2. Why Model?
Knowing reality means constructing systems of transformations that
correspond, more or less adequately, to reality.
—Jean Piaget
In this chapter, we define types of models. Models are often described as
simplifications of the world. They can be, but models can also take the form
of analogies or be fictional worlds mined for ideas and insights. We also
describe the uses of models. In school, we apply models to explain data. In
practice, we can also use models to predict, design, and take actions. We
can use models to explore ideas and possibilities. And we can use models to
communicate ideas and understandings.
The value of models also resides in their ability to reveal conditions
under which results hold. Most of what we know holds only in some cases:
the square of the longest side of a triangle equals the sum of the squares of
the other sides only if the longest side is opposite a right angle. Models
reveal similar conditions for our intuitions. With models we can parse out
when diseases spread, when markets work, when voting leads to good
outcomes, and when crowds make accurate predictions. None of those is a
sure thing.
This chapter consists of two parts. In the first, we describe the three
types of models. In the second, we cover the uses of models: to reason,
explain, design, communicate, act, predict, and explore. These form the
acronym REDCAPE, a notso-subtle reminder that many-model thinking
endows us with superpowers.1

Types of Models
When constructing a model, we take one of three approaches. We can aim
for realism and follow an embodiment approach. Such models include the
important parts and either strip away unnecessary dimensions and attributes
or lump them together. Models of ecological glades, legislatures, and traffic
systems take this approach, as do climate models and models of the brain.
Or we can take an analogy approach and abstract from reality. We can
model crime spreading like a disease and the taking of political positions as
choices on a left-right continuum. The spherical cow is a favorite classroom
example of the analogy approach: to make an estimate of the amount of
leather in a cowhide, we assume a spherical cow. We do so because the
integral tables in the back of calculus textbooks include tan(x) and cos(x)
but not cow(x).2
While the embodiment approach stresses realism, the analogy approach
tries to capture the essence of a process, system, or phenomenon. When a
physicist assumes away friction but otherwise makes realistic assumptions,
she takes the embodiment approach. When an economist represents
competing firms as different species and defines product niches, she makes
an analogy. She does so using a model developed to embody a different
system. No bright line differentiates the embodiment approach from the
analogy approach. Psychological models of learning that assign weights to
alternatives lump together dopamine responses and other factors; they also
invoke the analogy of a scale on which we balance alternatives.
A third approach, the alternative reality approach, purposely does not
represent or capture reality. These models function as analytic and
computational playgrounds in which we can explore possibilities. This
approach allows us to discover general insights that apply outside our
physical and social world. They help us to understand the implications of
real-world constraints: What if energy could be sent safely and efficiently
through the air? And they allow us to run impossible experiments: What if
we tried to evolve a brain? This book contains a few such models, notably
the Game of Life, which consists of a checkerboard whose squares are
classified as either alive (black) or dead (white) that switch between alive

and dead according to fixed rules. Though unrealistic, the model produces
insights into self-organization, complexity, and, some argue, even life itself.
Whether embodying a more complex reality, creating an analogy, or
building a made-up world for exploring ideas, a model must be
communicable and tractable. We should be able to write the model in a
formal language such as mathematics or computer code. When describing a
model, we cannot toss out terms like beliefs or preferences without
providing a formal description. Beliefs can be represented as a probability
distribution over a set of events or priors. Preferences can be represented in
several ways such as a ranking over a set of alternatives or as a
mathematical function.
How tractable something is means how amenable it is to analysis. In the
past, analysis relied on mathematical or logical reasoning. A modeler had to
be able to prove each step in an argument. This constraint led to an aesthetic
that valued stark models. English friar and theologian William of Ockham
(1287–1347) wrote, “Plurality must never be posited without necessity.”
Einstein summed up this principle, known as Ockham’s Razor, as follows:
everything should be made as simple as possible, but not simpler. Today,
when we run up against the constraint of analytic tractability, we can turn to
computation. We can build elaborate models with many moving parts
without concern for analytic tractability. Scientists take this approach when
constructing models of the global climate, the brain, forest fires, and traffic.
They still pay heed to Ockham’s advice, but recognize that “as simple as
possible” might require a lot of moving parts.

