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Paper to be presented at the DRUID 2012
on
June 19 to June 21
at
CBS, Copenhagen, Denmark,
Does Size Matter? Parallel Search and the Efficacy of Experiential
LearningHart Posen
University of MichiganRoss School of Business
Dirk MartignoniUniversity of Zurich
Department of Business [email protected]
Daniel Levinthal
University of PennsylvaniaWharton School
AbstractExperiential learning is a central idea in the management literature ? and its general efficacy is a taken-for-grantedelement of management thought. However, we also know that history is not generous with experience, and thisexperience constraint engenders errors and myopia in the process of experiential learning. We consider the implicationsof relaxing the experience constraint ? by increasing the number of agents in the organization to make possible parallelsearch. Employing a computational model, we find three stylized facts about larger, less constrained, organizations: (a)they explore less than smaller organizations, (b) they are less likely to discover the very best alternative, and (c) they
exhibit superior performance on average, even though their chosen alternatives may not be the very best. Thesestylized facts seem to reflect important elements of reality, yet arise from the simple assumption that organizations arecomposed of multiple agents that pool their knowledge and experience as they engage in experiential learning. Thus,while parallel search relaxes the experience constraint, it may amplify errors and myopia, and on some metrics, reducesthe efficacy of experiential learning.
Jelcodes:M10,-
Does Size Matter?
Parallel Search and the Efficacy of Experiential Learning
Hart E. Posen
Ross School of Business
University of Michigan
Dirk Martignoni
Department of Business Administration
University of Zurich
Daniel A. Levinthal
Wharton School
University of Pennsylvania
Very Preliminary
Please do not cite or distribute without permission.
February 26, 2012
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Does Size Matter?
Parallel Search and the Efficacy of Experiential Learning
Abstract
Experiential learning is a central idea in the management literature – and its general efficacy is a
taken-for-granted element of management thought. However, we also know that history is not generous
with experience, and this experience constraint engenders errors and myopia in the process of experiential
learning. We consider the implications of relaxing the experience constraint – by increasing the number of
agents in the organization to make possible parallel search. Employing a computational model, we find
three stylized facts about larger, less constrained, organizations: (a) they explore less than smaller
organizations, (b) they are less likely to discover the very best alternative, and (c) they exhibit superior
performance on average, even though their chosen alternatives may not be the very best. These stylized
facts seem to reflect important elements of reality, yet arise from the simple assumption that organizations
are composed of multiple agents that pool their knowledge and experience as they engage in experiential
learning. Thus, while parallel search relaxes the experience constraint, it may amplify errors and myopia,
and on some metrics, reduces the efficacy of experiential learning.
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1. Introduction
A central tenet of research in the Carnegie School tradition is that boundedly rational agents learn
from experience. Yet, “history is not generous with experience” (March, Sproull, and Tamuz 1991, p.1).
This paucity of experience often makes learning slow and prone to errors (March 2010), and engenders a
myopic tendency towards exploitation (Levinthal and March 1993, Denrell and March 2001). Collecting
multiple learning agents inside an organization should alleviate these problems because it makes possible
parallel learning – agents can augment their own experience with that of others in the organization. Yet,
as we will demonstrate, augmenting experience through parallel search generates unintended
consequences that may in fact amplify errors and myopia, rather than diminish them.
We develop a simple computational model that gives rise to three stylized facts about the efficacy of
experiential learning by organizations that differ only in the number of agents engaged in parallel search
(which, for brevity, we refer to as “size”).1 Larger organizations (those with more agents learning in
parallel) explore fewer policy alternatives than smaller organizations. As a consequence of more myopic
search, larger organizations are less likely to discover the very best alternative. From this, one might be
tempted to conclude that larger organizations are inferior to smaller organizations in terms of their ability
to learn from experience. However, the same mechanism that magnifies this myopic behavior also enables
larger organizations to, on average, identify better policy alternatives, even though their chosen
alternatives may not be the very best. These stylized facts seem to reflect important elements of reality,
yet arise not from mechanisms such as elaborated organizational structure or incentive misalignment, but
1 We purposefully abstract from other notions of organizational size, as we focus solely on the number of agents in an
organization engaging in parallel search. That said, there is a broad literature that examines the performance implications of size, focusing on the benefits of scale and scope economies (e.g., Zenger 1994), the costs associated with misaligned incentives (e.g., Holmstrom 1989), extensive organizational structure (Hannan and Freeman 1984), and hierarchical decision making (Siggelkow and Rivkin 2006).
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rather, from one simple assumption: organizations are composed of multiple agents that pool their
knowledge and experience as they engage in experiential learning.2
The intuition underlying these conclusions rests on the observation that, for many important
problems, the true value of an alternative cannot be known with certainty (e.g., Denrell and March 2001,
Knudsen and Levinthal 2007, March 1996, Posen and Levinthal 2011). Consider an organization engaged
in drug discovery. It has a consideration set of molecules that represent potential cures for a particular
disease. Scientists in the organization engage in a process of experiential learning. Each scientist trials a
different molecule by analyzing it in silica or in vivo (employing a computational or animal model of the
disease), and generates an evaluation of efficacy. However, an individual’s trial of a molecule does not
provide definitive evidence of efficacy. Any particular experience is likely to be misleading to the extent
there is variation in possible outcomes; and small samples of experience will have greater sampling errors
than larger samples. Clearly, larger organizations can explore a greater number of alternatives, each
alternative can be explored multiple times, and individuals in the organization have a broader base of
experience to rely on in making choices. However, naïve intuition on the implications of size breaks down
because, in the process of experiential learning, increasing the number of agents searching in parallel
engenders a trade-off – it decreases the quantity of exploration undertaken but also increases the quality
of alternatives explored. The returns to larger size then hinge on the relative strength of these two effects.3
We anchor our formal development on the multi-armed bandit model. This model is the canonical
representation of exploration and exploitation under conditions of uncertainty (Holland, 1975). In the
management literature, March formulates much of his discussion of learning in terms of the bandit model
2 There is a literature that examines the implications of size for the efficacy of team/group decision-making (Haleblian and
Finkelstein 1993) and creativity (e.g., Mullen, Johnson, and Salas 1991). However, this research typically does not study learning. Learning requires feedback from the environment from which the group or team updates beliefs, making potentially different choices in subsequent periods – this is absent from most research on teams and groups.
