untitled


Epistemic Landscapes, Optimal Search,
and the Division of Cognitive Labor

Jason McKenzie Alexander, Johannes Himmelreich,

and Christopher Thompson*y

This article examines two questions about scientists’ search for knowledge. First, which
search strategies generate discoveries effectively? Second, is it advantageous to diver-
sify search strategies? We argue pace Weisberg and Muldoon, “Epistemic Landscapes
and the Division of Cognitive Labor” ðthis journal, 2009Þ, that, on the first question, a
search strategy that deliberately seeks novel research approaches need not be optimal.
On the second question, we argue they have not shown epistemic reasons exist for the
division of cognitive labor, identifying the errors that led to their conclusions. Further-
more, we generalize the epistemic landscape model, showing that one should be skep-
tical about the benefits of social learning in epistemically complex environments.

1. Introduction. A well-known example of the benefits conferred by the
division of labor appears in Adam Smith’s The Wealth of Nations. In his
discussion of the pin factory, Smith noted that the efficiency gains derived
from specialization could yield an increase of productivity between 240-
and 4,800-fold from that of a single individual. Whereas a single worker
might endeavor to produce 20 pins in a day, a group of 10 in a single factory
had been seen to produce upward of 12 pounds.
Does the same hold true for the division of cognitive labor? Would there

be more discoveries or would discoveries come faster if scientists divided
their labor? For a number of reasons, the answer is obviously yes: undivided
cognitive labor would lead to unnecessary repetition, scientists would fail to

Received May 2014; revised October 2014.

*To contact the authors, please write to Johannes Himmelreich: Department of Philosophy,
Logic, and Scientific Method, London School of Economics and Political Science; e-mail:
johannes@mflour.com.

yJohannes Himmelreich is grateful for the hospitality of the School of Philosophy at the
Australian National University at which a first version of this article was written.

Philosophy of Science, 82 (July 2015) pp. 424–453. 0031-8248/2015/8203-0005$10.00
Copyright 2015 by the Philosophy of Science Association. All rights reserved.

424



benefit from the unequal distribution of skill and talent, and, finally, com-
plex projects would become unmanageable given only a single worker.
However, some have argued for a positive answer for other reasons. In par-
ticular, Weisberg and Muldoon ð2009Þ suggest that diversity of research
strategies may “stimulate . . . greater levels of epistemic production” ð225Þ
and contend that even small steps toward a more diverse community of sci-
entists “massively boosts the productivity of that population” ð246–47, ital-
ics addedÞ.
The argument Weisberg and Muldoon provide for this claim uses a for-

mal model of search strategies on an “epistemic landscape,” a natural rein-
terpretation of the idea of a fitness landscape from evolutionary biology. In
what follows, we show that, contrary to what they report, a careful exam-
ination of their formal model does not actually support many of the conclu-
sions they attempt to draw regarding the division of cognitive labor. There
are three main reasons. First, for the particular epistemic landscape they
consider, the purported benefits of cognitive diversity are exaggerated due
to a failure to consider a broad enough comparison class of search strate-
gies. We provide several examples of homogeneous populations that prove
surprisingly efficient at searching the space and identifying the points of
epistemic interest. Second, the apparent benefits of cognitive diversity re-
ported largely derive from implementation errors in two of the three search
strategies they discuss. And third, if the model of epistemic landscapes is
generalized to more rugged, higher-dimensional landscapes whose overall
topography is not discernible by the individuals,1 social learning and the
division of cognitive labor only helps in particular circumstances. The up-
shot is that, although there clearly are real benefits from the division of cog-
nitive labor, the reasons have nothing to do with the epistemic reasons sug-
gested by Weisberg and Muldoon’s formal model.
The overall structure of the article is as follows. In section 2 we briefly

revisit the original epistemic landscape model. In section 3 we derive results
providing an upper bound for efficient search strategies on that landscape;
essentially, no rational scientist should perform worse than this value. We
then show that some of the search strategies investigated by Weisberg and
Muldoon fare far worse than this constraint. Section 4 shows why the search
strategies considered by Weisberg and Muldoon performed so badly. Sec-
tion 5 considers two key hypotheses that they claim to have substantiated,
and we show that our reexamination of the model invalidates both hypoth-
eses, effectively undermining their attempts to provide epistemic reasons
for the division of cognitive labor. Section 6 demonstrates that homoge-
neous populations can do even better than heterogeneous populations, in

1. We use the general framework of NK-fitness landscapes of Kauffman and Levin
ð1987Þ.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 425



some cases. And, finally, in section 7 we generalize the epistemic landscape
model and argue that whether the division of cognitive labor is advanta-
geous depends on features of the landscape that may well be unknowable.

2. Epistemic Landscapes. The basic idea of an epistemic landscape de-
rives from Sewall Wright’s ð1932Þ insight in population biology that one
can represent the fitness values of genotypes in terms of an abstract land-
scape. There, a particular genotype corresponds to a point in a highly mul-
tidimensional landscape, with the fitness value of that genotype as its
“height” and “nearby” points on the landscape being genotypes accessible
via point mutation, recombination, and so on.2 Analogously, we can think
of an “epistemic landscape,” where a point of the landscape represents a
particular research approach to investigating a topic of inquiry. A research
approach consists of the composite set of research questions, instruments,
techniques, and methods used, as well as background theories a scientist or
group of scientists rely on.
Not all research approaches to a topic of inquiry are equally fruitful,

though. Some research approaches bring better results or more publications
or more useful applications than others. Following Weisberg and Muldoon,
we can treat each approach as having a significance between 0 and 1. This
generates an abstract landscape over the various research approaches, with
the height of each approach corresponding to its significance. The entire land-
scape itself represents a single topic of inquiry.3

How do they proceed to model this? First, they fix the form of the epi-
stemic landscape. For simplicity, they work with a discrete 101 × 101 lat-
tice, wrapping at the edges to form a torus, with two peaks as illustrated in
figure 1. Second, they must operationalize the concept of finding points of
epistemic significance on this landscape. This can be broken down into two
components:

Epistemic Success: The time required to visit the two peaks.

Epistemic Progress: The percentage of the significant regions explored
after a given time.

2. Of course, this description ignores the fact that frequently one cannot meaningfully
speak of the fitness of a genotype separate from the distribution of genotypes/pheno-
types in a population. For the purposes of investigating their model, we share this ideal-
ization with Weisberg and Muldoon. However, we agree that this assumption is highly
dubious, and future work should investigate the consequences of dropping it.

3. This can be seen from the fact that the same set of research methods may have very
different degrees of fruitfulness over different topics of inquiry: randomized controlled
trials are highly fruitful for determining the efficacy of various drugs, less so for pur-
poses of literary theory.

426 JASON MCKENZIE ALEXANDER ET AL.



Given these two aims, how should scientists search the space? Since a point
in two dimensions represents a research approach, this becomes the ques-
tion of how a scientist should move about the landscape in light of the in-
formation available to her.4 A variety of different kinds of information exist
that one might use. There is epistemic information, such as the significance
of one’s current approach. There is also social information, such as how
often a certain approach has been tried before or whether it has been tried at
all. Finally, there is the possibility of using metric information, such as how
far away is the nearest scientist.5

Weisberg and Muldoon consider three search strategies: a “control” and
two others that they call “mavericks” and “followers.” The control strategy,
also referred to as the “HE rule” ðshort for “hill-climbing with experimen-
tation”Þ only uses epistemic information, whereas mavericks and followers
use both epistemic information and social information. Essentially, the HE

4. In turn, this requires specifying just exactly what information is, in fact, available to a
scientist. This purportedly simple model has a lot of detail that needs to be posited before
one can begin to make any headway with the two questions.