The Seven Uses of Models
The academic literature describes dozens of uses of models. Here, we focus
on seven categories of uses: to reason, explain, design, communicate, act,
predict, and explore.

The Uses of Models (REDCAPE)
Reason: To identify conditions and deduce logical implications.
Explain: To provide (testable) explanations for empirical phenomena.
Design: To choose features of institutions, policies, and rules.
Communicate: To relate knowledge and understandings.
Act: To guide policy choices and strategic actions.
Predict: To make numerical and categorical predictions of future and
unknown phenomena.
Explore: To investigate possibilities and hypotheticals.

REDCAPE: Reason
When constructing a model, we identify the most important actors and
entities along with relevant characteristics. We then describe how those
parts interact and aggregate, enabling us to derive what follows from what,
and why. In doing so, we improve our reasoning. While what we can derive
depends upon what we assume, we uncover more than tautologies. Rarely
can we infer the full range of implications of our assumptions from
inspection alone. We need formal logic. Logic also reveals impossibilities
and possibilities. With it, we can derive precise and sometimes unexpected
relationships. We can discover the conditionality of our intuitions.
Arrow’s theorem provides an example of how logic reveals
impossibilities. The model addresses the question of whether individual
preferences aggregate to form a collective preference. This model
represents preferences as ordinal rankings over alternatives. If applied to
five Italian restaurants, denoted by the letters A through E, the model
allows any of the 120 orderings. Arrow required that the collective ordering
be monotonic (if everyone ranks A above B, then so does the collective),
independent of irrelevant alternatives (if no person’s relative rankings of A
and B are unchanged but rankings of other alternatives change, then the
order of A and B in the collective ranking does not change), and
nondictatorial (no single person should decide the collective ordering).
Arrow then proved that if any preferences are allowed, then no collective
ordering necessarily exists.3
Logic can also reveal paradoxes. Using models we can show the
possibility of each subpopulation containing a larger percentage of women
than men but the total population containing a larger percentage of men, a
phenomenon (Simpson’s paradox). This actually happened: 1973, the
University of California, Berkeley, accepted a larger percentage of women
in most departments. Overall, it accepted men at a higher rate. Models also
show that it is possible for two losing bets, when played alternately, to
produce a positive expected return (Parrondo’s paradox). With models, we
can show that it is possible to add a node to a network and reduce the total
length of the edges needed to connect all the nodes.4

We should not dismiss these examples as mathematical novelties. Each
has practical applications: efforts to increase the population of women could
backfire, combinations of losing investments could win, and the total length
of a network of electric lines, pipelines, ethernet lines, or roads could be be
reduced by adding more nodes.
Logic also uncovers mathematical relationships. Given Euclid’s axioms,
a triangle can be uniquely determined by any two angles and a side, or by
any two sides and an angle. With standard assumptions about consumer and
firm behaviors, in markets with a large number of competing firms, price
equals marginal cost. Some results are unexpected: among them the
friendship paradox, which states that in any friendship network, on average,
people’s friends have more friends than they do.
The paradox arises because highly popular people have more friends.
Figure 2.1 shows Zachary’s Karate Network. The person represented by the
dark circle has six friends, denoted by gray circles. His friends have nine
friends on average. These people are represented by white circles. Over the
entire network, twenty-nine of the thirty-four people have friends who are
more popular than they are.5 Later we show that if we make a few more
assumptions, most people’s friends will also be, on average, better-looking,
kinder, richer, and smarter than they are.

Figure 2.1: The Friendship Paradox: A Person’s Friends Have More Friends

Last, and most important of all, logic reveals the conditionality of truths.
A politician may claim that lowering income taxes increases government
revenue by spurring economic growth. A rudimentary model in which
revenue equals the tax rate times the income level proves that revenue
increases only if the percentage growth in income exceeds the percentage
cut in taxes.6 Thus, a 10% cut in income taxes increases revenue only if it
causes income to grow by more than 10%. The politician’s logic only holds
given certain conditions. Models identify those conditions.

