17. Markov Models
History is a cyclic poem written by time upon the memories of man.
—Percy Bysshe Shelley
Markov models capture systems that transition probabilistically between a
finite set of states. A political system might transition between democratic
and dictatorial, a market between volatile and stable regimes, or a person
between happy, contemplative, anxious, and sad. In a Markov model, the
movements between states occur according to fixed probabilities. The
probability that a country transitions from authoritarian to democratic in a
given year might be 5%; the probability that a person transitions from
anxious to tired within an hour may be 20%. If, in addition, the system can
move from any one state to any other through a sequence of transitions and
if there exists no simple cycle of back and forth, a Markov model attains a
unique statistical equilibrium.
In a statistical equilibrium, individual entities continue to move between
states but the probability distribution across the states remains fixed. A
statistical equilibrium in a Markov model of ideology would allow for
people to transition between liberal, conservative, and independent, but the
proportions of people of each ideology would remain unchanged. When
applied to a single entity, a statistical equilibrium means that long-run
probability of the entity being in each state does not change. A person
could be in a statistical equilibrium in which he is happy 60% of the time
and sad 40% of the time. The person’s mental state could change from hour
to hour, but his long-run distribution across those states does not.
The unique statistical equilibrium implies that long-run distributions of
outcomes cannot depend on the initial state or on the path of events. In
other words, initial conditions do not matter, and history does not matter.
Nor can interventions that change the state matter. As time marches on, a
process that satisfies the assumptions inexorably heads to its unique
statistical equilibrium and then stays there. Here again, a model reveals
conditional logic: if the world fits the assumptions of a Markov model,
history does not matter in the long run. The Markov model does not say
that history can never matter. First, the model allows for outcome path
dependence—what happens next will depend on the current state. Second,
it allows for history to model in the long run as well, but for that to be the
case, one of the model’s assumptions must be violated.
Markov models have myriad applications. They can be used to interpret
dynamic phenomena such as democratic transitions, buildups to war, and
the success of drug interventions. They can also be used to rank webpages,
academic journals, and sports teams. They can even be used to adjudicate
authorship of books and articles. In this chapter, we cover each of these
applications. We begin with two examples and then state a general theorem
for a statistical equilibrium to exist. Then, in the third section, we turn to
the applications. In the discussion at the end of the chapter, we reengage the
question of how and when history matters in light of our knowledge of
Markov models.
Two Examples
A Markov model consists of a set of states and transition probabilities
between those states. In our first example, we characterize a person’s mood
on a given day as either mentally engaged or bored. These are formally
represented as the two states of the model. The transition probabilities then
characterize the probability of moving between states. For example, as
shown in the diagram below, we assume that when mentally engaged, a
person has a 90% chance of remaining in that state and a 10% chance of
becoming bored, and that when bored, a person has a 70% chance of
remaining bored and a 30% chance of becoming mentally engaged.
image
Figure 17.1: A Markov Process
Suppose that these transition probabilities hold for 100 students taking a
biology class. Initially half of the students are engaged and half are bored,
as shown in 17.1. Applying the transition probabilities described above, we
expect that on the next day, 5 (10%) of the mentally engaged students will
become bored, and 15 (30%) of the bored students will become mentally
engaged. This results in 60 mentally engaged students and 40 bored
students. The day following that, 6 of the 60 mentally engaged students
should become bored and 12 of the bored students should become mentally
engaged, resulting in 66 mentally engaged students and 34 bored students.
As we continue to apply the transition rules, the process converges to a
statistical equilibrium with 75 mentally engaged students and 25 bored
students. In that equilibrium, students continue to move between the two
states but the number of students in each state does not change.
If instead the process begins with all 100 students mentally engaged, the
next day only 90 students will be mentally engaged. On the following day,
that number falls to 84. If we continue to iterate the process, we find that in
the long run 75 students will be mentally engaged and 25 students will be
bored. The model attains the same statistical equilibrium.
In our second example, we categorize countries into three states: free,
partly free, or not free. Figure 17.2 shows the percentage of countries in
each of the three categories over the thirty-five-year period ending in 2010
based on Freedom House data. The figure shows a clear trend toward
increased democratization. Over the past thirty-five years, the percentage of
free countries has risen by 20%. If that linear trend continues, by 2040 twothirds of all countries would be free, and by 2080 eight out of nine
countries would be free.