5. Strictly speaking, the epistemic landscape provides a topological model rather than a
metric model, since the spatial “positions” are supposed to be abstract representations
of variations in some research approach. ðConsider, by way of comparison, the concept
of “distance” between two genomes identical except at one base. If the differing base
was, say, adenine instead of thymine, does that make the second genome closer or farther
than it would have been if the base had been guanine or cytosine? It is hard to make
sense of this question.Þ Nonetheless, one could impose a metric onto the epistemic space
by simply imposing a Euclidean metric onto the landscape. Whether this would mean
anything is, of course, unclear.

Figure 1. Three-dimensional and two-dimensional projections of the epistemic
landscape. Left, Weisberg and Muldoon ð2009Þ, fig. 1. Right, Significance hills are
indicated by shaded gradient, with the perimeter highlighted.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 427



rule instructs a scientist to try to climb a hill toward the peak ðwithout be-
ing able to detect the gradientÞ and otherwise to follow a straight line on the
landscape with, occasionally, a random change in direction. Followers, as the
name suggests, are intended to favor explored approaches that have been pre-
viously considered and only take significance into account as a secondary
consideration. The maverick strategy, however, is sensitive to research ap-
proaches that have been explored previously and deliberately seeks out un-
explored approaches at random.

3. Controls, Followers, Mavericks, and the Efficiency of Search. Before
getting into the details of the Weisberg and Muldoon model, let us first
establish some clear upper bounds on the search efficiencies one might ex-
pect to find. Recall that the control strategy attempts to hill climb, if in an
area of epistemic significance, and otherwise follows a straight line, with oc-
casional random changes in direction. If we dispense with the requirements
of hill climbing and the occasional random reorientation when in areas of
zero significance, we get a search strategy that can be proven to almost al-
ways visit both peaks and exhaustively search the entire space. How so?
Recall that, since the two-dimensional landscape wraps at the edges, it is
topologically equivalent to a torus. The Kronecker foliation of the torus is
obtained by projecting a straight line in the real plane with slope v onto the
surface of the torus. If the slope v is rational, the projected line forms a
closed loop; if the slope is irrational, the line will be dense in the surface of
the torus ðfig. A1, available online only, illustrates both types of foliationsÞ.6
Since the epistemic landscape under consideration is divided into discrete
cells, the fact that the Kronecker foliation is dense in the surface of the torus
guarantees that a single agent who simply follows a straight line will, in
finite time, search the entire space.7 Call an agent employing such a strategy
a “foliator.”8 A population of foliators will manage to explore the entire

6. Strictly speaking, the slope of the line needs to be classified as rational or irrational
relative to both the major and minor radii of the torus. Let the major radius have a length
of 1=2ð Þ

ffiffiffi
3

p
and the minor radius a length of 1/2. The surface of the torus is equivalent to

a rectangular lattice of height 1 and width
ffiffiffi
3

p
that wraps at the edge. A line in the plane

with a slope of 1= 1=3ð Þ
ffiffiffi
3

p
ðand hence irrationalÞ will, when mapped onto the torus, self-

intersect after three loops. However, if we take ‘irrational’ to mean ‘irrational with re-
spect to the major and minor radii of the torus’, then the claim holds.

7. Although the lattice is divided into discrete cells, the heading of an agent may vary
continuously.

8. Note that for the purpose of establishing a theoretical upper bound, strictly speaking,
we may not require the existence of a real world analog to this strategy. This strategy
requires to impose a Euclidian metric onto the landscape, which is a plausible assump-
tion when real world constraints are set aside ðsee our remarks in n. 5Þ. We discuss the
real world applicability of the epistemic landscape model at length in sec. 7.

428 JASON MCKENZIE ALEXANDER ET AL.



space more quickly than a single agent. And since almost all of the possible
slopes v that an agent may follow are irrational, a population of foliators
uses a simpler search strategy than Weisberg and Muldoon’s control agents
but can be proven to almost always succeed in achieving both epistemic
aims.
Let us estimate the efficiency of this search rule by simulation. Figure 2

illustrates one result from a simulation containing 10 foliators beginning at
random locations in the area of zero significance, which we call the “des-
ert.” Notice that the foliator strategy can be quite quick: within 500 steps,
one peak of the landscape had already been found and 37% of the entire
landscape explored. Out of 5,000 simulations run with 10 foliators and ran-
dom initial conditions, 4,988 managed to find both peaks within 50,000 steps.
The mean time required to find both peaks was 1,855 steps, with minimum
and maximum times of 43 and 32,167 steps, respectively, and a median time
of 1,430 steps.
Now compare these results with the results reported by Weisberg and

Muldoon ðsee table 1Þ. The discrepancy reveals something rather curious.
In a simulation with 100 repetitions of populations of 10 HE rule agents,
only 95 populations found both peaks within the time allowed ð50,000
stepsÞ. Of these 95 populations: “the time to finding the two significant
peaks varied considerably from a maximum of 43,004 cycles to a minimum
of 553 cycles. The mean for these runs was 6,075 with standard deviation
8,518 and the median was 2,553. More importantly, the length of runs is
distributed in aheavy-taileddistribution,with60%oftherunsbeingcompleted
in 4,000 cycles and 80% being completed in 10,000 cycles” ð2009, 236Þ.
In short, foliators—who never attempt to hill climb and who never pick a new
direction of travel—are both more effective at finding both peaks ðwith a
success rate of 99.7% instead of 95%Þ and faster ða mean time of 1,855 steps,
as opposed to 6,075Þ. Let us bracket this observation, for the moment, and
return to it at the end of this section.

Figure 2. Epistemic search by 10 foliators. Squares are ‘bread crumbs’ showing
approaches that have been visited. A, 50 iterations; B, 100 iterations; C, 200 iter-
ations; D, 500 iterations.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 429



Now consider, as an alternative search strategy, the case of a simple random
walk. Assume that at points of zero significance ðthe desertÞ the agent
randomly moves to one of its eight nearest neighbors, with all transition
probabilities equal; at points with positive significance, the agent follows
the gradient. This means that at points of zero significance, an agent’s move-
ment at a time is independent from her movement at all previous times and
that once the agent enters a region with positive epistemic significance, she
never leaves. Thus we can treat the two hills as a single absorbing state and
model the movement of the agent in the desert as a Markov process.
Let kn,m denote the expected time to absorption for an agent starting at

location ðn, mÞ. By construction, ki,j50 for all points ði, jÞ having positive
epistemic significance. Furthermore, for all points ðn, mÞ having zero epi-
stemic significance, we know that

kn;m 5 1 1
1

8
ðkn21;m11 1 kn;m11 1 kn11;m11

1 kn21;m 1 kn11;m 1 kn21;m21 1 kn;m21 1 kn11;m21Þ:

Figure 3A illustrates the local transition diagram for a point in the desert
bordering the perimeter of a hill. This gives a system of 10,201 simulta-
neous linear equations. If we only consider equations generated from points
having zero significance and the perimeter of the two epistemically signif-
icant regions, the system reduces to 8,273 simultaneous linear equations.
Figure 3B illustrates the expected hitting time of the absorbing state, for
each point in the landscape.9 The maximum expected hitting time is a frac-
tion over 1,600 steps. The average hitting time, for a single agent starting in
the desert, is 881.9 steps.
Since it is not feasible to obtain an analytic solution for a population of

10 agents engaged in independent random walks, we examined this via
simulation. Out of 5,000 populations of 10 agents, all starting from random
initial conditions, 4,944 found both peaks. In all remaining 56 cases, all

TABLE 1. EPISTEMIC SUCCESS

Strategy P Mean SD Min Max Median

HE rule .95 6,075 8,518 553 43,004 2,553
Random walk .9888 249 327 14 4,241 141
Foliator .9976 1,855 1,755 43 32,167 1,430

Note.—Comparison of simulation results reported in Weisberg and Muldoon on 10 agents using the HE
rule with results of agents using the random walk and Kronecker foliation search strategies. P denotes the
proportion of populations that found both peaks.