The power of conditionality becomes evident when we contrast claims
derived from models with narrative claims, even when the latter have
empirical support. Consider the management proverb first things first: the
idea that when facing multiple tasks, you should do the most important task
first. This rule is also known as big rocks first, because when filling a
bucket with rocks of various sizes, you should put the big rocks in first—if
you put the little rocks in first, the big rocks will not fit.
The rule big rocks first, inferred from expert observation, may be a good
rule most of the time, but it is unconditional. A model-based approach
would make specific assumptions about the task and then derive an optimal
rule. In the bin packing problem, a set of objects of various sizes (or
weights) must be allocated into bins of finite capacity. The objective is to
use as few bins as possible. Imagine, for example, you are packing up your
apartment and putting everything into 2-foot-by-2-foot boxes. Ordering
your possessions by size and putting each object in the first box with
sufficient space (known as the first fit algorithm) turns out to be quite
effective. Big rocks first works well. However, suppose that we consider a
more complex task: allocating space on the International Space Station for
research projects. Each project has a payload weight, a size, and power
requirements along with demands on the astronauts’ time and cognitive
abilities. Each also makes a potential scientific contribution. Even if we
came up with some measure of bigness as a weighted average of these
attributes, big rocks first would prove a poor rule given the dimensionality
of interdependencies. More sophisticated algorithms and possibly market
mechanisms would perform much better.7 Thus, under some conditions, big
rocks first is a good rule. Under other conditions, it is not. With models, we
can trace the boundaries of when we should place the big rocks first and
when we should not.
Critics of formalism claim that models repackage what we already
know, that they pour old wine into shiny mathematical bottles, that we do
not need a model to know that two heads are better than one or that he who
hesitates is lost. We can learn the value of commitment from reading of
Odysseus tying himself to the mast. That criticism fails to recognize that
inferences drawn from models take conditional forms: if condition A holds,
then result B follows (e.g., if you are packing bins and size is the only

constraint, pack the biggest objects first). Lessons drawn from literature or
proverbial advice from great thinkers often provide no conditions. If we try
to lead our lives or manage others by unconditional rules, we find ourselves
lost in a sea of opposite proverbs. Are two heads better than one? Or, do too
many cooks spoil the broth?
Proverb: Two heads are better than one
Opposite: Too many cooks spoil the broth
Proverb: He who hesitates is lost
Opposite: A stitch in time saves nine
Proverb: Tie yourself to the mast
Opposite: Keep your options open
Proverb: The perfect is the enemy of the good
Opposite: Do it well or not at all
Proverb: Actions speak louder than words
Opposite: The pen is mightier than the sword
While opposite proverbs abound, opposite theorems cannot. Within
models, we make assumptions and prove theorems. Two theorems that
disagree on the optimal action, make different predictions, or offer distinct
explanations must make different assumptions.

REDCAPE: Explain
Models provide clear logical explanations for empirical phenomena.
Economic models explain price movements and market shares. Physics
models explain the rate of falling objects and the shape of trajectories.
Biological models explain the distributions of species. Epidemiological
models explain the speed and patterns of disease spread. Geophysical
models explain the size distribution of earthquakes.
Models can explain point values and changes in their values. A model
can explain the current price of pork belly futures and why prices rose over
the past six months. A model can explain why a president appoints a
moderate Supreme Court justice and why a candidate moves to the left or
right. Models also explain shape: models of the diffusion of ideas,
technologies, and diseases produce an S-shaped curve of adoption (or
contagion).
The models we learn in physics, such as Boyle’s Law (a model stating
that the pressure of oxygen times the volume equals a constant (PV = k)),
explain phenomena unreasonably well.8 If we know the volume, we can
estimate the constant k, and then explain or predict pressure P as a function
of V and k. The model owes its accuracy to the fact that gases consist of
simple parts that exist in large numbers and follow fixed rules: any two
oxygen molecules placed in the identical situation follow the same physical
laws. They exist in such large numbers that statistical averaging cancels out
any randomness. Most social phenomena share none of these three
attributes: social actors are heterogeneous, interact in small groups, and do
not follow fixed rules. People also think. Even more problematic, people
respond to social influences, meaning that behavioral variations may not
cancel out. As a result, social phenomena are much less predictable than
physical phenomena.9
The most effective models explain both straightforward outcomes and
puzzling ones. Textbook models of markets can explain why an
unanticipated increase in the demand for a normal good like shoes or potato
chips increases the price in the short run, an intuitive result. These same
models explain why in the long run, demand increases have less of an effect