A Markov model leads to a different prediction. To make the predictions,
we set the length of a period to five years and loosely calibrate transition
probabilities based on the past data (see table 17.1).1
image
Table 17.1: Markov Transition Probabilities for Democratization
If we initialize the model using the percentages of countries in each
category in 1975, the calibrated model (as we would expect) almost
perfectly matches the 2010 distribution: 48% of countries are free, 31% are
partly free, and 21% are not free. The actual percentages in 2010 were
46%, 30%, and 24%. If we continue to run the model, it predicts that in
2080, 62.5% of countries will be free, 25% partly free, and 12.5% not free.
image
Figure 17.2: Freedom House: % Free, Partly Free, and Not Free
The less rosy prediction of the Markov model stems from the fact that a
linear projection fails to take into account that free countries can transition
to being partly free and also to being not free. As more countries become
free, the number of free countries becoming partly free increases. The
reasons for this are manifold. For one, operating a democracy requires
fiscal authority and institutions capable of implementing policies. To
borrow the phrasing of Flores and Nooruddin, democracy may not easily
take root in some places.2 In those places, we should expect transitions
from free to partly free. The Markov model captures these.
The Perron-Frobenius Theorem
Both of our examples converge to unique statistical equilibria. That was not
by chance. Any Markov model with a finite set of states, fixed transition
probabilities between them, the potential to move from any state to any
other in a series of transitions, and no fixed cycles between states converges
to a unique equilibrium.
The theorem implies that if those four assumptions are satisfied, the initial
state, history, and interventions that change the state cannot change the
long-run equilibrium. If nations move between dictatorships and
democracies according to fixed probabilities, then interventions that impose
or encourage democracies in some countries have no long-term effects. If
fluctuations in dominant political ideologies satisfy the assumptions, then
history cannot influence the long-run distribution over ideologies. And if a
person’s mental state can be represented as a Markov model, then words of
encouragement or supportive gestures have no long-run impact.
Perron-Frobenius Theorem
A Markov process converges to a unique statistical equilibrium provided it
satisfies four conditions:
Finite set of states: S = {1, 2,…, K}.
Fixed transition rule: The probabilities of moving between states are
fixed, for example, the probability of transitioning from state A to state B
equals P(A, B) in every period.
Ergodicity (state accessibility): The system can get from any state to any
other through a series of transitions.
Noncyclic: The system does not produce a deterministic cycle through a
sequence of states.
The takeaway from the theorem should not be that history cannot matter
but that if history does matter, one of the model’s assumptions must be
violated. Two assumptions—the finite number of states and no simple cycle
—almost always hold. Ergodicity can be violated, as when allies go to war
and cannot transition back to an alliance. Such examples notwithstanding,
ergodicity generally holds as well.
That leaves the restriction to fixed transition probabilities between states as
the assumption least likely to hold. Thus, the model says that when history
matters, underlying structural forces must change transition probabilities
(or change the set of states). Consider the challenge of helping families to
escape poverty. The forces that create social inequality have proven
immune to policy interventions.3 In Markov models interventions that
change families’ states—such as special programs for underperforming
students or a one-day food drive—can provide temporary boosts. They
cannot change the long-run equilibrium. In contrast, interventions that
provide resources and training that improve people’s ability to keep jobs,
and therefore change their probabilities of moving from employed to
unemployed, could change long-run outcomes. At a minimum, the model
gives us a terminology—the distinction between states and transition
probabilities—along with a logic to see the value of changing structural
forces rather than the current state.
The Sales-Durability Paradox
The sales-durability paradox states that the prevalence of a product (or an
idea) depends less on its relative sales than on its durability. Markov
models can explain the paradox by letting states represent the proportion of
people who own a type of good. Here, we consider two types of floor
coverings: tile (the durable good) and linoleum (the good with higher
sales). The paradox arises when the good with higher sales, in this case
linoleum, is less prevalent.
In our model, we assume that linoleum outsells tile by a ratio of 3 to 1. To
capture durability differences, we assume that each year 1 in 10 people
replace their linoleum floors, while only 1 in 60 people replace their tile
floors. The resulting Markov model has an equilibrium in which two-thirds
of floors are tile.4
The same logic that underpins the sales-durability paradox explains the
positive relationship between market share and brand loyalty (the
likelihood that someone switches brands). If we write a Markov model,
lower brand loyalty must imply lower market share in equilibrium because
loyalty operates just like durability. This empirical regularity is known as
the double jeopardy law. If you have low brand loyalty, you tend to have
low sales.5
Markov: One-to-Many
Markov models can be applied in a variety of contexts. We can use them to
model genetic drift between the four nucleic acids: adenine (A), cytosine
(C), thymine (T), and guanine (G). If each nucleic acid has a small and
equal probability of becoming one of the other three types, we can write a
transition matrix for drift. We can use them to model health trajectories by
letting states represent health categories such as excellent, moderate, and
poor. The model can evaluate how interventions such as drug protocols,
behavioral changes, and surgeries change the transition probabilities and
equilibrium distributions over outcomes. Interventions that produce better
equilibria—that is, more people in excellent health—merit pursuing.6
Markov models can also be used to identify patterns in international crises
and distinguish between transitions that lead to war and those that produce
peace and reconciliation.7 This application requires us to estimate two
different models, one using cases where crises led to war, and one using
cases in which reconciliation occurred prior to war. If the transition
probabilities in the two models differ significantly, then we can compare
existing patterns, such as bombing, hostage taking, no exchange of
prisoners, and escalated posturing, and see which process better fits the
data.