3 Early models of search and experiential learning (e.g., Levinthal and March 1981, Nelson and Winter 1982), and more recent models of search in complex environments (e.g., Gavetti and Levinthal 2000, Siggelkow and Rivkin 2005), highlight the implications of the exploration-exploitation trade-off. However, the models typically treat the organization as a single unitary actor (e.g., Herriott, Levinthal, and March 1985, Levinthal 1997, Rivkin 2000). Thus, these models ignore the issue of parallel learning in organizations of differing sizes and implications for the efficacy of learning.
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(Denrell and March, 2001, March 1996, 2003, 2010), and Posen and Levinthal (2011) extend the model to
examine the implications of environmental turbulence. The bandit model takes its name from a slot
machine analogy in which the agent seeks to maximize the flow of returns over time. In the context of a
unitary organization (i.e., a singular learning agent), management consists of making choices under
uncertainty. In each period, she chooses from a set of policy alternatives, with the payoff to a choice
reflecting a draw from a probability distribution with an unknown, alternative-specific mean. The agent
chooses based on her beliefs about the expected returns to each of the alternatives. Thus, she is portrayed
as possessing a mental model or cognitive map (derived from her own prior experience) that encapsulates
her understanding of the merits of the available set of choices.
We extend this standard single-agent bandit model to a multi-agent model of an organization
consisting of multiple learning individuals. We assume that agents within an organization form beliefs
about the relative merits of alternatives not only based on their own experience, but also on the experience
of others in the organization. As such, individual agents may explore alternatives that, based on their own
experience, appear inferior, because the alternative is believed to be superior based on the experience of
others in the organization.4
This paper proceeds as follows. In the next section, we describe the theoretical background and
setup of our multi-agent multi-armed bandit computational model. In section 3, we present the results of a
simulation experiment in which we examine the properties of the model of experiential learning as a
function of organizational size (number of agents engaged in parallel search). In the final section, we
discuss the implications of these results for theory and practice.
4 Kogut and Zander (1992) argue that the defining feature of an organization is that knowledge flows more freely within the
organization than across its boundaries. Similarly, Herriott et al. (1985) argue that “in a social environment, learning from direct experience is supplemented by the diffusion of experience, that is, by copying others. From a rational perspective, copying can be seen as a way of increasing (on average) the amount of experience from which an individual draws … From a behavioral perspective, it can be seen as a standard way by which adaptive systems deal with uncertainty and ambiguity” (p.299).
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2. Model
The bandit model, which is the basis of our analysis, has been the subject of significant study
because its underlying structure closely resembles a variety of realistic economic situations ranging from
research settings such as R&D (Hardwick and Stout 1992), to strategic issues, such as product pricing
(Bergemann and Välimäki 1996), and consumer choice (Gans, Knox, and Croson 2007). There are two
common features underlying economic problems that are modeled in a bandit framework. First,
information about the returns to an alternative can only be gathered by trialing it. Second, feedback from
trials is subject to uncertainty that gives rise to variation in possible outcomes, and as such, any particular
experience is likely to be misleading.
Formally, the bandit model reflects a sequential choice problem where, at each point in time, t, an
agent must choose among N alternatives. The realized outcome of a particular choice is a draw from an
unknown (alternative specific) probability distribution. If the process is Bernoulli, then the choice of
alternative n results in an outcome of r = (1, 0), reflecting a positive outcome of r = 1 with probability pn
and a negative outcome r = 0 with probability 1-pn. As such, the state of the environment can be described
by the vector of payoffs to the alternatives, P =[p1,…, pN].
Consider agent m’s beliefs, qm,n,t, about alternative n at time t, which are encapsulated by the vector
Qm,t=[qm,1,t ,…, qm,N,t] where 0 ≤ qm,n,t ≤ 1. To refine her beliefs and maximize the value of the stream of
rewards, the agent engages in learning. In the initial period t=0, agent m has prior beliefs, Qm(t=0) =
[qm,1,…, qm,N], about the value of the N reward outcome probabilities P =[p1,…, pN]. In each subsequent
period, the agent makes a choice from the set of alternatives. By acting – making a choice of an
alternative – the agent m receives feedback from the environment in the form of an outcome signal as a
success or failure, rm,n=(1,0).
We assume that beliefs at any given point in time reflect the average reward over an agent’s entire
history of trials of a given alternative (March 1996, Sutton and Barto 1998). This simple average updating
is a special case of the more general Bush-Mosteller fractional adjustment updating methodology. Thus,
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for an agent, m, her belief about the merits of alternative n at time t, reflect the average reward history
(yielding rewards 𝑟!, 𝑟!,… , 𝑟!!) associated with her 𝑘! trials of that alternative. As such:
𝑞!,! =!!,!
!!!!!!!
, (1)
where kn denotes the number of times alternative n has been trialed by period t. In any period, t, only the
belief about currently trialed alternative, n, is updated; for all other alternatives, j, beliefs remain
unchanged such that 𝑞!,! = 𝑞!,!!! .