on price than the marginal cost of producing the good. Increases in demand
can even produce reductions in price that result from increased returns to
scale in production, a more surprising result. The same models can explain
paradoxes such as why diamonds, which have little practical value, have
high prices, but water, a necessity for survival, costs little.
As for the claim that models can explain anything: it is true, they can.
However, a model-based explanation includes formal assumptions and
explicit causal chains. Those assumptions and causal chains can be taken to
data. A model that claims that high levels of criminal behavior can be
explained by low probabilities of being caught can be tested.

REDCAPE: Design
Models aid in design by providing frameworks within which we can
contemplate the implications of choices. Engineers use models to design
supply chains. Computer scientists use models to design web protocols.
Social scientists used models to design institutions.
In July 1993, a group of economists met at Caltech in Pasadena,
California, to design an auction to allocate the electronic spectrum for
cellular phones. In the past, the government had allocated spectrum rights to
large companies for modest fees. A provision within the Omnibus Budget
Reconciliation Act of 1993 allowed for auctioning the spectrum to raise
money.
The radio signal from a tower covers a geographic range. Therefore, the
government sought to sell licenses for specific regions: Western Oklahoma,
Northern California, Massachusetts, Eastern Texas, and so on. This created
a design challenge. The value of any given license for a company depended
on the other licenses that company won. The license for Southern California
would be worth more to a company that also owned the license for Northern
California, for example. Economists refer to these interdependent
valuations as externalities. The externalities had two main sources:
construction and advertising. Holding neighboring licenses meant lower
construction costs and the potential to exploit overlapping media markets.
The externalities created a problem with holding simultaneous auctions.
A company trying to win a bundle of licenses might lose one license to
another bidder and therefore lose the externalities. That company might
then want to back out of its bids on other licenses. Sequential auctions had a
different shortcoming. Bidders would underbid in early auctions to hedge
against losing subsequent licenses.
A successful auction design had to be immune to strategic manipulation,
generate efficient outcomes, and be comprehensible to participants. The
economists used game theory models to analyze whether features could be
exploited by strategic bidders, computer simulation models to compare the
efficiency of various designs, and statistical models to choose parameters
for experiments with real people. The final design, a multiple-round auction

that allowed participants to back out of bids and prohibited sitting out early
periods to mask intentions, proved successful. Over the past thirty years, the
FCC has raised nearly $60 billion using this type of auction.10

REDCAPE: Communicate
By creating a common representation, models improve communication.
Models require formal definitions of the relevant features and their
relationships that we can then communicate with precision. The model F =
MA relates three measurable quantities, force, mass, and acceleration, and
does so in equation form. Each term is expressed in measurable units that
can be communicated without fear of mis-interpretation. By comparison,
the claim that “bigger, faster things generate more power” offers far less
precision. Much can get lost in translation. Does bigger mean weight or
size? Does faster mean velocity or acceleration? Does power mean energy
or force? And how do bigger and faster combine to produce power?
Attempts to formalize the claim could result in any of several forms: power
could be written incorrectly as weight plus velocity (P = W + V), weight
times velocity (P = WV), or weight plus acceleration (P = W + A).
When we formally define an abstract concept like political ideology
using a reproducible methodology, those concepts take on some of the same
features as physical qualities such as mass and acceleration. We can use a
model to say that one politician is more liberal than another based on their
voting records. We can then communicate that claim with precision.
Liberalness is well defined and measurable. Someone can use the same
method to compare other politicians. Of course, voting records may not be
the only measure of liberalness. We might construct a second model that
assigns ideologies based on textual analysis of speeches. With that model as
well, we can communicate with clarity what we mean by more liberal.
Many underappreciate the impact of communication on progress. An
idea that cannot be communicated is like a tree falling in a forest with no
one around to notice it. The remarkable economic growth in the Age of
Enlightenment was due in no small part to the transferability of knowledge,
often in model form. In fact, the evidence suggests that the transferability of
ideas may have contributed more to economic growth during that time than
did levels of education: city-level growth in eighteenth-century France
correlates more strongly with the number of subscriptions to Diderot’s
Encyclopédie than with literacy rates.11