9. The system of equations was programmatically generated and then solved using
Mathematica. Since the system of equations is both linear and described with a sparse
coefficient matrix, a solution is found quite quickly.

430 JASON MCKENZIE ALEXANDER ET AL.



10 agents had found the same peak. ðWith the random walk search strategy,
when an agent finds a peak it stays there.Þ The mean time required to find
both peaks was 249 steps, with a minimum and maximum time of 14 and
4,241 steps respectively, and a median of 141 steps. After 500 iterations on
average 15.6% of the total landscape had been explored.
In summary, we have shown that if we are interested in both effective-

ness ðwere both peaks found?Þ and efficiency ðhow long did it take to find
both peaks?Þ, the HE rule did worse than pure populations of foliators or
random walkers. ðTable 1 has a comparison of all three strategies.Þ In par-
ticular the HE rule—despite taking epistemic information into account by
hill climbing—did worse on average by a factor of three when compared
to the foliator strategy, which simply ignored this information. Furthermore,
the HE rule did worse than the random walk strategy by a factor of roughly
24. Admittedly, our random walk strategy would be expected to do better,
in regions of positive significance, because it simply followed the gradi-
ent, whereas the HE rule used a probe-and-adjust method to hill climb.
But that surely does not explain everything about why it did over 24 times
worse.

4. A Closer Look. Why did the control scientists perform so badly? Let us
examine the exact statement of the HE rule as described in the original
article. It is as follows:

Figure 3. Analytically solving for the expected waiting time of the random walk
search strategy. A, Portion of the Markov chain transition diagram for the random
walk search strategy. Gray areas represent absorbing states. For simplicity, only the
state transitions exiting the 3 × 4 block of points are shown. B, Plot of the expected
number of steps required for an agent to encounter a region of epistemic signifi-
cance via a random walk ðheight of each column represents the expected number of
stepsÞ.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 431



HE Rule:
1. Move forward one patch.
2. Ask: Is the patch I am investigating more significant than my

previous patch?

If yes: Move forward one patch.
If no: Ask: Is it equally significant as the previous patch?
If yes: With 2% probability, move forward one patch with a ran-

dom heading. Otherwise, do not move.
If no: Move back to the previous patch. Set a new random heading.

Begin again at Step 1. ðWeisberg and Muldoon 2009, 231–32Þ

Scope ambiguities regarding the nested conditionals in step 2 make the rule,
as stated, open to multiple interpretations. The pseudocode representation
provided in figure 4A makes precise the scope relations between the else-
clauses of the conditionals under one interpretation. Here, step 1 of the pub-
lished version of the HE rule corresponds to the forwardðÞ command in line 2.
Step 2 corresponds to the nested conditionals and commands in lines 3–20.
There is nothing corresponding to the “Begin again at Step 1” instruction
because we assume each agent calls HE_ruleðÞ at the start of each iteration.
From this, it is clear that our foliator rule approximates the HE rule quite

well: when exploring areas of zero significance, the test on line 3 fails, and

Figure 4. Two implementations of the HE rule search strategy. Variables curr_sig
and prev_sig refer to the significance of the site currently occupied by the agent and
the significance of the site previously occupied by the agent, respectively. A, As in
the article; B, as implemented.

432 JASON MCKENZIE ALEXANDER ET AL.



the test on line 7 succeeds. As the test on line 8 succeeds only 2% of the
time, the remaining 98% of the time the HE rule will not reset its heading.
Thus, when the forwardðÞ command on line 2 is invoked at the start of the
next iteration, the agent continues to move forward according to its previous
heading ði.e., in a straight lineÞ. Furthermore, this shows that the stay_putðÞ
command on line 13 is not, strictly speaking, necessary, since the agent has
already taken one step forward during the current iteration. ðIndeed, this
serves to highlight a small error in the Weisberg-Muldoon statement of the
HE rule: when in areas of increasing significance, the combination of the
forwardðÞ command on line 2 and the successful test on line 3 ensures that
the agent will step forward twice in the same iteration.Þ
Inspection of the original code used in the Weisberg-Muldoon simulation,

which they made available, revealed that their actual implementation was
as in figure 4B. There are several things to note. First, the absence of a for-
wardðÞ command at the beginning means that agents are not guaranteed to
move at least once each iteration. Second, the test condition at line 2 con-
tains the ≥ operator. When exploring regions of zero significance, this means
the test at line 2 will succeed, dropping us immediately into the test at line 3,
which will also succeed. A control agent, given lines 4–6, moves forward at
a random heading with a 2% probability and otherwise remains stationary
98% of the time.
In other words, whereas the description of the HE rule in the article

suggests that agents ought to behave rather like foliators, Weisberg and Mul-
doon’s actual implementation has those agents behaving like lethargic ran-
dom walkers. This explains the difference between our baseline results and
those reported by Weisberg and Muldoon. However, it also calls into ques-
tion the meaningfulness of the comparison between control agents who use
the HE rule and other search strategies. Since control agents do nothing 98%
of the time, unless a similar delay is incorporated into the definition of any
compared search strategies, we are comparing rules that operate on funda-
mentally different timescales.
Let us now turn to the follower strategy. Here is the definition, as in the

original article:
Follow[er] Rule:

Ask: Have any of the approaches in my Moore neighborhood been
investigated?
. . .

If yes: Ask: Is the significance of any of the investigated approaches
greater than the significance of my current approach?

If yes: Move towards the approach of greater significance. If there is
a tie, pick randomly between them.

If no: If there is an unvisited approach in the Moore neighborhood,
move to it, otherwise, stop.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 433



If no: Choose a new approach in the Moore neighborhood at ran-
dom. ðWeisberg and Muldoon 2009, 239–40Þ

In our version ðsee fig. 5Þ, there is no need to specify the tie-breaking rule
explicitly because we have already selected, at line 3, a random visited
neighbor with maximum significance.
As interpreted, the follower rule performs a biased random walk that can

get stuck. In the presence of sites that have been previously visited, and that
are of greater significance, the rule moves to one of those sites at random.
When no visited sites are of greater significance, it moves to a random un-
visited site. However, when entirely surrounded by visited sites of equal
significance—as can happen in the desert—the follower rule will end up
getting trapped in a cycle of length 2, flipping back and forth between two
visited states.
How does this interpretation compare with the results Weisberg and

Muldoon report? They write:

With only 10 followers, not a single population managed to find both ap-
proaches of maximum significance and only 3% managed to find at least one
approach of maximum significance. . . . With 200 followers, a single
approach of maximum significance was found 60% of the time, with both
approaches being found only 12% of the time. However, when the popula-
tions of followers did find both peaks, this happened very rapidly with an
average time to converge . . . of 56 cycles, which suggests that the randomly

Figure 5. Follower rule.