In the context of a multi-agent organization, an agent may not only rely on his own experiences but
also on the experiences of all other agents in the organization. We denote ρ as the extent to which the
agent h weights her own experience compared to the aggregated experiences of all agents in the
organization. Agent h’s estimate of the value of alternative n in period t is given by the linear combination
of its individual experience with this alternative (first term in equation (2), weighted by 1-ρ), and the
aggregated experience all agents in the organization (second term in equation (2), weighted by ρ):
𝑞!,!,! = (1 − 𝜌) !!,!,!!!,!!!!!!,!
+ 𝜌 !!,!,!!!,!!!!
!!!!
!!,!!!!!
, (2)
where m indexes individual agents in the organization of size M. This belief updating mechanism implies
that all agents’ experiences with a given alternative contribute equally to belief formation.
As a result, when ρ=0, all agents make choices based on beliefs that arise from their own experience
alone. In contrast, when ρ=1, all agents in an organization make choices based on beliefs that arise from
the aggregated experience of all agents in the organization.5 From a practical perspective, ρ>0 has two
interpretations. On one hand, it represents the simplest hierarchy, as in March (1991a), in which a
superior compiles the information gathered by each agent, aggregates it, and instructs her subordinates to
5 Note that when ρ=0, a multi-agent organization is identical to an equal number of atomistic agents.
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act on the basis of this pooled belief. On the other hand, we might conceptualize a flat organization in
which agents simply pool experience.
In each period, each agent, h, in the organization independently chooses alternatives to trial based
on its beliefs, qh,n,t. While there is a wide variety of plausible exploration strategies, perhaps the simplest
and best-known strategy is that of selecting, at each point in time, the alternative with the highest belief,
max(q1,…, qN), reflecting the highest expected reward (Auer, Cesa-Bianchi, and Fischer 2002, Sutton and
Barto 1998). This rule is “greedy” (Sutton and Barto 1998) in the sense that at each point in time it
maximizes the expected reward in the next period. Agents select a different alternative only when there
are multiple alternatives that share the maximum belief. We assume the agents choose the alternative with
the highest estimated value (belief). If they are indifferent across alternatives, because several alternatives
appear equally attractive, they randomly select one of them (Sutton and Barto 1998).6 In subsequent
experiments, we relax the greedy assumption, and allow agents to engage in either random exploration
that is uninformed by beliefs (as in Csaszar and Siggelkow 2010, Ethiraj and Levinthal 2009, Levinthal
1997) or exploration where random choice is weighted by beliefs about the relative merits of alternatives
(Posen and Levinthal 2011, Sutton and Barto 1998).
We use the multi-agent and multi-armed bandit model described above to analyze the effect of
organizational size on learning and exploration. The structure of the simulation requires initializing both
the opportunity structure of the task environment and organizational beliefs about the merits of
alternatives.
The opportunity structure of the environment is defined by initializing the payoff probabilities for
each alternative. We formulate a 50-arm bandit model such that the vector of alternatives’ payoff
probabilities is P = [p1,…, p50]. Each alternative is allocated a payoff probability, pn, that is a draw from a
uniform distribution [0,1]. This produces a distribution of payoff probabilities (across alternatives) that is
6 In our sensitivity analysis, we examine the possibility that individual agents employ less greedy choice rules. Our results are
robust to this possibility.
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symmetrical with a mean of 0.50 and standard deviation of 0.29. The choice of a particular alternative n
leads to a positive/negative outcome with probabilities (pn, 1-pn), and an associated reward of ri,t =(1,0).
Given this task environment, we assume agents have initial beliefs across the alternatives, Qm(t=0) =
[qm,1,…, qm,50], that are homogeneously set to 0.5, which is equal to the mean value of the actual
distribution of payoffs. Under this assumption, the belief updating process employing the average reward
history (per Equation 1) is equivalent to Bayesian updating.7
We examine organizations ranging in size from M=1 through M=10 agents. Each organization
searches on a uniquely specified environment. Within an organization, agents pool their experience and
update beliefs according to equation 2, where ρ=1. To average over the stochastic outcome of any single
organization, we examine twenty thousand iterations of each organizational size. A simulation runs for
200 periods by which time steady state is reached. To make sensible comparisons across organizations of
different sizes, we normalize by the number of agents in the organization.
3. Analysis
In the following analysis, we begin by examining the impact of organizational size (the number of
agents searching in parallel and pooling experience) on performance outcomes. We examine the
underlying mechanisms of these outcomes, by focusing on how organizational size alters the dynamics of
exploration and the efficacy of learning. In the first experiment, we make two key assumptions about
agents in organizations. First, agents are greedy, exploiting the alternative they deem to currently be the
best, only exploring if they are indifferent among a set of such alternatives.8 Second, agents act on the
basis of beliefs that result from the aggregation of the experience of all agents in the organization (ρ=1).
7 In particular, the initial priors are assumed to have arisen from two pre-sample trials that generated one positive and one
negative result. As such, qn=0.5 and kn =2 at t=0. This assumption implies that the agents have rational expectations and update their beliefs according to Bayesian principles.
8 In the bandit model, agents are assumed to make choices based on their beliefs about the relative merits of alternatives. For simplicity, in this analysis we focus on agents pursuing a greedy strategy, always choosing the alternative on which they have the highest belief, and choosing randomly between alternatives that are believed to be equally good. Relaxing the greedy assumption by allowing for additional exploration (e.g., ε probability of choosing a different arm at random, a strategy called ε-greedy) increases the level of exploration but does not change the basic intuition.
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In the second experiment, we relax these assumptions, allowing agents to explore more broadly and to act
on their own beliefs.