REDCAPE: Act
Francis Bacon wrote, “The great end of life is not knowledge but action.”
Good actions require good models. Governments, corporations, and
nonprofits all use models to guide actions. Whether it be raising or lowering
prices, opening a new location, acquiring a company, offering universal
health care, or funding an after-school program, decision-makers rely on
models. On the most important actions, decision-makers use sophisticated
models. Models are linked to data.
In 2008, as part of the Troubled Asset Relief Program (TARP), the
Federal Reserve gave $182 billion in financial assistance to bail out the
multinational insurance company American International Group (AIG).
According to the US Department of the Treasury, the government chose to
stabilize AIG “because its failure during the financial crisis would have had
a devastating impact on our financial system and the economy.”12 The
purpose of the bailout was not to save AIG but to prop up the entire
financial system. Businesses fail every day, and the government does not
intervene.13
The particular choices made within TARP were based on models. Figure
2.2 shows a version of a network model produced by the International
Monetary Fund. The nodes (circles) represent financial institutions. The
edges (the lines between the circles) represent correlations between the
values of the holdings of those institutions. The color and width of an edge
corresponds to the strength of the correlation between the institutions, with
darker and thicker lines implying greater correlation.14
AIG occupies a central position in the network because it sold insurance
to other firms. AIG held promises to pay other firms if those firms’ assets
lost value. If prices fell, then AIG owed those firms money. By implication,
if AIG failed, so too would the firms connected to AIG. A cascade of
failures might ensue. By stabilizing AIG’s position, the government could
prop up the market values of other firms in the network.15

Figure 2.2: Correlation Graph Between Financial Institutions

Figure 2.2 also helps to explain why the government let Lehman
Brothers fail. Lehman did not occupy a central position in the network. We
cannot rerun history, so we cannot know if the Federal Reserve took the
correct action. We do know that the financial industry did not collapse as a
result of Lehman’s failure. We also know that the government earned a $23
billion profit on its loan to AIG. So, we can infer that the policy choices—
based on many-model thinking—were not a failure.
Models that guide action, such as policy models, often rely on data, but
not all do. Most policy models also use mathematics, though that was not
always true. In the past, policymakers built physical models as well.
Phillips’s hydraulic model of the British economy was used to think through
policy choices in the mid-twentieth century, and a physical model of San
Francisco Bay was instrumental in the decision not to dam the bay for fresh
water.16 The Mississippi River Basin Model Waterways Experiment
Station, which covers nearly 200 acres near Clinton, Mississippi, is a
miniature replica of the river’s basin built on a horizontal scale of 1:100.
The model can test the upstream and downstream effects of building new
dams and reservoirs. The released water follows the laws of physics within
the physical structure. In these physical models, the entities themselves are
analogs of the real world. The models are logical because they follow the
laws of physics.
Our examples so far have considered organizations using models to act.
People can do the same. When taking important actions in our personal

lives, we should also use models. In deciding to purchase a home, take a
new job, return to graduate school, or buy or lease a car, we can use models
to guide our thinking. Those models may be qualitative rather than tied to
data. Even in those cases, the models will oblige us to ask relevant
questions.