Using Markov models to discriminate between patterns in this way can
adjudicate authorship controversies. Given an author’s known writings, we
can estimate the probability that one word follows another. In the text of
this book, the word “the” follows the word “for” four times as often as does
the word “example.” We could represent that information as transition
probabilities in a large matrix. The matrix for this book would look
different than the matrix for a book written by someone else. If we were to
construct separate word transition matrices for Melville, Morrison, and
Mao, we would see differences in their transitions between word pairs.8
Using a technique like this, we can use models to aid in assigning
authorship of the Federalist Papers—eighty-five essays written in 1787
and 1788 by Alexander Hamilton, John Jay, and James Madison to
convince New Yorkers to support the United States Constitution. Each
essay was signed with the pen name Publius. Though the authorship of
most of the essays has been settled, several remain a matter of dispute. A
Markov model assigns all of the disputed essays to James Madison.9
Hamilton or Jay could have written those essays, but if either did, he wrote
in the style of Madison. A similar analysis of four discourses and twelve
short unattributed essays discovered by Arlene Saxonhouse showed that at
least three could be attributed to Hobbes with high probability.10 In neither
of these cases does the model necessarily give the correct answer. The
model produces knowledge. We rely on our wisdom to decide how to
weigh this model against other models or intuition.
For our last application, we describe how Markov models were used to
create Google’s original PageRank algorithm. PageRank transformed
search on the World Wide Web.11 The World Wide Web consists of a
network of websites connected by links. To estimate the importance of each
site we could count the links to and from a site. In the network of sites in
figure 17.3, sites B, C, and E each have two links, A has one link, and D
has no links. This method provides a crude estimate of importance, but it
has flaws. Sites B, C, and E all have two links, but site E seems more
important than site B given its position in the network.
PageRank considers each site to be a state in a Markov model. It then
assigns a positive transition probability between two sites if they share a
link. For the moment, we assign equal probability to any link; that is, we
assume that a searcher at A would be equally likely to move to B or E. If
our searcher goes to E, she then alternates between C and E forever.
Alternatively, if she chooses B, she goes to C, and again starts alternating
between C and E. In fact, beginning at any site results in alternation
between C and E. Again, C and E appear to be the most important sites.
Unfortunately, this model does not fit two assumptions of the PerronFrobenius theorem. The system cannot get from any site to any other: there
is no way to get from C to D. In addition, the transition probabilities create
a loop between C and E.
image
Figure 17.3: Linkages Between Sites on the World Wide Web
image
Figure 17.4: Adding in Random Movements Between Sites
To fix both problems, Google added in a small random probability of
moving from any site to any other as shown in figure 17.4. The model now
satisfies all assumptions of the theorem and has a unique statistical
equilibrium. Sites can be ranked by their probabilities in that equilibrium.
A searcher who begins at A will most likely end up at C or E within a few
searches. Once there, she will bounce back and forth between those two
sites until trying a random site. If she goes to A or D, the path back to C
will most likely go through B or E. It follows that site B should have a
higher ranking than A or D, but that all three should be unlikely. In the
unique statistical equilibrium shown in figure 17.5, that happens to be the
case. A, B, and D are all rarely visited, but B is the most visited of the
three.
image
Figure 17.5: Statistical Equilibrium of PageRank Model
PageRank can be thought of as a combination of the random walk model
and a Markov model. If we think of PageRank as an algorithm, we realize
that we can use it to produce rankings of any network. We can let nodes
represent baseball or soccer teams and transition probabilities denote the
percentage of time that one team defeats another.12 If the teams play only
once, the transition probabilities can be assigned based on margin of
victory. The resulting ranking, though not definitive, complements
subjective expert assessments. We can also use PageRank to compute
species’ importance using food web data.13
Summary
Markov models describe dynamic systems that move between states
according to fixed transition probabilities. If we additionally assume that
the process can move between any two states and that the process does not
produce a cycle, then a Markov model attains a unique statistical
equilibrium. In the equilibrium people or entities move between states in
such a way that the probability distribution across states does not change. It
follows that as a process nears that equilibrium, the changes in the
probabilities diminish. Represented as a graph, the slope of the curve
flattens. Recall our earlier discussion of California’s population growth
when we learned linear models. California’s population growth has slowed
because as the population of California has grown, the number of people
leaving California has increased. That result holds true even if the
proportion of Californians leaving does not change.