Experiment 1: Baseline Model of Organizational Size
In Figure 1, we plot two performance metrics (averaged over the first 200 periods). Organizational
performance is measured as the cumulative reward stream for all agents in the organization. Discovery of
the best alternative is measured as the cumulative probability that at least one agent in the organization
has trialed the best alternative.
< Insert Figure 1 about here >
The key result in the figure above is the paradox of size – larger organizations (those with more
agents searching in parallel) are less likely to discover the best alternative, but on average generate higher
performance (with the exception of two-agent organizations, which we will return to later in the analysis).
The paradox of size emerges even though we hold effort constant across different size organizations (we
divide both the cumulative rewards and the probability of discovering the best alternative by the number
of agents in the organization).
Our objective in the subsequent analysis how the pooling experience gives rise to this paradox of
size. We focus our analyses on two key factors that might alter the efficacy of search and learning: the
quantity of exploration and the quality of alternatives explored.
Quantity of Exploration
We begin by examining differences in the extent to which firms of different sizes explore the set of
alternatives – which we term the quantity of exploration. Because we model agents as greedy exploiters,
they are likely to under-explore the set of alternatives. As such, we consider the possibility that pooling
experience in an organization increases the quantity (extent) of exploration, moving the organization
closer to the optimal balance.
< Insert Figure 2 about here >
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In Figure 2 we examine two metrics of the quantity of exploration. In Panel A, we plot the extent to
which choices taken by an agent within an organization are exploratory by measuring the cumulative
count of events in which an agent chooses a different alternative in period t+1 than it chose in period t.
An agent in a one-agent organization explores 2.98 times (during the 200 periods until steady state is
reached), while each agent in a 10-agent organization explores only 1.76 times. The key result of this
figure is that agents in larger organizations explore less – less frequently changing alternative choices –
than do agents in smaller organizations. In particular, the quantity of exploration for a ten-agent
organization is 40% less than that of a one-agent organization.
To explain why agents in larger organizations are less likely to change alternatives over time, we
also track the extent to which, conditional on changing alternatives, whether the alternative to which they
change is new to the organization. For one-agent organizations, exploration usually entails the trial of
alternatives untried in the past (the black portion of the bar is 0.5 times out of nearly 3 change events). In
contrast, when agents in a large organization explore by choosing an alternative different from its current
choice, this new alternative is rarely new to the organization. In a large organization, the positive
experience of one agent with an alternative leads to an increased probability that others in the
organization will trial that alternative. Likewise, when one agent has a negative experience with an
alternative, others in the organization are less likely to pursue it. Thus, pooling experience in an
organization tends to generate directed exploration, which leads to the early identification of a relatively
good alternative within the restricted consideration set. Yet, at the same time, it also leads to a rapid
decrease in the level of exploration.
In Figure 2 Panel B, we examine the implications of directed exploration for the breadth and depth
of exploration. To normalize results across organizations of different sizes, we measure the quantity of
exploration of ten agents organized in different arrangements – one ten-agent, five two-agent, two five-
agent, and one ten-agent organization. The two polar cases of organizing ten agents generate substantially
different patterns of exploration. The ten one-agent organizations explore more than 28 different
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alternatives (of the 50 available), while one ten-agent organization explores less than ten alternatives.
Indeed, larger organizations rarely explore beyond the set of alternatives that they trial in the first period.9
Pooling experience in organizations gives rise to directed exploration, reducing the quantity of
exploration by 66% (far more than the 40% reduction in exploration measured by alternative changes).
Moreover, larger organizations devote a disproportionate share of their exploration effort to testing an
alternative only once. If we consider only alternatives trialed at least twice, ten-agent organizations
explore 78% fewer alternatives than those explored by one-agent organizations.
Quality of Exploration and Average Performance
If the quantity of exploration decreases with increased organizational size, it is insufficient to
explain why larger organizations exhibit superior average performance. We conjecture that because
experiences are pooled in organizations, larger organizations also explore better quality alternatives.
In Figure 3, we plot the distribution of alternatives trialed, conditional on changing alternatives. The
horizontal axis indicates the quality of the alternative trialed (ranked from best, rank=1, to worst,
rank=50). The vertical axis indicates the probability that an alternative of a given rank is selected. For
one-agent organizations, the choice of alternatives is uniform across the distribution of alternative
payoffs. In contrast, larger organizations tend to choose better alternatives when they explore. In this
sense, larger organizations exhibit a higher quality of exploration.
< Insert Figure 3 about here >
The quality of exploration increases with organizational size for two reasons. First, agents in the
organization respond to rewards received by other agents in the organization. When an alternative is
trialed, a positive reward suggests that the alternative is unlikely to be extremely bad, while a negative
reward suggests that it is unlikely to be extremely good. Pooling such experience guides agents in the
9 In the first period, agents are by construction indifferent among the alternatives. Thus, ten agents examine approximately 9.1
alternatives in the first period (independent of the way they are organized).
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organization toward good alternatives and away from bad ones. This effect increases with organizational
size.
Second, by concentrating exploration on a narrow set of alternatives, directed exploration from
pooled experience generates more accurate beliefs about the relative merits of alternatives. Because the
challenge faced by firms is one of evaluative uncertainty, multiple trials of an alternative will provide
more accurate estimates (beliefs) of its true value. Larger organizations explore a smaller quantity of
alternatives, but on the alternatives they explore, they engage in repeated trials (per Figure 2 Panel B).
This generates more accurate beliefs and gives rise to a virtuous cycle as search effort is further refined
and focused on alternatives that appear to be of high value.10 Moreover, directed exploration and repeated
trials tend to mitigate the “hot stove effect,” in which alternatives generating a single negative reward are
unlikely to be sampled again by a unitary agent (Denrell and March 2001, March, Sproull, and Tamuz
1991).
