N j / j N j ), white (relative mean maximum amount
P
of dancing for site j, E[L̄∗j ]/( j E[L̄∗j ])), light gray (relative mean total number of bees visit site
P
P
P
j, E[ k B(j, k)]/( j E[ k Q(j, k)]), dark gray (proportion times chosen). Bottom-right: quantile
plot of number of bees that dance for zero, one, two, etc. sites.
test (Treisman and Gelade 1980) with one “target” (the best site) that should be chosen, and a
variable number of distractors. The results in Figure 7 show that as more distractors of such a low
quality are added there is little effect on the percent of correct choices relative to when there are
P
only two distractors; however, this comes at the cost of an increased mean Lt for the swarm to
try to resolve the differences. Treisman and others took this as evidence of early parallel neural
processing of alternatives, early enough that it was not at the level of consciousness.
The main result, however, comes from comparing to the case where everything is the same
but we let N 6 = 0, that is, when the target is removed. In standard feature search tests human
subjects are asked to decide if the target is there or not, something we cannot request from the
swarm. Instead, we let the swarm come to a decision for this case and compare to the last case.
23
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
1
%choice(blk);%qual(gray)
30
20
a
T , hrs
25
15
10
0.8
0.6
0.4
0.2
5
0.1
0.4
0
0.5
Split(x),no dec(o)(right)
0.2
0.3
0.4
0.5
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
1.358
33.6
0.9053
22.4
0.4527
11.2
0.1
0.2
0.3
0.4
Differential quality
0.5
0
44
45
33
34
22
22
11
11
e
44.8
0
0.1
56
1.8106
0
0
0
0
0.1
0.2
0.3
0.4
Differential quality
0.5
c
t
Mean (.)/std(+)Σ L
0.3
MeanB (x)MeanB (+)
4
x 10
0.2
Mean#Bees visit
0
0
Figure 6: Amplified discrimination effect (see Figure 3 caption for axes explanation).
The results in Figure 8 show that as expected, with two distractors each is chosen 50% of the time,
3 are each chosen about 33% of the time, 4 are chosen around 25% of the time, and 5 about 20% of
P
the time. The mean Lt increases with more distractors but the median Ta goes up, then comes
down since it becomes easier to make the errors that occur with 5 distractors. Also, comparing
Figures 7 and 8, the median Ta values are significantly higher for the case when there is no target
compared to when the target is present. An analogous result is obtained in tests for humans, and
Treisman hypothesized that humans switched to a “sequential search mode” where by a process of
elimination they decided that the target was not present (Treisman and Gelade 1980; Gazzaniga
et al. 1998). For the swarm such a mode switch is not possible. The swarm simply takes longer to
decide due to the internal dynamics of the decision-making process being slowed by a more lengthy
evaluation of the evidence gathered.
Discrimination-Distraction Interactions
Earlier tests were primarily designed to illustrate isolated swarm discrimination abilities and distractor effects. In other nest-site quality landscapes, however, both effects are present and interact
with each other as we show in this section.
24
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
%choice(blk);%qual(gray)
25
5
4
Mean (.)/std(+)Σ L
t
2.2557
x 10
4
Split(x),no dec(o)(right)
1.3534
8.4
0.9023
5.6
0.4511
2.8
3
4
#Distractors
5
2
3
4
5
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
49
14
11.2
2
0.2
0
5
1.8046
0
0.4
0
43
40
32
30
22
20
11
10
e
3
Mean#Bees visit
2
0.6
0
2
3
4
#Distractors
5
c
10
0.8
MeanB (x)MeanB (+)
15
a
T , hrs
20
0
Figure 7: Effect of number of distractors, with perfect site present (see Figure 3 caption for axes
explanation).
A Distractor Can Attenuate Discrimination
First, we show that distraction can attenuate discrimination. To illustrate this, let all sites have
zero quality, except let site 4 have a quality of 0.5 and let sites 5 and 6 both start out at a quality
of 0.75 and differentially move to 0.5 and 1. We consider site 4 to be a distractor since it should
not be chosen. Notice that in Figure 9 the region of “generalization” (i.e., quality range where the
swarm treats qualities as similar) (Gazzaniga et al. 1998) grows relative to Figure 3 and the slope
of the line representing correct choices is about 1 (lower than the cases in Sections 4.1 and 5.3) so
that discrimination is attenuated by the relatively low quality distractor.