434 JASON MCKENZIE ALEXANDER ET AL.



placed agents were near the boundary of significance at the beginning of the
simulation. ð2009, 240Þ

Even though the biased random walk performed by the followers can get
stuck in the desert, it seems strange that a population of 200 followers only
found one peak 60% of the time, and both peaks only 12% of the time.
Further cause for concern should arise when one reads that when followers
did find both peaks it happened “very rapidly.” How can a search strategy
perform so badly at searching, generally, yet succeed so rapidly when it
does?
Inspection of the code used by Weisberg and Muldoon showed that their

implementation was functionally equivalent to our interpretation ðsee fig. 5Þ
except for the test at line 4, which in their version was

if ðsignificanceðnÞ >5 current_significanceÞ f

The use of the ≥ operator instead of the > operator is a serious error. Given a
sparse distribution of followers, where each agent is at least three squares
distant from every other in regions of zero significance, the agents get stuck
in a loop.
Figure 6 illustrates how this happens in detail. In the initial state shown in

the upper left, the if-test at line 2 of the follower rule fails, but the if-test at
line 8 succeeds, resulting in the agent moving to an adjacent site selected at
random. But then, in a configuration like that shown in the upper right, the
if-test at line 2 now succeeds, and because the Weisberg-Muldoon imple-
mentation of the follower rule has the ≥ operator at line 4, the agent simply
moves back to its previous position.10 From this point on, the follower os-
cillates between the two visited sites. Instead of following others, the agent
‘chases his own tail’.
Thus we see why only 3% of the 100 simulations of 10 followers man-

aged to find a single peak: most of the time they were in regions of zero
significance, which resulted in the followers getting trapped in a cycle as
described in figure 6. It also explains why larger populations of followers,
when they managed to find both peaks, found both peaks so quickly: the
random initialization positioned a few followers in a site of zero signifi-
cance that was adjacent to a region of positive significance. If the follower
happens to randomly move into an area of positive significance, it will then
proceed to climb up the hill via a random walk. With this in mind, consider

10. This happens because there is exactly one previously visited neighbor, and so it is
guaranteed to be chosen when we select “a previously visited neighbor with maximum
significance.”

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 435



the one graphic from the original Weisberg and Muldoon article that
showed the paths traveled by a population of 300 followers, reproduced in
figure 7. Here we see that only agents positioned near the region of epi-
stemic significance eventually managed to climb toward the top of the peak.
Furthermore, the vast majority of paths traveled by followers consist of
mere oscillations between two adjacent squares, exactly as one would
expect given the code analysis.

5. On the Division of Cognitive Labor. Weisberg and Muldoon note that
“modern science requires the division of cognitive labor” ð2009, 225Þ and
claim that their simulation results illustrate the epistemic benefit so con-
ferred. But what, exactly, do we mean by the “division of cognitive labor”?
Consider the following disambiguation. On one hand, we have the phenom-
enon that scientists choose different approaches to investigate a research

Figure 6. Implementation of the follower rule typically results in cycles of length 2.
ðAll shown sites have zero significance.Þ Agent moves from the initial configura-
tion ðupper leftÞ to a randomly selected unvisited square ðupper rightÞ. It then re-
turns to a previously visited site whose significance is greater than or equal to the
significance of the current site ðbottom leftÞ and is stuck in a loop ðbottom rightÞ.

436 JASON MCKENZIE ALEXANDER ET AL.



topic. In the epistemic landscape ðwhich represents the research topicÞ, the
agents can occupy different points ðwhich represent research approachesÞ.
Scientists specialize in different approaches. This is one meaning of divi-
sion of cognitive labor. It describes a phenomenon of coordination. On the
other hand, we have the phenomenon that scientists use different strategies
to choose their research approach. The agents move across the epistemic
landscape according to different strategies. Some scientists employ methods
simply because they are trending ðthese would be the followersÞ, whereas
others favor approaches because they are exotic or unusual ðthese would
be the mavericksÞ. This is a second meaning of division of cognitive labor.
It describes a phenomenon of diversity. Let us call the phenomenon where
different people work on different projects epistemic coordination and the
latter, where different people have different reasons to work on different
projects, cognitive diversity.
When Kitcher ð1993Þ and Strevens ð2003Þ use the Marginal Contribution

and Reward model to explain why scientists pursue different research ap-
proaches, they seek to explain the phenomenon of epistemic coordination.
They show that research behavior of scientists is coordinated when scien-
tists are sensitive to social and epistemic information. Very roughly, scien-
tists seek to maximize the rewards that accrue from scientific discoveries.
They consider both the likelihood that an avenue will generate results ðepi-
stemic informationÞ and the number of other scientists working on that ave-
nue ðsocial informationÞ. This yields the phenomenon that the scientific com-
munity spreads out across different possible avenues for research. However,

Figure 7. Weisberg and Muldoon ð2009Þ, fig. 8, showing the exploration of the
epistemic landscape by 300 followers before ðAÞ and after ðBÞ movement began. Tails
plot the path they followed. Note that the vast majority of followers are trapped in
cycles of length 2.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 437



because Kitcher and Strevens assume that all scientists choose among pos-
sible avenues for research in essentially the same way, they do not address
cognitive diversity.
In contrast, an example of research on cognitive diversity can be found

under the headings of “swarm intelligence” or “wisdom of the crowds.”11

For example, similar to Weisberg and Muldoon, Hong and Page ð2004Þ de-
velop a model of agents searching for local maxima in a space. The agents
are cognitively diverse in two ways. Not only do they use different strate-
gies to explore the epistemic space, but they also represent the epistemic
space differently; each individual has its own language to describe the points
in the space. This is an example of a model of agents that are cognitively
diverse.12 More generally, models of cognitive diversity can be found in areas
ranging from complex systems research and theoretical biology ðBonabeu,
Dorigo, and Theraulaz 1999; Krause et al. 2011Þ and management and or-
ganization studies ðThomas and Ely 1996; Polzer, Milton, and Swarm 2002;
Jackson, Joshi, and Erhardt 2003Þ to psychology ðKerr and Tindale 2004Þ,
computer science ðClearwater, Huberman, and Hogg 1991Þ, and economics
ðHong and Page 2004; Arrow et al. 2008Þ.
The epistemic landscape of Weisberg and Muldoon models division of

cognitive labor in both senses. Agents take different approaches on a re-
search topic ðepistemic coordinationÞ, and they use different strategies in
choosing those approaches ðcognitive diversityÞ. The central observation of
“general trends about the division of cognitive labor” ð2009, 249Þ made by
Weisberg and Muldoon is that cognitive diversity gives a scientific com-
munity an epistemic advantage. An increase in epistemic performance en-
sues if a community of researchers uses different epistemic search strategies.
Weisberg and Muldoon argue that “to be maximally effective, scientists need
to really divide their cognitive labor” ð227Þ and that a “healthy number of
followers with a small number of mavericks” would provide an “optimal
way” ð251Þ to do so. They write that this is because there is a “very signif-
icant indirect affect that mavericks have on the research progress via their
ability to stimulate the followers” ð249Þ. That sounds like a fine example of
epistemic benefits of cognitive diversity.13 Is it true?

11. However, a very different research project also uses the same label. It surrounds the
phenomenon that large numbers of people are better at solving epistemic tasks. “Wis-
dom of the crowds” in this sense is related to the Condorcet Jury Theorem and not so
much to cognitive diversity.

12. Hong and Page ð2004Þ use the term “functional diversity.”
13. This assumes that each stimulated follower would do better than a maverick would
in her place. Otherwise, why should we ideally not have only maverick scientists to
begin with?