Combined Effect of the Quantity and Quality of Exploration
The pooling of experience in organizations tends to reduce the quantity of exploration, but increase
its quality. Independently, these two effects function in opposite directions. Decreasing the quantity of
exploration alone reduces performance, while increasing the quality of exploration increases performance.
Over a broad range of sizes, from two through ten agents, the net of this trade-off is an increase in average
performance. However, recall from Figure 1 that increasing organizational size from one to two agents
decreases average performance, suggesting that the marginal effect of a change in quantity or quality of
exploration may not be constant.
< Insert Figure 4 about here >
10 This further reinforces the process that limits the quantity of exploration in larger organizations. Not only do larger
organizations develop more accurate beliefs early in the search process, they also focus on superior alternatives. A more accurate belief about a good alternative is higher in magnitude than an accurate belief about a mediocre alternative.
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To examine this possibility, in Figure 4 we graph the individual contributions of the quantity and
quality of exploration, and their combined effect. In the top panel, the quantity of exploration (right scale)
is measured by the number of times an agent chooses a different alternative in period t than she chose in
period t-1. It is decreasing with organizational size, from 2.98 for a one-agent organization to 1.76 for a
ten-agent organization. The quality of exploration (left scale) is measured by the average difference
between the actual value of the new alternative explored and the alternative abandoned, conditional on
exploring. For a one-agent organization, an exploration event leads to the selection of an alternative that is
on average 0.11 better than the prior alternative, while for a ten-agent organization, the average
improvement is 0.19.
The combined returns to increasing organizational size reflect the product of the quantity and
quality of exploration, which we plot in the lower panel of Figure 4. We observe that an increase in size
from one-agent organizations to two-agent organizations decreases average performance (per the result in
Figure 1) because the benefits of increased quality of exploration are more than offset by the decrease in
quantity of exploration. Further increases with organizational size have combined positive average
performance implications because quality effects dominate.
In sum, increasing organizational size has two competing effects on experiential learning in
organizations – it decreases the quantity of exploration and increases the quality of exploration. When the
latter dominates the former, increasing organizational size can lead to superior average performance while
also decreasing the level of exploration.
Quality of Exploration and the Discovery of the Best Alternative
In the discussion above, we examine the mechanisms that give rise to the observation that larger
organizations, on average, find superior solutions. We have yet to examine why larger organizations are
less likely to discover the very best solution.
The pooling of experience in organizations is a double-edged sword with respect to the efficacy of
experiential learning. On one side of the blade, pooling experience in organizations directs exploration to
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a deep search of a relatively small portion of the set of alternatives. This is beneficial when the problem is
one of uncertainty in evaluation and shallow sampling of an alternative may generate beliefs that do not
accurately reflect the true value of an alternative.
On the other side of the blade, pooling experience reduces the quantity of exploration and decreases
its breadth. This decline in exploration quantity and breadth accounts for the decreased discovery
probability with increasing organizational size. Moreover, the observed reduction in discovery probability
understates the cost associated with pooling experience in larger organizations. Increased size not only
reduces discovery, but also increases the probability that, conditional on discovery, the best alternative
will be abandoned.
In Figure 5, we report the probability that an agent explores a different alternative in period t+1,
conditional on having chosen the best alternative in period t (averaged over the first 10 periods). While
one-agent organizations rarely abandon the best alternative (less than one percent), the probability of
abandoning the best alternative increases with organizational size, reaching nearly nine percent for ten-
agent organizations.
< Insert Figure 5 about here >
Why do organizations discover but then (sometimes) abandon the best alternative? Consider a one-
agent organization. A positive reward on an individual agent’s choice always reinforces current behavior,
reducing the probability of exploring (abandoning the current preferred alternative). Likewise, a negative
reward on an individual agent’s current choice always increases the likelihood of exploring. Both of these
mechanisms are independent of organizational size.
A second dynamic happens in a multi-agent organization, where the rewards received by other
agents also affect subsequent choice by the focal agent. When one agent in a multi-agent organization
gets a positive reward after choosing the best alternative, but by chance, another agent in the organization
gets a positive reward after choosing a mediocre alternative. Holding initial beliefs constant, the agent
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who had previously chosen the best alternative is now less likely to do so again in the subsequent period,
because the mediocre seems equally attractive.11
To examine this mechanism, we disaggregate the abandonment phenomenon by splitting it into
abandonment of the best alternative that occurs after the agent has a negative reward from the best arm
(shaded black), and those abandonments that occur after it has a positive reward (shaded grey) in Figure
5. For a one-agent organization, abandonment can only occur after a chance negative reward on the best
alternative. The impact of abandonment based on a negative signal from the best alternative is reduced
with size (declining height of the black bars) because directed exploration generates deeper sampling of
the explored alternatives, and, thus, more accurate beliefs. In contrast, for larger organizations almost all
abandonment events occur after the agent gets a positive reward on the best alternative. This phenomenon
increases with organizational size.
The discussion above is predicated on the evolution of beliefs in an organization. We wish to
graphically illustrate how beliefs and choices change over time. To do so, we track the temporal pattern of
beliefs of one ten-agent organization. The graphs, in Figure 6, provide two examples that illuminate the
pattern of belief formation and the choices of agents. The vertical axis reflects time periods, while the
horizontal axis reflects the alternative payoffs ranked from best (rank=1) to worst (rank=50). The color
reflects the organization's beliefs about the alternatives. Darker grey reflects alternatives that are believed
to be better (have a higher payoff), while lighter grey reflects those alternatives believed to be worse
(lower payoff).