Discrimination Tries to Overcome Distraction
Let quality of site 5 be at 0.75, let the quality of site 4 vary as N 4 = D ∈ [0, 1] and consider site 4
to be a distractor for the range D ∈ [0, 0.75] since it should not be chosen for that range of values,
but it is the best site for D ∈ (0.75, 1] when you can view site 5 as the distractor (this is case
of nonlinearly decreasing differential quality). Of course, you could view the basic task as one of
25
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
0.5
%choice(blk);%qual(gray)
30
20
a
T , hrs
25
15
10
0.4
0.3
0.2
0.1
5
4
3
4
5
984.8
44
49
2.0929
738.6
33
37
1.3952
492.4
22
24
0.6976
246.2
11
12
2
3
4
#Distractors
5
0
e
2.7905
0
Split(x),no dec(o)(right)
2
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
55
x 10
Mean#Bees visit
t
0
5
1231
3.4881
Mean (.)/std(+)Σ L
4
0
2
3
4
#Distractors
5
c
3
MeanB (x)MeanB (+)
2
0
Figure 8: Effect of number of distractors, without perfect site present (see Figure 3 caption for axes
explanation).
discriminating between the two sites. The results are in Figure 10. First, note that discrimination
level is asymmetric in the sense that the swarm is better at discriminating when D ∈ [0, 0.75], but
discrimination is not as good for D ∈ (0.75, 1] (mistakes are not as costly in that region). There is a
region of generalization around 0.75. It is interesting that the median Ta values are relatively high
P
until the quality of the sites approach each other, and then the amount of dancing Lt increases in
order to discriminate between the sites, but then decreases as the sites move apart again. Also, the
median Ta value decreases significantly in the range D ∈ [0.8, 1] (since it is easier to make a quick
but incorrect decision) and note that there are fewer outliers. There are many no-decision cases
when site 4 has a low quality; this is due to the difficulties of finding the single relatively good site
5. Overall, this shows that distraction tends to have an effect on choice performance degradation
that discrimination tries to overcome.
26
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
%choice(blk);%qual(gray)
30
20
15
10
5
0.1
0.2
0.3
0.4
0.6
0.4
0.2
0
0.5
13961
14.4
10471
10.8
6981
7.2
3490
3.6
0
0.1
0.2
0.3
0.4
Differential quality
0.2
0.3
0.4
0.5
0.5
0
46
36
35
27
23
18
12
9
e
18
0
0.1
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
17451
Mean#Bees visit
Mean (.)/std(+)Σ Lt
Split(x),no dec(o)(right)
0
0
0
0.1
0.2
0.3
0.4
Differential quality
0.5
c
0
0.8
MeanB (x)MeanB (+)
a
T , hrs
25
0
Figure 9: Distraction can attenuate discrimination (see Figure 3 caption for axes explanation).
Context Dependence
Transitivity, Similarity, and Comparison Effects
We tested a broad range of nest-site quality landscapes to see if the swarm would ever misorder
the percentages of choices for sites in comparison to the order of relative nest-site qualities. Our
tests included all the simulation results shown in the discrimination and distraction tests above,
the general class of landscapes described at the beginning of the next section, the cases considered
in (Passino and Seeley 2006), and many specific landscapes many not reported here. We were
never able to “trick” the swarm: the percentage choice order is always given by the ordering of the
relative qualities of the nest sites (to see this, review all plots of choice performance in this paper).
Hence, the swarm never violates “strong stochastic transitivity” (Luce and Suppes 1965) which
can lead to choice errrors. Moreover, the “similarity” (Tversky 1972) and “comparison” effects
(Simonson 1989; Simonson and Tversky 1992) discussed in (Roe et al. 2001), both which represent
choice performance degradations, are not exhibited for the swarm.
27
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
1
%choice(blk);%qual(gray)
30
20
a
T , hrs
25
15
10
0.8
0.6
0.4
0.2
5
0.8
0
1
Split(x),no dec(o)(right)
1
39
22
26
19
11
13
0
0
0.853
38
0.4265
1
0.8
33
57
0.4
0.6
0.8
Distractor quality
0.6
51
1.2795
0.2
0.4
45
76
0
0.2
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
64
95
1.706
0
0
e
t
Mean (.)/std(+)Σ L
0.6
0
0.2
0.4
0.6
0.8
Distractor quality
1
c
4
x 10
0.4
MeanB (x)MeanB (+)
0.2
Mean#Bees visit
0
0
Figure 10: Nonlinear differential quality (see Figure 3 caption for axes explanation).