438 JASON MCKENZIE ALEXANDER ET AL.



Weisberg and Muldoon observe that the epistemic performance of a pop-
ulation of followers increases when mavericks are added to it ð247–48Þ.14
Does this vindicate the thesis that cognitive diversity improves epistemic per-
formance? It does not if the improvement rests on a defective implemen-
tation of the search strategies. And it does not vindicate the thesis if the im-
provement is only due to the epistemic performance of the agents that have
been added. It seems that both are the case here.
Furthermore, Weisberg and Muldoon describe that the improvement in

epistemic performance is due to the following “indirect affect.” They write:
“Mavericks help many of the followers to get unstuck, and to explore more
fruitful areas of the epistemic landscape” ð2009, 247Þ. Maverick scientists
may help follower scientists get unstuck, but the followers should not have
been stuck in the first place. As shown above, the search strategy of follower
scientists suffered from a defective implementation such that they ended up
chasing their own tail. The beneficial “indirect affect” that Weisberg and
Muldoon describe requires a population of follower scientists that has hardly
left the place where they started. If a search strategy performs so direly, the
result that a complementary strategy improves overall performance is hardly
surprising. It does not vindicate the thesis that there is an epistemic reason
for cognitive diversity in any interesting sense.
If the follower strategy worked properly, in that it did not get stuck almost

immediately in a cycle, would there still be an indirect affect to vindicate the
thesis that there is an epistemic reason for cognitive diversity? We show that
this is not the case. It turns out that the improvement in epistemic perfor-
mance is exclusively due to the performance of the mavericks that are added
to the population of followers. It should not be surprising that the epistemic
performance of a population increases when agents are added. In particular,
this is true when these agents are mavericks, who have been shown to per-
form quite well.
We set up a simulation to record the epistemic progress of followers and

mavericks separately as the mavericks are added to a population of follow-
ers. In these simulations, we used a correct implementation of the follower
search strategy. This separate bookkeeping enables us to identify which sub-
population caused the increase in the total epistemic progress of the mixed
population. We ran simulations with 100 repetitions for each condition,
for populations of followers with an initial size of 100, 200, 300, and 400.
We observed how the epistemic progress of this population changes as

14. More precisely, Weisberg and Muldoon consider two settings. In one setting the size
of the mixed population remains fixed and merely the proportion of followers to mav-
ericks changes. In the second setting mavericks are added to a population of followers.
We focus on the second because it is more instructive. Our findings apply to both.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 439



maverick scientists are added. Replicating the experiment from Weisberg and
Muldoon, we added 10, 20, 30, 40, and 50 maverick scientists. We measured
the epistemic progress after 1,000 iterations, recording the total epistemic
progress of the mixed population and the progress of each subpopulation
ðsee fig. 8Þ.
For all initial sizes of follower populations, the epistemic progress of the

followers remains virtually unaffected by the presence of mavericks. In the
case of the initial population size of 100 followers, the epistemic progress
of the follower subpopulation even decreased. While a pure population of
100 followers managed to explore 23% of the significant points after 1,000
steps, when mavericks are added to the population this number goes slightly
downward. The epistemic progress of the follower subpopulation in the pres-
ence of 50 mavericks reaches only 21%.
Thus, the increase in the epistemic progress of the mixed population can

be solely accounted for by the epistemic progress of the maverick subpop-
ulation. Indeed, if anything, the followers seems to get in the way of the mav-
ericks. The mavericks are doing particularly well when only a few followers
are present. Notice that the epistemic progress of a maverick subpopulation
of the size of 50 is 19% in the presence of a follower subpopulation of a size
of 100. The epistemic progress of the maverick subpopulation decreases as
the size of the follower subpopulation increases. The maverick subpopula-
tion of the same size achieves an epistemic progress of 15% when the fol-
lower subpopulation has a size of 200, 14% when it has a size of 300, and
12% when it has a size of 400.

6. Efficient Search in Homogeneous Populations. The Weisberg and
Muldoon epistemic landscape model features three different kinds of infor-
mation: epistemic, social, and metric. Although control agents use episte-
mic and followers and mavericks use both epistemic and social information,
none of the three strategies considered use metric information. To an extent,
using metric information ðhow far away other nearby scientists are and
where they are goingÞ can be interpreted as a “strategic follower” strategy:
it pays attention to where other scientists are going rather than seeking parts
of the landscape where they have been. Since there is good reason to sus-
pect that some scientists behave in roughly similar ways ðin that they
consciously align their “research brand” with what is trendingÞ, let us con-
sider how such a search strategy performs. Let us call this strategy the
“swarm.” Stated informally, a swarm scientist receives information about
what others in her area are working on via journals and conferences and
adjusts her own approach such that it is always similar but yet distinct to the
approaches pursued by others in her area. Furthermore, when she observes
that many of her colleagues incorporate a certain turn into their approaches,
she will try to imitate this change.

440 JASON MCKENZIE ALEXANDER ET AL.



Figure 8. Epistemic progress of a population as mavericks are added. Increase of
the mixed population ðAÞ is almost entirely due to the maverick subpopulation ðCÞ.
Epistemic progress of the follower subpopulation ðBÞ remains unchanged while
mavericks are added.



There are at least four interesting parallels between the epistemic search
of a scientific community and the foraging behavior of animal groups. First,
individuals in a school of fish or a flock of birds need to coordinate their
behavior to avoid occupying the same space at the same time. In scientific
research, this is the problem of epistemic coordination: we do not want every-
one to be attempting to do the same thing at the same time, as such redun-
dancy would often be a waste of effort. Second, the animal group often has
a common goal, such as finding food or traveling to a nesting site. Analo-
gously, the scientific community has epistemic aims, such as determining a
high-yield, cost-effective way of manufacturing graphene. Third, informa-
tion is distributed differently among individuals in a group: only a small
subset of individual animals in a group may have information about the lo-
cation of particular food site. Analogously, only a small subset of research-
ers has experience with a particular approach to the research topic. Finally,
just like how some herd-based animals follow others who take the lead, a
similar behavior may be found in the scientific community.15

Theoretical biology has a rich literature on collective behavior ðsee
Sumpter 2010, chap. 5, for an overviewÞ. The swarm search strategy we use
is a simplified version of one by Couzin et al. ð2005Þ and similar to the
Boids model ðReynolds 1987Þ. Roughly, if another agent gets too close,
then the agent swerves to avoid collision. Otherwise, the agent aligns its
direction of travel with the other agents around it. More precisely, the space
surrounding the agent is divided into two different “zones,” as shown in
figure 9, a zone of repulsion and a zone of orientation. If there is another
agent in the front half of the agent’s zone of repulsion, then the agent
changes its direction to the right if the closest individual is ahead and to the
left, and the agent changes its direction to the left if the closest individual is
ahead and to the right. If there are individuals in an agent’s zone of ori-
entation but not its zone of repulsion, then the agent adjusts its direction by
the mean of the differences between its own direction and the directions of
the individuals in the zone of orientation.
Although figure 9 assumes that the radius of the zone of repulsion is

smaller than that of the zone of orientation, this need not be the case. If the
radius of the zone of repulsion is greater or equal to the radius of the zone of
orientation, the resulting swarm has considerably different properties. To
distinguish between these two cases, let us introduce some terminology: call
the latter case a “repulsing swarm” and the former case a “flocking swarm.”

15. An instructive example on the last two points are honeybees ðsee Seeley 2010Þ.
Once a decision about a nest site has been made, “up to around 10,000 bees of which
only 2% or 3% are informed of the location of the nest site fly as a single swarm to the
site” ðSumpter 2010, 123Þ. See also Ward, Krause, and Sumpter ð2012Þ.