< Insert Figure 6 about here >
In Panel A, we examine a case in which pooling of experience leads to the exploration of higher-
quality alternatives. At the start of t=0, the agents in the organization have uninformative priors, and as
such, they are indifferent across alternatives. They randomly try alternatives number 3, 9, 13, 21, 26, 30,
11 Even if we introduce a status-quo bias (see Figure 9), i.e. if an agent is indifferent, she sticks to her current choice, the
probability of abandoning decreases but the paradox of size persists.
17
42, 46 (twice), and 48.12 Recall that the alternatives are ranked from best to worst (left to right on the x-
axis). The organization receives a positive reward on alternatives 3, 13, 21, and 26 (on alternative 46, the
organization receives one positive and one negative reward). As a result, the organization updates the
beliefs on these alternatives upwardly as indicated by the slightly darker shading. The organization
receives negative rewards on the rest of the alternatives as indicated by the slightly lighter shading.
Because the agents explore greedily, in t=2 they select only from the set of alternatives 3, 13, 21, and 26,
for which they hold superior beliefs. In this trial, alternatives 3, 13, and 21 each generate two positive
rewards, while alternative 26 generates zero net reward (two positive and two negative). Thus, in the third
period, the organization only selects from alternatives 3, 13, and 21. Alternative 3 generates three positive
rewards, alternative 13 a zero net reward (one positive and one negative), and alternative 21 one net
positive reward (three positive and two negative). Now, the organization is no longer indifferent among
several alternatives – alternative 3 has consistently generated only positive rewards. As a result, it has the
highest estimated value and all ten agents in the organization converge to alternative 3, which continues
to generate net positive rewards in subsequent periods. In this way pooled experience in the organization
guides agents towards better alternatives.
In Panel B, we examine a case in which pooling experience leads to both exploration of higher
quality alternatives and, sometimes, to myopic behavior in which the best alternative is tried and then
rejected. In the first period, all agents are indifferent across all alternatives and choose randomly. Here,
one agent in the organization tries the optimal choice, alternative 1, and garners a positive reward. Other
agents initially choose alternatives 12, 14, 18, 24, 29 and 33, also garnering positive rewards. The
remaining agents try alternatives 25, 42, and 47, which generate negative rewards. In the next period, all
ten agents select among the alternatives that generated positive rewards in the prior period, because they
have identical high beliefs on these alternatives. They randomly select alternatives 12, 14, 18, 24, 29, and
12 Alternative 46 is trialed by two agents, generating one positive and one negative reward; beliefs remain unchanged for their
initial level. The alternative takes the same color as the background grey.
18
33. In doing so, they abandon alternative 1 even though it is the best alternative, and despite the fact no
agent received a negative reward from this alternative. They do so because other agents in the
organization generated positive rewards for other (inferior) alternatives. The ten agents get one positive
reward on alternatives 12 and 14, net zero positive rewards on 18, and two negative rewards on 24, net
one positive reward on alternative 29, and one negative reward on alternative 33. Now both alternatives
12 and 14 are believed to be superior. The process continues, and the agents in the organization converge
on alternative 12. While this alternative is relatively good,13 it is not the best alternative, which was tried
and abandoned.
In sum, pooling experience in organizations guides organizational search towards better alternatives.
Not only does this reduce the rate of exploration, it also increases average performance. However, these
gains are not without costs, as the same process tends to deflect larger organizations away from the very
best alternatives.
Experiment 2: Individual Choice Behavior
In the discussion above, we imposed two important assumptions on the individual behavior of
agents in organizations. First, agents were greedy, always exploiting the alternative currently deemed
best, only exploring if they are indifferent among a set of such alternatives. Second, agents acted on the
basis of beliefs that result from the aggregation of the experience of all agents in the organization. In this
experiment, we relax these assumptions.
Exploration Strategy
Earlier, we found that larger organizations explore less than smaller organizations. Would
increasing the level of exploration increase performance further? To examine this possibility we induce
additional exploration by allowing agents in the organization to pursue an overt exploration strategy
13 With 50 arms drawn from a uniform distribution [0.0,1.0], the alternative ranked first has an expected payoff of 0.98 and the
alternative ranked last has an expected payoff of 0.02. Thus, rank=12 alternative translates to a long-run expected performance of 0.98-11*0.02=0.76.
19
whereby they engage in random exploration (Csaszar and Siggelkow 2010, Ethiraj and Levinthal 2009,
Levinthal 1997). In particular, a simple variation on the greedy strategy (employed in the earlier analysis)
is to allow agents to behave greedily a majority of the time, but with some small probability, ε, select an
alternative at random, independent of beliefs (Sutton and Barto 1998). This exploration strategy is
typically referred to as ε-greedy (and the results in Experiment 1 reflect the greedy strategy of ε=0).
In Figure 7 we examine the implications of the pursuit of a non-greedy exploration strategy for the
paradox of size. We plot the results for ε=0.05 indicating that agents pursue an exploration strategy that
generates random search, independent of beliefs, every twenty periods (on average). In the bottom panel,
we plot the quantity of exploration (measured as change of alternatives). In terms of the average level of
exploration across size, increasing ε from ε=0 (in Experiment 1) to ε=0.05 here, we observe a ten-fold
increase in the quantity of exploration (from an average of 2.5 with ε=0 represented in Figure 2, to
approximately 20 with ε=0.05). As with a greedy strategy, exploration decreases in organizational size.
< Insert Figure 7 about here >
The results in the top panel of Figure 7 indicate that average performance increases with size, while
the probability of discovering the best alternative decreases with size. Thus, the paradox of size persists,
independent of the level of exploration strategy.
Of note in the top panel of Figure 7 is the slope of average performance across organization size.