Attraction Effect
Let sites 5 and 6 have qualities 0.7 and 0.75, respectively, then take site 4 and let its quality vary
as N 4 = D ∈ [0, 1] to see its effect on the ability of the swarm to discriminate between sites
5 and 6. This test is designed to evaluate if the swarm exhibits the “attraction effect” (Huber
et al. 1982) even though there is only a single quality dimension that is being compared. Let
P (i|i, j, ℓ, ...) represent the probability of choosing site i when sites i, j, ℓ, ... are available. The
attraction effect is present if for some i, j, and ℓ, P (i|i, j) < P (i|i, j, ℓ) so that the probability of
choosing i goes up with the introduction of ℓ (Roe et al. 2001). If the attraction effect is present,
the “regularity principle” (i.e., that a preference cannot go up by adding dominated alternatives)
of decision making is violated. Let i = 6, j = 5, and ℓ = 4 and notice that the attraction effect
arises in the region around D = 0.3 to D = 0.5 in Figure 11. Even though the differential quality
between the two best sites is constant across that region, a range of distractor qualities around 0.4
amplifies the discrimination to a great enough extent that the best site is chosen more often than if
the distractor is not present (the D = 0 case). Essentially, the distractor acts more strongly on site
5, which assists the swarm in its discrimination task for sites 5 and 6. In terms of the underlying
mechanisms and dynamics, it seems that with an appropriate quality level a distractor site can
capture recruits and then “dump” them at a point in the process where it is unlikely that they will
28
go explore and hence likely that they will then get recruited. When they are recruited they will be
preferentially recruited by bees advertising for the best site. This surge in recruits for the best site
then makes it more likely that it is chosen.
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
15
10
5
0
0.2
4
0.8
0.3
0.2
0.1
0
1
0.2
0.4
0.6
0.8
1
17.6
43
43
1.3812
13.2
32
32
0.9208
8.8
22
21
0.4604
4.4
11
11
0
0.2
0.4
0.6
0.8
Distractor quality
1
0
e
1.8416
0
Split(x),no dec(o)(right)
0
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
54
x 10
Mean#Bees visit
t
0.6
0.4
22
2.302
Mean (.)/std(+)Σ L
0.4
0.5
0
0
0.2
0.4
0.6
0.8
Distractor quality
1
c
20
a
T , hrs
25
0.6
MeanB (x)MeanB (+)
%choice(blk);%qual(gray)
30
0
Figure 11: Attraction effect (see Figure 3 caption for axes explanation).
Overall, then we get the counter-intuitive result that the distractor improves choice performance.
In other words, the attraction effect corresponds to improved choice performance. Considering the
above results, this means that distraction can improve or degrade discrimination. Clearly, the
challenges presented by an arbitrary nest-site quality landscape are complex and interconnected.
Adaptive Tuning of Swarm Cognition Processes
To gain more insights into the mechanisms of swarm decision making we study their adaptation.
First, we study the effect of the amount of individual bee nest-site assessment noise on the swarm’s
choice performance and speed-accuracy trade-off. Second, we study the adaptive tuning of the
parameters of the individual bees’ decision-making. This helps to further validate the model we
use, and serves to show how speed accuracy trade-offs emerge from the adaptive tuning of the
swarm’s cognition process.
29
To study the adaptive tuning of individual bee-level behavioral parameters we change (“pseudomutate”) their values from experimentally-determined ones and determine average time/energy
costs and choice performance. We use six sites with qualities uniformly distributed on [ǫt , 1],
where ǫt = 0.2 is the threshold quality. We order the randomly generated qualities so that site
1 is the lowest quality, site 2 is the second best, and so on, and make site 6 the best site. Each
nest-site quality landscape generated this way is highly likely to produce interacting distraction
and discrimination effects and hence generally more challenging choice tests than in (Passino and
Seeley 2006). We consider seven values of each behavioral parameter, and for each of these, 1000
nest-site selection processes, each for a randomly generated landscape. Performance is characterized
by statistics of the 1000 runs for each parameter.