When applying Markov models to explain phenomena or predict trends, a
modeler’s selection of the states proves critical. The choice of states
determines the transition probabilities between those states. A Markov
model of drug addiction could assume two states: being a user or being
clean. A more elaborate model might distinguish users by frequency of use.
Regardless of the choice over states, if the four assumptions hold (and in
this instance, the key test would be whether transition probabilities remain
fixed), then the system will produce a unique statistical equilibrium. Any
one-time change in the state of a system has at most a temporary effect.
Reducing drug use in equilibrium would require changing transition
probabilities.
Continuing with that same logic, we can infer that a one-day event to spur
interest in education may lack meaningful impact. Volunteers coming into a
community and cleaning up a park may produce few long-term benefits.
Any one-time influx of money, regardless of its size, will dissipate in its
effect unless it changes transition probabilities. In 2010, Mark Zuckerberg
donated $100 million to the Newark, New Jersey, public schools, an
amount that was matched by other donors. That one-shot donation, which
amounted to approximately $6,000 per student, has produced few
measurable effects on test scores.14
Markov models guide action by distinguishing between policies that
change transition probabilities, which can have long-term effects, and those
change the state and can only have short-term effects. If transition
probabilities cannot be changed, then we must reset the state on a regular
schedule to change outcomes. A person’s work life may create transition
probabilities that lead toward competitive, selfish, and stressful mental
states. Daily exercise, meditation, or religious practice may move a person
into a more grateful, compassionate, and relaxed state to start each day.
Weekends perform a similar function, as do regular date nights for married
couples. Both temporarily move a person’s state away from the
equilibrium.
Not every dynamic system satisfies the assumptions of the Markov model.
For those that do not, history, interventions, and events can have long-term
consequences. In the Polya process, outcomes change the long-run
equilibrium. A large intervention or shock to a system can change transition
probabilities or even the set of states. Major technological improvements
such as the steam engine, electricity, the telegraph, or the internet change
the set of possible states of the economy. Political and social movements
that define new rights or create new policies also change the set of states.
We might therefore think of history as a sequence of Markov models rather
than as a single process moving toward an inevitable equilibrium.
Markov Decision Models
A Markov decision model amends a Markov model by including actions.
An action generates a reward, which is conditional on the state and also
affects the transition probabilities between states. Given the effect of an
action on transition probabilities, the optimal action does not always
maximize the immediate reward.
As an example, we consider students who have a choice between two
actions: surfing the internet and studying. Surfing the internet always
produces the same payoff. Studying produces a high payoff when the
student is engaged and a low payoff when the student is bored. To add in
the effect of actions on transition probabilities, we assume that a bored
student who surfs the internet remains in the bored state and that an
engaged student who surfs the internet becomes bored half of the time. A
student who studies has a 75% chance of being mentally engaged in the
next period regardless of her present state:
Actions: surf the internet (U), study (S)
States: bored (B), mentally engaged (E)
Reward Structure
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Transition Mapping
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A solution to a Markov decision problem consists of an action to be taken
in each state. Myopic best response behavior, which we encountered earlier,
selects the reward-maximizing action in each state. In the example, that
corresponds to surfing when bored and studying when mentally engaged.
The myopic solution results in the student falling into the bored state. Once
that occurs, she chooses to surf the internet and remains bored in all
remaining periods. Thus, her long-run average reward equals 6. The
always-study solution puts her in the engaged state 75% of the time and in
the bored state 25% of the time, producing an average return of 7. This
solution produces a higher average payoff because she is more often in the
mentally engaged state.
As seen in this example, framing a choice as a Markov decision problem
can produce better actions. By taking into account the consequences of an
action on our state, we choose more wisely: Sleeping late produces a higher
immediate reward than getting up early and exercising. Buying an
expensive coffee produces a higher reward than making our own cup. Yet,
in the long run we may be happier exercising and saving money on coffee.
Did we need a model for this? We might have instead turned to Proverbs
21:17: “Whoever loves pleasure will be a poor man; he who loves wine and
oil will not be rich.” That may be true, but had we turned to Ecclesiastes
8:15: “And I commend joy, for man has nothing better under the sun but to
eat and drink and be joyful,” we would have found an opposite proverb. By
embedding our choices within a Markov decision framework, we can use
logic to decide which commonsensical advice makes sense in a given
setting.