Increasing organizational size from one agent to ten agents, average performance increases by one
percent, while in Experiment 1 (Figure 1 where ε=0) average performance increases by three percent. It
seems that the average performance benefits of increasing size are less sensitive to size when
organizations are pursuing a more exploratory strategy (higher ε).
This finding is the result of two related mechanisms. First, the performance implications of
exploration depend, in part, on the extent to which an agent chooses an alternative that is better than its
currently preferred alternative. Exploration induces a cost (decreases performance) if the alternative
20
explored is inferior to the current preferred alternative. Indeed, this problem is more severe as the value of
the currently preferred alternative improves (and as such, this cost is more severe in later periods).
Second, the performance benefits of a more exploratory strategy are a function of the likelihood that
having explored a superior alternative, the organization chooses to exploit it. As organizational size
increases, any positive signal from one agent’s newly explored alternative may be swamped by the
rewards and experiences of other agents in the organization. Thus, larger organizations are less likely to
exploit a good alternative found via random exploration. As such, larger organizations capture fewer
benefits of random exploration associated with exploiting a good alternative, but still incur the same level
of costs associated with the more exploratory strategy.
This analysis suggests that the decrease in the marginal effects of organizational size on average
performance is a function of the intelligence of the exploration strategy. A more intelligent exploration
strategy will explore better alternatives, which reduces the costs of negative rewards, and increases the
probability that an explored alternative will subsequently be exploited, which increases the benefits.
We examine organizations pursuing an exploration strategy that weights the alternative chosen for
exploration by the current beliefs regarding the merits of the alternative. We employ the softmax choice
rule attributable to Luce (1959) and employed widely (Camerer and Ho 1999, Gans et al. 2007, Sutton
and Barto 1998, Vermorel and Mohri 2005, Weber et al. 2004).14 The softmax choice rule is tunable (like
ε-greedy), ranging from purely exploitative when τ=0 (equivalent to ε=0) to increasingly exploratory as τ
increases (τ>>0). The results (for a modestly exploratory strategy of τ=0.05) are presented in Figure 8.
Employing this more intelligent exploratory strategy increases average performance and the probability of
discovering the best alternative (relative to the results from Experiment 1), while the paradox of size
persists.
< Insert Figure 8 about here > 14 This strategy formulation takes the form of random choice based on a Gibbs (Boltzmann) distribution. The probability of
selecting alternative i, mi, is defined as 𝑚! = 𝑒 !!/! 𝑒 !!/!!!!! .
21
In the bottom panel, we observe that the average quantity of exploration (across organizations of
different sizes) is approximately 50, reflecting a 24-fold increase from the results in Figure 2. More
interesting is the fact that the quantity of exploration, measured as changing alternatives, is now
increasing, rather than decreasing, with organizational size (in contrast to the results in Experiment 1).
This change in the pattern of exploration across size is a direct function of the intelligence of exploration.
Under random exploration, beliefs don’t affect the exploration process – given the decision to
explore, an alternative is picked at random. With a more intelligent exploration strategy, beliefs affect the
exploration process in two ways. First, the probability of exploring is a function of the heterogeneity in
beliefs across alternatives – when alternatives have similar beliefs, exploration is more likely. Second,
alternatives with high estimated values are more likely to be explored than alternatives with low estimated
values. As a consequence, parallel exploration by more agents (larger organizational size) decreases
heterogeneity in beliefs across alternatives that are viewed as relatively good, engendering an increase in
the quantity of exploration.
Pooling of Experience
The pattern of exploration can also be altered by changing the level of experience pooling. Recall
from Equation 2 that an agent’s belief about the value of an alternative is given by the linear combination
of its individual experience (weighted by 1-ρ), and the aggregated experience of all agents in the
organization (weighted by ρ). In the earlier results, because we set ρ=1, agents update their beliefs
employing the aggregate experience of all agents in the organization. We relax this assumption and
examine ρ=0.80. Our analysis of the implications of this change builds on the intuition from Experiment
1.
In the bottom panel of Figure 9, we examine the quantity of exploration, measured as changing
alternatives. Because they weight their own experience more highly than that of other individuals in the
organization, agents in the organization act somewhat independently, nevertheless the quantity of
exploration decreases slightly. This decrease in exploration occurs because, when ρ is reduced from 1 to
22
0.8, an agent is less likely to be prompted to change alternatives (explore) based on the experience of
other agents in the organization.
This moderate decrease in the pooling of experience leads to a substantial increase in average
performance and a small increase in the probability of discovering the best alternative (top panel in Figure
9). The increase in both average performance and discovery, along with a reduction in the quantity of
exploration (i.e., implying an increase in the efficiency of exploration) occurs because the moderate
reduction in experience pooling reduces the likelihood that an agent will prematurely abandon a relatively
good alternative.15 Nevertheless, the paradox of size is robust to this reduction in experience pooling.
< Insert Figure 9 about here >
In sum, we find that pooling experiences within organizations affects both the quantity of
exploration and the quality of exploration. Although pooling experience suppresses exploration, it also,
guides exploration to more attractive alternatives. This is the paradox of size – an explicit trade-off in
which size confers both an advantage to average performance and a disadvantage to the probability of
discovering the best alternative. This result is insensitive to relaxing the assumptions about the agents’
exploration strategy, or how they weight their own experience relative to that of others in the
organization.
4. Discussion
Experiential learning is a central idea in the management literature – and its general efficacy is a
taken-for-granted element of management thought. Yet, we also know that history is not generous with
experience, and this experience constraint engenders error and myopia in the process of experiential
learning.
15 While the results here examine the implications of a moderate reduction in experience pooling (ρ), in results available from the
authors, we examine the full range from ρ=0 to ρ=1. We find that performance takes an inverse u-shaped relationship with organization size, with the optimal level of experience pooling increasing in organizational size.