Effect of Bee Assessment Noise Magnitude
The effect of varying the magnitude w of the bee assessment noise wi ∈ [−w, w] is shown in
Figure 12. If w increases, the choice error rate does not degrade or improve much over the case
where w = 0.1, the value from (Passino and Seeley 2006). It is impossible for the individual bees to
have w = 0 and there is a small choice performance degradation when w increases to w = 0.1. For
w > 0.1 the choice performance slightly increases (and most importantly, does not decrease), but
P
the swarm needs a higher median value of Ta and mean Lt to reach agreement. This is due to a
slowing of the decision process due to resolving the differences due to noise. For increasing noise
magnitudes, the number of split decisions goes down (since it becomes more unlikely there will be
simultaneous agreement) and the number of no-decision cases generally goes up (due to too much
confusion caused by the noise). Also, an increasing noise magnitude results in more cross-inhibition
as seen in the mean number of bees that visit two sites. This occurs since the noise perturbs the
system away from a quick decision and thereby avoids “locking” onto a low quality site. Noise
results in more deliberation so that on average better sites will be found (i.e., deliberation allows
more time for search and consideration).
From a swarm cognition perspective, since group memory is more accurate when more bees are
committed to a site, and choices are made based on group memory, the swarm effectively filters
individual level assessment errors especially for the chosen site (even for unrealistically large quality
assessment error magnitudes such as w = 0.5). However, natural selection seems to have favored a
reduction in individual bee site assessment noise magnitude because that leads to shorter agreement
times and less dancing, without much degradation in choice accuracy.
Effect of Bee Assessment Noise Magnitude on Discrimination
The results in Section 5.3 for the effects of noise magnitude are for a very general class of nest-site
quality landscapes. The effect can be amplified for specific and common landscapes. For instance,
it seems likely that the swarm will often face discrimination problems for relatively close quality
sites. To study the effect of noise magnitude on discrimination we take the landscape from Figure 3
and specifically for the case where all site qualities are zero, except N 6 = 0.76 and N 5 = 0.74, a
differential quality of only 0.02, which by Figure 3 results in around 50% of the time the swarm
choosing each of the two sites. That is, the sites are in the region of generalization and it is difficult
for the swarm to discriminate between the two since their quality is so close. Figure 13 shows that
by increasing the amount of individual assessment noise, choice performance for the best site can
increase somewhat as the noise magnitude increases. Notice that in this case we are performing
30
20
15
10
5
0
0.1
4
0.4
0.2
0.1
0
0.5
0.1
0.2
0.3
0.4
0.5
2.3096
220.8
41
37
1.7322
165.6
31
28
1.1548
110.4
20
19
0.5774
55.2
10
9
0
0
Split(x),no dec(o)(right)
0
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
51
46
x 10
0.1
0.2
0.3
Noise mag.
0.4
0.5
0
Mean#Bees visit
t
0.3
0.3
276
2.887
Mean (.)/std(+)Σ L
0.2
0.4
0
0
0.1
0.2
0.3
Noise mag.
0.4
0.5
e
a
T , hrs
25
0.5
c
30
MeanB (x)MeanB (+)
%choice(blk);%qual(gray)
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
0
Figure 12: Effect of w (see Figure 3 caption for axes explanation).
1000 simulations for each parameter value so that at w = 0.1 there is about a 5% difference in
the choice rates for the two sites (the increased number of simulations gives an accurate estimate
of the choice percentages for low differential quality values). In the region above w > 0.3 there is
about a 10% difference in the rates of choice for the two sites; hence, the noise has increased the
ability of the swarm discriminate between these close-quality sites. This increase in performance
P
comes, however, at the expense of a higher median Ta and mean
Lt , and an increased number
of no-decision cases. These results further confirm the conclusions reached for the general class of
landscapes.
Effect of Quorum Threshold Size
Figure 14 shows that small ǫq values result in fast decisions (upper-left plot) and relatively few
dances (bottom-left plot), but relatively frequent errors (upper-right plot) since only a few bees
evaluate the chosen site. High ǫq values result in slower decisions, more dancing, and relatively
low error rates since many bees evaluate the chosen site. The experimentally-determined quorum
threshold value (in the range of 10-20 (Seeley and Visscher 2003)) is the adaptive result of balancing
P
the trade-off between keeping the median Ta and mean Lt values low and the percent of correct
31
%choice(blk);%qual(gray)
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
30
20
15
10
5
0.1
0.4
Split(x),no dec(o)(right)
0
0.1
0.2
0.3
0.4
0.5
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
1280
1024
1.626
768
1.084
512
0.542
256
0
0.1
0
0.5
2.1681
0
0.2
0.1
0.2
0.3
Noise mag.