REDCAPE: Predict
Models have long been used to predict. Weather forecasters, consultants,
sports handicappers, and central bankers all predict using models. Police
agencies and the intelligence community use models to predict criminal
behavior. Epidemiologists used models to predict the spread of COVID as
well as the effects of social distancing. As data has become more available
and granular, this use of models has grown. Twitter feeds and internet
searches are used to predict consumer trends and social uprisings.
Models can predict individual events as well as general trends. On June
1, 2009, Air France flight AF 477, en route from Rio de Janeiro to Paris,
crashed over the Atlantic. In the days following, rescuers found floating
debris but could not locate the fuselage. By July, the batteries in the plane’s
acoustic beacons were depleted, halting search efforts. A year later, a
second search led by the Woods Hole Oceanographic Institution using US
Navy side-scan sonar vessels and autonomous underwater vehicles also
proved unsuccessful. The French Bureau d’Enquêtes et d’Analyses
eventually turned to models. They applied probabilistic models to ocean
currents and identified a small rectangular region as being most likely to
contain the fuselage. Using the model’s prediction, searchers found the
wreckage within a week.17
In the past, explanation and prediction tended to go hand in hand.
Electrical engineering models that explain voltage patterns can also predict
voltages. Spatial models that explain politicians’ past votes can also predict
future votes. In perhaps the most famous example of applying an
explanatory model to predict, the French mathematician Urbain Le Verrier
applied the Newtonian laws created to explain planetary movements to
evaluate the discrepancies in the orbit of Uranus. He discovered the orbits
to be consistent with the presence of a large planet in the outer region of the
solar system. On September 18, 1846, he sent his prediction to the Berlin
Observatory. Five days later, astronomers located the planet Neptune
exactly where Le Verrier had predicted it would be.
That said, prediction differs from explanation. A model can predict
without explaining. Deep-learning algorithms can predict product sales,

tomorrow’s weather, price trends, and some health outcomes, but they offer
little in the way of explanation. Such models resemble bomb-sniffing dogs.
Even though a dog’s olfactory system can determine whether a package
contains explosives, we should not look to the dog for an explanation of
why the bomb is there, how it works, or how to disarm it.
Note also that other models can explain but have little value as
predictors. Plate tectonics models explain how earthquakes arise but do not
predict when they occur. Dynamical systems models can explain hurricanes,
but they cannot predict with much success when hurricanes will form or
what paths they will take. And while ecology models can explain patterns
of speciation, they cannot predict new types of species.18

REDCAPE: Explore
Last, we use models to explore intuitions and possibilities. These
explorations can be policy-related: What if we make all city buses free?
What if we let students choose which assignments determine their course
grades? What if we put signs on people’s lawns showing their energy
consumption? Each of these hypotheticals can be explored with models. We
can also use models to explore unrealistic environments. What if Lamarck
had been correct and acquired traits could be passed on to our offspring, so
the children of parents with orthodontically corrected teeth would not need
braces? What happens in such a world? Asking that question and exploring
its implications can help to reveal the limits of evolutionary processes.
Abandoning the constraints of reality can spur creativity. For this reason,
advocates of the critical design movement engage in speculative fictions to
generate new ideas.19
Exploration sometimes consists of comparing common assumptions
across domains. To understand network effects, a modeler might begin a
collection of stylized network structures and then ask whether and how
network structure affects cooperation, disease spread, or social uprisings. Or
a modeler might apply a collection of learning models to decisions, twoperson games, and multiperson games. The purpose of these exercises is not
to explain, predict, act, or design. It is to explore and learn.
When we apply a model in practice, we may use it in any of several
ways. The same model may explain, predict, and guide action. As an
example, on August 14, 2003, sagging trees leaning on power lines near
Toledo, Ohio, created a localized power outage that spread when a software
failure prevented an alarm from alerting technicians to redistribute power.
Within a day, more than 50 million people in the northeastern United States
and Canada had lost power. That same year, a storm knocked out a power
line between Italy and Switzerland, leaving 60 million Europeans without
power. Engineers and scientists turned to models that represent the power
grid as a network. The models helped to explain how the failures occurred,
offered predictions of regions where future failures might be likely, and also
guided actions by identifying locations where new lines, transformers, and

power supplies would enhance the robustness of the network. Putting one
model to many uses will be a recurrent theme in this book. As we see next,
one-to-many is a necessary complement to our central theme of applying
many models to make sense of complex phenomena.