23
Naive wisdom would suggest that, if the challenges of experiential learning are driven by a paucity
of prior experience (an experience constraint), allowing agents to draw on a larger pool of experience in
the learning process will enhance the efficacy of experiential learning. In our model, relaxing the
experience constraint is reflected in an increase in organizational size – which allows agents to search in
parallel and augment their own experience with that of others in the organization.
Our main result is the paradox of size for experiential learning. Effort to relax the experience
constraint, by pooling the experience of multiple agents in an organization, is a double-edged sword.
Pooling experience directs organizations towards exploring superior alternatives, but also reduces the
extent of exploration and the likelihood of discovering and exploiting the very best alternative. Thus,
efforts to relax the experience constraint relieve some symptoms of myopia while exacerbating others.
In conceiving of organizations as collections of agents who pool experience in the learning process,
we reduce organizational size to the number of agents within the organization. Our notion of
organizational size is one of many ways of thinking about size.16 Moreover, “of the various structural
variables, size is perhaps the most pervasive in terms of the number of suggested relationships with other
organizational features” (Gooding and Wagner 1985: 462). This research has focused on issues related to
size such as market power, resources, complementary assets, incentives, economies of scale and scope,
organizational structure, and agency problems.
Surprisingly, prior research has not cast attention on the implications of organizational size for the
efficacy of experiential learning. This is not to say that the literature has not looked at the interaction of
agents within an organization. March (1991) focuses squarely on the issue of interactions across agents in
an organization and the implications for exploration and exploitation. Related work adds organizational
structure: Fang, Lee, and Schilling (2010) examine the role of structure in March’s model, and Rivkin and
Siggelkow (2006) examine the benefits of the division of learning into sub-problems. Yet this work does
16 For reviews, see Gooding and Wagner (1985), Damanpour (1992), Haveman (1993), and Camisón-Zornoza et al. (2004).
24
not examine the implications of changing the number of learning agents grouped into an organization. In
contrast, while there is a large body of research on the implications of size for the efficacy of team/group
decision-making (e.g., Haleblian and Finkelstein 1993) and creativity (e.g., Mullen, Johnson, and Salas
1991), this research typically does not study learning.17
By focusing on the number of agents learning in parallel, we find that some of the basic dynamics of
exploration and exploitation change substantially. For example, in a unitary organization, exploration is
often triggered by the agent’s negative experience with the currently preferred alternative (as in the “hot
stove” effect of Denrell and March 2001). In contrast, in a multi-agent organization, exploration is also
triggered by the positive experiences of others. As a consequence, organizational size alters the quantity
of exploration, and its quality, in dynamic and often complex ways.
In sum, organizational learning is a complex process that does not reduce to the sum of the learning
of individuals in the organization. Our study attempts to begin understanding the implications of
organizational size for the efficacy of experiential learning by “organizations.” The issues surrounding
this process remain a fertile and important line of inquiry for organizational theorists.
17 Learning requires feedback from the environment from which the group or team updates beliefs, making potentially different
choices in subsequent periods – this is absent from most research on teams and groups.
25
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FIGURES Figure 1: Trade Off between Average Performance and Optimal Choice
29
Exploration Breadth and DepthAlternatives Explored and Alternatives Explored Only Superficially �
0
5
10
15
20
25
30
1 2 5 10Organizational Size
Uni
que
Alte
rnat
ives
Exp
lore
d
Explored At Least Twice
Explored Only Once
Figure 2
Panel A: Quantity of Exploration Measured as Agent Level Change of Alternative Choice
Panel B: Quantity of Exploration Measured as Breadth and Depth of Set of Alternatives
Extent of ExplorationNumber of Different Alternatives an Agents Explores
(Cumulative)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1 2 5 10
Organizational Size
Cou
nt o
f Cha
nge
Even
ts (p
er a
gent
)New Alternatives
Old Alternatives
30
Figure 3: Quality of Exploration
Quality of ExplorationRank of Alternative Explored
0%
1%
2%
3%
4%
5%
6%
1 6 11 16 21 26 31 36 41 46
Alternative Ranked (Best=1 to Worst=50)
Prob
abili
ty o
f Exp
lorin
g
Size=1
Size=2
Size=5
Size=10
31
Figure 4: Trade-Off between the Quality and Quantity of Exploration
Quality and Quantity of ExplorationExamining the Trade-Off
0.1
0.12
0.14
0.16
0.18
0.2
1 2 5 10
Organizational Size
Perf
orm
ance
Incr
ease
per
Cha
nge
Even
t
1.5
2.0
2.5
3.0
Cou
nt o
f Cha
nge
Even
ts
Quality of Exploration(left scale)
Quantity of Exploration (right scale)
0.32
0.33
0.34
0.35
1 2 5 10
Organizational Size
Com
bine
d
32
Figure 5: Rejecting the Best Alternative
Probability of Rejecting the Best AlternativeTriggers of Rejecting the Best Alternative
(First 20 periods)
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
1 2 5 10Organizational Size
Rej
ectin
g th
e B
est A
ltern
ativ
e
Positive Experience of Others(Own) Negative Experience
33
Figure 6: Illustration of Belief Updating in Ten-Agent Organizations
Per
iod
(A) Better Choices Pooling of Experience
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
1
2
3
4
5
6
0.5
0.55
0.6
0.65
0.7
0.75
0.8
(B) Myopia Through Pooling of Experience
Alternatives (ranked from best to worst)
Per
iod
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
1
2
3
4
5
6
34
Figure 7: Altering the Quantity of Exploration via Random Exploration
35
Figure 8: Altering the Quantity of Exploration via Intelligent Exploration
36
Figure 9: Altering the Extent of Pooling of Experience