442 JASON MCKENZIE ALEXANDER ET AL.



The flocking swarm can be thought of as being composed of “strategic
followers,” with the repulsing swarm being composed of “strategic mav-
ericks.”
Note that swarm scientists also use epistemic information but in a man-

ner somewhat different from the HE rule. Since some scientists have the
ability to intuit the correct way to develop a theory—think of Newton, Ein-
stein, von Neumann, and Feynmann—we incorporate this into the model
by assigning to each agent a probability of being “clairvoyant.” That is, each
agent who happens to be in a region of positive significance has a probabil-
ity of guessing the direction to the top of the hill.16 When this happens, the
clairvoyant agent adjusts its heading to point in the direction of its insight.
Clairvoyance does not last, though, and so the initial flash of insight might
disappear as the agent further adjusts its behavior to the rest of the sur-
rounding swarm. Brilliant ideas may go unrecognized.
Compare this with the complexity of the behavior of the other agents. The

HE rule, maverick, and follower agents all change their behavior when they
enter an area of positive significance. Each of them uses epistemic infor-
mation in each move to climb toward the top of the hill. The swarm strategy
exhibits no richer behavior or greater complexity than these strategies.
In short, in each iteration a swarm scientist does as follows. If the agent is

on a point of positive significance, it has a one-off moment of clairvoyance
with probability p, which causes it to change its direction toward the closest

16. The probability is the same for all agents.

Figure 9. Focal agent of a flocking swarm with its zone of orientation ðouter circleÞ
and repulsion ðinner circleÞ.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 443



peak, taking one step forward. Otherwise, it adjusts its direction to align
with all the other agents in its zone of orientation, or it swerves away from
the closest nearby agent in its zone of repulsion, if there is any. After
aligning or swerving, the agent moves one step forward.
We ran simulations starting with populations of 10 agents, increasing the

population by 10 to up to 400 agents with 100 repetitions each.17 We inves-
tigated the flocking swarm strategy and the repulsing swarm strategy.18 The
probability of clairvoyance was 0.03 for the flocking swarm and 0.015 for
the repulsing swarm. We compare our results with similar simulations us-
ing mavericks and the follower scientists.19 The results for 10, 200, and 400
agents can be found in table 2.20

On epistemic success, we found that mavericks and followers perform
better than the swarm strategy only if the populations are sufficiently small
and the population of maverick or follower agents actually manages to find
both peaks. We found that as the size of the population increases beyond 30,
this result no longer holds. Then the swarm strategy performs better than the
maverick and the follower strategy. The repulsing swarm configuration per-
forms particularly well. Consider the median time to find both peaks for pop-
ulations of size 100. Mavericks, followers, and the flocking swarm con-
figuration have a median time between 50 and 60 iterations to find both
peaks, whereas half of all the repulsing swarm populations find both peaks
already after 37 iterations.
On epistemic progress, as previously shown, we found that the maverick

scientists have a greater epistemic performance than follower agents. How-
ever, swarm scientists have an even greater epistemic performance than the
maverick scientists; the repulsive swarm configuration seems to do slightly

17. Let the number of agents be n, the radius of the zone of repulsion r, and the number
of groups s. The agents were placed in s 5 3 1 n = 10 groups that were randomly lo-
cated in the desert. The radius for each group in which each agent was randomly placed
is

ffiffiffiffi
rs

p
. To start the simulation without a pre-run to form stable swarms, all members of a

group were given the same random heading.

18. The radii were set to 3 and 1, respectively. Note that when the zone of repulsion is
greater than the zone of orientation, the agents never align their direction with that of the
agents in their vicinity.

19. We used a repaired implementation of the follower strategy, not the defective im-
plementation that was originally used, as discussed above.

20. Comparisons between the two swarm strategies and the correct implementation of the
Weisberg and Muldoon strategies regarding epistemic progress are not likely meaningful
because controls, followers, and mavericks remain at epistemic peaks, once found. That
said, it is worth noting that the control strategy—when implemented as intended—per-
forms very well in terms of epistemic progress compared with pure populations of mav-
ericks or followers ðthereby again undermining the results of the original articleÞ. Com-
parisons between all strategies regarding epistemic success, though, are meaningful.

444 JASON MCKENZIE ALEXANDER ET AL.



better than the flocking swarm configuration. The take-home message is
that cognitively homogeneous populations of agents can do very well.

7. A Generalized Epistemic Landscape Model. One further concern with
the Weisberg and Muldoon model derives from the simplicity of the epi-
stemic landscape considered. Although we are generally ignorant about the
shape of the epistemic landscapes underlying real scientific research, it is
clear that they have at least two properties that are largely absent from
the Weisberg and Muldoon landscape. First, on real epistemic landscapes,
it is much easier to get trapped at a local optimum and much harder to
identify the global optimum. And second, when we consider the “epistemic
fitness” conferred by a combination of scientific methods, theories, tech-
niques, and so on, there is a much greater degree of interdependency than a
two-dimensional landscape would allow. If we consider more “realistic”

TABLE 2. EPISTEMIC SUCCESS AND EPISTEMIC PROGRESS OF DIFFERENT STRATEGIES

Epistemic Success
Epistemic
Progress

Strategy P Mean SD Min Max Median 200 500

10 agents:
Maverick .75 121 58 52 299 109 7.74 8.79
Follower .79 123 63 50 509 110 6.76 7.24
Control 1 107 71 37 668 93 19.32 27.41
Swarm-F 1 167 110 34 602 133 17.94 39.51
Swarm-R 1 151 88 17 488 131 17.80 39.42

100 agents:
Maverick 1 54 8 37 75 54 36.95 36.99
Follower 1 52 7 38 73 52 22.92 22.92
Control 1 42 10 23 71 41 67.35 74.25
Swarm-F 1 64 29 21 154 57 80.85 97.54
Swarm-R 1 41 18 12 111 37 86.49 99.32

200 agents:
Maverick 1 49 6 37 66 48 51.23 51.23
Follower 1 43 5 31 59 43 29.10 29.10
Control 1 33 8 20 54 32 84.24 88.70
Swarm-F 1 45 20 20 135 40 93.87 99.78
Swarm-R 1 32 11 13 66 31 98.09 99.99

Note.—‘Swarm-F’ and ‘Swarm-R’ are, respectively, the flocking and the repulsing configuration of
swarm scientists. The follower and control ði.e., HE ruleÞ strategies used here are as intended: followers
do not necessarily get stuck in a cycle of length 2 when isolated ðalthough they might get stuck eventu-
allyÞ, and controls do not have an artificial time delay ðnor do they move twice per iteration in regions of
increasing significanceÞ. P is the proportion of populations that found both peaks within the time allotted,
and 200 and 500 is the percentage of significant landscape that has been explored after 200 and 500 iter-
ations, respectively. Simulations were stopped after any number of iterations equal to 0 modulo 500 if no
epistemic progress had been made in the last 500 iterations.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 445



epistemic landscapes, what, if anything, can we infer about the benefits of
cognitive diversity?
We can begin to consider this question by reinterpreting the NK-

landscape model of Kauffman and Levin ð1987Þ and Kauffman and Wein-
berger ð1989Þ as an epistemic landscape. The idea is straightforward: sup-
pose we have a set of N scientific propositions, where these propositions may
consist of both abstract general statements of high theory as well as specific
statements of particular laboratory technique. The belief state of an indi-
vidual scientist can be represented by a vector~b 5 b1; : : : ; bNih , where bi 5
0 if the scientist does not believe the ith proposition, and bi 5 1 if the
scientist does believe the ith proposition.
The reason why NK landscapes are useful for thinking about epistemic

landscapes is that they allow one to model interdependencies between the
various propositions believed ðor not believedÞ by a scientist. That is, the
fitness contribution of bi may depend on not just the value of bi ð0 or 1Þ but
the value of several other entries in the scientist’s overall belief state. The fitness
function, in a word, may have varying degrees of epistasis. Let 0 ≤ K ≤ N
denote the number of interdependencies contributing to the fitness contri-
bution of bi. ðSee fig. 10 for an illustration of two different epistatic regions.Þ
One can think of the amount of epistasis in a fitness function for an epistemic
landscape as a formal model of the Quinean web of belief.
Figure 11 illustrates how the fitness of a belief vector is calculated for a

bit string of length 8 and epistasis 2. The fitness function f is defined in
terms of eight other functions f1, . . . , f8, where function fi is used to de-
termine the fitness contribution of bit bi. Since the degree of epistasis is 2,
the fitness of bit bi also depends on the values of bi−1 and bi11 ðwhere, at the
end of the bit string, we wrap around the ends to avoid edge effectsÞ. The
individual fitness functions f1, . . . , f8 are defined using the lookup table in
figure 11A. In general, if N is the length of the bit string ~b, then

f ð~bÞ 5 o
N

i51

fi bi2 K2 : : : bi : : : bi1 K2
� �

:

Although we consider, simply for reasons of simplicity, only two possible
values for each bi ðfull belief or full denialÞ, there is no reason why we could

Figure 10. Two different regions of epistasis for the bit bi.