0.4
0.5
0
41
40
31
30
21
20
10
10
0
e
t
Mean (.)/std(+)Σ L
0.3
0.3
0
0.1
0.2
0.3
Noise mag.
0.4
0.5
c
4
x 10
0.2
0.4
MeanB (x)MeanB (+)
0
Mean#Bees visit
a
T , hrs
25
0.5
0
Figure 13: Effect of w, N 6 = 0.76 and N 5 = 0.74 case.
choices high. This range of ǫq also keeps (bottom-right plot) enough bees involved in the process by
visiting sites and evaluating them, enough explorers in the role of searching for sites, yet relatively
few bees that visit two sites since that can lead to degraded performance in some cases.
Effect of Initial Dance Strength and Model Validation
For the effect of variation on γ see Figure 15. This shows that the value of 150 waggle runs for the
initial dance strength from an excellent site found in experiments (Seeley 2003) is the result of a
P
trade off between keeping the median Ta and mean Lt values low (high γ values) and the percent
of correct choices high (low γ values), while at the same time avoiding split and no-decision cases.
This provides a more complete verification that the values we used in (Passino and Seeley 2006)
(and here) are in the range settled on by evolution since the class of quality landscapes considered
here is considerably broader. Similar results are found for ǫs and σ. The results here also help
to verify the model in (Passino and Seeley 2006) since ǫq , ǫs , and γ (a parameter not studied in
(Passino and Seeley 2006)) are the ones found in experiments (Seeley and Buhrman 1999 2001;
Seeley 2003; Seeley and Visscher 2003 2004b 2004a).
Overall, from a swarm cognition perspective, the results here show that individual-level bee
32
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
20
15
10
5
0
10
4
0.2
0.1
10
20
30
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
83
85
3.8563
171.2
68
67
2.8922
128.4
51
50
1.9282
85.6
34
33
0.9641
42.8
17
17
0
10
20
ε
30
0
e
214
Mean#Bees visit
t
0.3
0
30
Split(x),no dec(o)(right)
x 10
Mean (.)/std(+)Σ L
20
0.4
0
q
10
20
ε
30
c
a
T , hrs
25
0.5
MeanB (x)MeanB (+)
%choice(blk);%qual(gray)
30
0
q
Figure 14: Effect of ǫq (see Figure 3 caption for axes explanation).
behavioral parameters related to early (γ and ǫs ) and late (ǫq ) processing have values that are the
result of balancing a swarm-level choice speed and accuracy trade-off.
Effects of Other Behavioral Parameters
Results for considering the tendency to seek to observe dances, pm ∈ [0, 1] show that this parameter
has little effect on most variables. Increasing it does, however, increase the number of split decisions
and decrease the number of no-decision cases since there is an increase in coupling in the process
that leads to build-up for similar sites to be closer, and helps to ensure that some site will have
enough bees to reach a quorum. If site qualities are generated on [0.2, 1] but we consider ǫt ∈ [0, 0.4],
there is little effect on the choice performance since higher values of ǫt simply eliminate inferior
alternatives that the swarm is already quite capable of eliminating. Simulations show that all low
values of pd have no significant impact on choice performance.
33
Marker(site): *(6),o(5),sq(4),dia(3),tU(2),tR(1)
20
15
10
5
100
120
4
180
0.3
0.2
0.1
0
200
120
140
160
180
200
1116.8
42
40
1.8273
837.6
31
30
1.2182
558.4
21
20
0.6091
279.2
10
10
100
120
140
γ
160
180
200
0
e
2.4364
0
Split(x),no dec(o)(right)
100
Marker(#sites):x(6)*(5)o(4)sq(3)dia(2)tU(1)tR(0)
52
50
x 10
Mean#Bees visit
t
160
0.4
1396
3.0455
Mean (.)/std(+)Σ L
140
0.5
0
100
120
140
γ
160
Figure 15: Effect of γ (see Figure 3 caption for axes explanation).
34
180
200
0
c
a
T , hrs
25
0.6
MeanB (x)MeanB (+)
%choice(blk);%qual(gray)
30