446 JASON MCKENZIE ALEXANDER ET AL.



not allow more finely grained credal states. If we denote the number of
credal states by A, then we see that the Weisberg-Muldoon epistemic land-
scape model is simply an NK landscape with N 5 2, K 5 2, and A 5 101
and a particular fitness function. ðSee the appendix for further details regard-
ing the calculation of fitness functions on NK landscapes.Þ
In order to see whether social learning and cognitive diversity help people

reach the peak of greatest epistemic fitness on NK landscapes, let us con-
sider—as a baseline result—how a single independent agent would fare.
Assume that the agent searches via probe and adjust as follows. The agent
starts with a randomly selected belief vector of length N. In each iteration,
the agent probes one randomly selected belief by considering its alternate
value ð0 if 1, and 1 if 0Þ. If changing that belief yields an overall increase in
fitness, the agent keeps the change; if changing that belief decreased the
fitness, the change is rejected. Table 3 illustrates the simulation results for a
range of values of N and K. For each set of values of N and K, 100 sim-
ulations were performed, running for 1,000 iterations. Each simulation used
a randomly generated uncorrelated fitness function.21 The mean fitness over
all simulations ðand the standard deviationÞ are shown.

Figure 11. Applying a fitness function defined via a lookup table to a bit string. A,
Lookup table for the fitness function f ; B, applying the fitness function f to the bit
string 01001101.

21. According to Kauffman and Weinberger ð1989, 216Þ, a fitness function is said to
be uncorrelated if “the fitness of 1-mutant neighbors ½is� assigned at random from some

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 447



Now consider the possibility of social learning. Suppose we have a pop-
ulation consisting of some fixed number of agents. At the end of each iter-
ation, each individual polls every other. If agent M changed bi such that it
yielded greater fitness to M, then all other agents incorporate that change.22

As table 3 shows, social learning makes absolutely no difference in cases of
uncorrelated fitness functions. Furthermore, social learning may actually be
detrimental for the performance of the agents, given epistasis.23

The crucial difference between the NK model just described and the
Weisberg and Muldoon epistemic landscape is this. Expressed as an NK
model, Weisberg and Muldoon’s model assumes fitness functions are highly
correlated. Let us suppose, for the sake of argument, that the fitness function
used is similarly highly correlated. In particular, assume that the fitness of bi
is simply the relative frequency of the current value of bi in its epistatic re-
gion ðsee fig. 12Þ. The results for simulations where the fitness functions are
correlated in this way, as in table 4, show that, in this case, there is indeed a
positive effect of social learning on the rate of epistemic progress.
Why does this matter? It matters because the Weisberg and Muldoon

model builds into the basic topology of the epistemic landscape correlations
that make social learning advantageous. As such, we should not be sur-
prised to find, in the case they consider, that cognitive diversity and social
interactions between agents can be beneficial. But, as the generalization to
NK landscapes shows, social learning is not always beneficial. Whether
social learning is beneficial or harmful depends on the topology of the epi-
stemic landscape, a point of which we know very little. In some cases we
might well suspect that social learning will be advantageous ðe.g., Does the
problem decompose into subproblems? Does the problem require a diverse
set of skills possessed by no single individual?Þ, but we cannot be sure that

23. Suppose agent M finds a fitness-enhancing change for bi. The reason why it may be
detrimental for other agents to adopt this change is because the fitness increase, for M,
depends on the particular value of the other beliefs in the region of epistasis around bi. If
other agents in the population do not have the same K other beliefs that agent M has,
there is no guarantee that changing the value of bi for them will have the same effect.

fixed underlying distribution.” For the purpose of this article, an uncorrelated fitness
function assigns the fitness to 1-mutant neighbors at random using the uniform distribu-
tion over ½0, 1�. More precisely, the fitness function fi, specifying the fitness contribution
for bit bi and its surrounding epistatic region, is defined on the substring bi2 K=2ð Þ : : :
bi : : : bi1 K=2ð Þ of ~b, which has length K 1 1. The fitness contribution of each of the
possible 2K11 arguments to fi is set to a randomly chosen value in ½0, 1�, drawn from the
uniform distribution. With such an uncorrelated fitness function, knowledge of the val-
ues of the entries in the region of epistasis around some bi gives no information as to
whether the 1-mutant neighbor will have greater or lower fitness than the current be-
lief vector.

22. Think of this as a model in which each agent publishes the result of each experiment,
and people always trust each other’s results.

448 JASON MCKENZIE ALEXANDER ET AL.



social learning will be advantageous because we will not know the epi-
stemic landscape’s topology.
We stress this point because, although Weisberg and Muldoon acknowl-

edge that “landscapes can be made more rugged, they can contain more in-
formation,” and so on, they suggest that some of their findings do, in fact,
generalize. They state that “even with our current models and current land-
scape, we have observed a number of very interesting general trends about
the division of cognitive labour.” What are some of these general trends?
For one, that “followers seem very well suited for puzzle solving—the
simple articulation of details of a paradigm. Mavericks can partially fulfill
this role, but their search patterns through the epistemic landscape are not
particularly well suited for the kind of long term analyses required, for

TABLE 3. SOCIAL LEARNING AND GLOBAL PERFORMANCE

N 5 8 N 5 16 N 5 24 N 5 48 N 5 96

No social learning:
K 5 0 .66 ð.08Þ .67 ð.05Þ .66 ð.05Þ .67 ð.04Þ .66 ð.03Þ
K 5 2 .70 ð.07Þ .70 ð.05Þ .70 ð.04Þ .70 ð.03Þ .71 ð.02Þ
K 5 4 .69 ð.06Þ .70 ð.04Þ .70 ð.03Þ .70 ð.03Þ .70 ð.02Þ
K 5 8 .67 ð.06Þ .68 ð.04Þ .68 ð.03Þ .68 ð.02Þ .68 ð.02Þ
K 5 16 .64 ð.03Þ .65 ð.03Þ .66 ð.02Þ .66 ð.02Þ
K 5 24 .62 ð.03Þ .64 ð.02Þ .64 ð.02Þ
K 5 48 .60 ð.02Þ .61 ð.02Þ
K 5 96 .58 ð.01Þ

Social learning:
K 5 0 .66 ð.10Þ .67 ð.06Þ .67 ð.05Þ .67 ð.04Þ .67 ð.03Þ
K 5 2 .73 ð.07Þ .72 ð.04Þ .72 ð.04Þ .71 ð.02Þ .71 ð.02Þ
K 5 4 .70 ð.06Þ .72 ð.04Þ .72 ð.03Þ .71 ð.02Þ .72 ð.02Þ
K 5 8 .66 ð.05Þ .68 ð.04Þ .70 ð.03Þ .70 ð.02Þ .70 ð.02Þ
K 5 16 .64 ð.04Þ .65 ð.03Þ .66 ð.02Þ .66 ð.02Þ
K 5 24 .62 ð.03Þ .64 ð.02Þ .65 ð.02Þ
K 5 48 .60 ð.02Þ .61 ð.01Þ
K 5 96 .58 ð.02Þ
Note.—Social learning makes no difference to the global performance given uncorrelated fitness

functions. When K 5 N, the fitness function fi for bi depends on the entire belief state of the agent. SD in
parentheses.

Figure 12. Correlated fitness function. Local fitness of any bit bi is the relative
frequency of the current value of bi 5 1 in its epistatic region.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 449



example, to add one more decimal place to a known constant” ð2009, 249Þ.
And also: “We have also seen that in mixed populations, mavericks can
provide pathways for followers to find the base of the peaks on the epi-
stemic landscape. Once the followers find these bases, they are reasonably
efficient at finding the tops. And mavericks can also stimulate followers to
engage in pure puzzle solving, ensuring that the landscape is fully explored
to find hidden significant approaches. Therefore, mixed populations of mav-
ericks and followers are valuable divisions of cognitive labor” ð250Þ. And,
finally: “As we showed, individual mavericks find the peaks extraordinarily
quickly and indeed the whole population converges rapidly on those peaks.
This means that if one wants to search the landscape rapidly for the most
significant truths, one should employ a population of mavericks, at least as
opposed to followers or controls” ð250, italics addedÞ.
Each of these claims would be uninteresting if “the ½epistemic� land-

scape” only referred to the epistemic landscape modeled in the article. The
reason why these claims are interesting is that they gesture toward general
properties of scientific practice and suggest fruitful ways of organizing sci-
entific research. Yet it only makes sense to say that one should employ a
population of mavericks in cases where the epistemic landscape is such that
the maverick strategy would be beneficial, and it is far from obvious that
the maverick strategy will prove to be beneficial on an arbitrary epistemic
landscape or when the model is adjusted to allow for greater realism by, say,
incorporating observation error.24 So, although the general trends have a
certain degree of intuitive appeal, it is unclear to what extent these claims
are, in general, justified by the Weisberg and Muldoon epistemic landscape
model.

24. One way this could be done would be to incorporate a probability that agents in-
correctly perceive the true epistemic fitness of the point they currently occupy on the

TABLE 4. SOCIAL LEARNING AND EPISTEMIC PROGRESS

No Social Learning Social Learning

Mean Mean Steps Mean Mean Steps

K 5 2 .92 ð.01Þ 994.43 .94 ð.02Þ 508.14
K 5 4 .93 ð.01Þ 997.74 .95 ð.02Þ 277.66
K 5 8 .90 ð.01Þ 994.77 .93 ð.03Þ 253.35
K 5 16 .90 ð.02Þ 989.29 .92 ð.04Þ 215.37
K 5 24 .90 ð.02Þ 927.92 .92 ð.06Þ 197.83
K 5 48 .93 ð.04Þ 916.71 .97 ð.08Þ 183.61
K 5 96 1.00 ð.00Þ 600.94 1.00 ð.00Þ 185.58
Note.—When correlation exists for the fitness function, social learning makes a positive difference in

the rate at which epistemic progress occurs. N 5 96; SD in parentheses.

450 JASON MCKENZIE ALEXANDER ET AL.



8. Conclusion. The Weisberg-Muldoon model has received a considerable
amount of attention regarding its purported claim to show that there are
epistemic reasons for the division of cognitive labor. In particular, Weisberg
and Muldoon alleged to show that the “maverick” research strategy is far
better than its competitors,25 and one of the “general trends” ð2009, 249Þ
they observed is that “to be maximally effective, scientists need to really
divide their cognitive labor” ð227Þ. We have argued that these two claims are
not true. Maverick scientists do not perform far better than their competitors,
such as the HE rule, once the implementation errors that handicapped the
other search types have been corrected. By proper bookkeeping, we have
shown that the increase in the performance of the mixed population is only
due to the performance of the added mavericks. As for the benefits of cog-
nitive diversity, we have constructed at least one other search strategy, the
“swarm scientist,” which, in some cases, outperforms the maverick scien-
tists.
In saying this, we do not wish to be understood as arguing that there are

no epistemic reasons for cognitive diversity. We are simply pointing out
that, despite its intuitive appeal, the Weisberg and Muldoon model does not
succeed in showing that there are epistemic reasons for cognitive diversity.
Furthermore, since so much in their methodology turns on assumptions
regarding the specific nature of the epistemic landscape—something whose
very nature is beyond our ken—we are skeptical as to whether their par-
ticular method of arguing for the epistemic benefits of cognitive diversity
can ever succeed.

Appendix

Generating Uncorrelated Fitness Functions on Large NK Landscapes

Following Kauffman and Levin ð1987Þ, fix integers N and K such that 0 ≤
K ≤ N. The total fitness of a bit string b1b2 . . . bN is calculated using fitness
functions f1, . . . , fN, where fi is applied to bit bi and the surrounding region

25. “If one wants to search the landscape rapidly for the most significant truths, one
should employ a population of mavericks, at least as opposed to followers or controls”
ðWeisberg and Muldoon 2009, 250Þ.

landscape. Suppose agents “publish” their findings, and this incorrect report will be the
epistemic fitness ascribed to that point on the landscape by other agents. ðIf agents visit
that same point on the landscape, they have the chance to perceive correctly the “true”
value, of course.Þ If the change of experimental error is sufficiently high, the maverick
strategy, in these cases, could well prove disadvantageous: mavericks, by avoiding parts
of the landscape already explored, could miss finding peaks for a very long time if that
peak was explored previously but incorrectly identified.

EPISTEMIC LANDSCAPES, OPTIMAL SEARCH 451



of epistasis consisting of the K bits flanking bi on the left and right. ðTo pre-
vent edge effects, we assume the bit string wraps at the edges to ensure that
all bits have regions of epistasis the same size.Þ For small N and K, the
fitness function can be defined using a lookup table specifying all 2K11

values for each of the N fitness functions, as shown in figure 11.
For large N and K, it is not feasible to define fitness functions using a

lookup table. If N 5 100 and K 5 60, an uncorrelated fitness function would
have 100 × 261 different values. However, it is possible to procedurally
define an uncorrelated fitness function using a trick similar to that used by
programmers of the 1984 video game Elite. ðThat game had to fit 8 different
universes, each containing 256 planets with unique properties, into 16 kilo-
bytes of memory.Þ We used a common 19937 Mersenne Twister RNG to
procedurally generate the fitness functions.
Given fixed values of N, K with K < N, let S 5 ðs1, . . . , sNÞ be a list of N

different salt values. Denote the region of epistasis around bit i at time t by
hti 5 fhti;1; : : : ; hti;K11g. The fitness contribution of bi at t is determined by
seeding the RNG with the bits of si � hti ðzeroing out the remaining bits in
the state space of the RNGÞ and generating a random real number between
0 and 1. As long as the length of si � hti ≤ 19; 968, we obtain a unique seed.
This is because the state of the 19937 Mersenne Twister is 624 words, each
with 32 bits, so there are 624 × 32 5 19,968 bits in the state. If the length of
si � hti is sufficiently less than the length of the state space, the random value
will be unique ðand reproducibleÞ. This yields a completely uncorrelated
fitness function.

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