14. Path Dependence
No man ever steps in the same river twice, for it’s not the same river and he’s
not the same man.
—Heraclitus
In this chapter, we cover models of path dependence. In any domain in which
people base their behavior on the actions of others, be it international affairs,
art, music, sports, business, religion, technology, or politics, we should expect
some degree of path dependence. A college student’s choice of courses point
her toward some career paths over others. An endorsement of a candidate may
launch a political career. A friendship may lead to other social connections.
The clothes we wear, the books we read, the movies we watch, and the
activities that consume our time all exhibit some degree of path dependence.
Path dependence also exists on grander scales. Common-law rulings
establish and reinforce precedents, influencing future rulings.1 Early
institutional forms impact later institutional choices. The decision in the
United States to provide health insurance through private firms resulted in a
large private health insurance industry, health maintenance organizations, and
a mix of public and private hospitals.2 Institutions also induce behavioral
patterns, such as selfish or cooperative tendencies, that can in turn influence
the efficacy of future institutions.3
In this chapter, we build dynamic urn models that produce sequences of
outcomes that exhibit path dependence. These models extend the Bernoulli
urn model by allowing the distribution of balls within the urn to change as a
function of past outcomes. With these models to structure our thinking, we
then provide a formal definition of path dependence and distinguish pathdependent outcomes from path-dependent equilibria. These formal definitions
differentiate path dependence from tipping points, which are more abrupt
changes in outcomes.
The chapter consists of four parts. The first two cover the Polya process
and the balancing process. The Polya process assumes positive feedbacks and
produces both path-dependent outcomes and equilibria. Many of the canonical
examples of path dependence, including the growth of the QWERTY
typewriter, are based on positive feedbacks, also known as increasing returns.
The balancing process assumes negative feedbacks and produces pathdependent outcomes but not path-dependent equilibria. The third part defines
a measure of path dependence based on entropy. The final section discusses
further applications of the models.
Polya Process
The Polya process captures positive feedbacks using an extension of the
Bernoulli urn model in which we add a ball to the urn that matches the ball
chosen. This process generates outcome path dependence, where outcomes in
each period depend on previous outcomes. It will also be true that the longrun distribution over outcomes—equilibrium path dependence—depends on
outcomes.4 The distinction between these two types of path dependence will
be central to what follows. A process that is equilibrium path dependent must
be outcome path dependent. If outcomes in the long run depend on the path,
then so must outcomes along the way. A process can be outcome path
dependent but not equilibrium path dependent. What happens now could
depend on the past, but the long-run equilibrium might be determined at the
outset.
The Polya Process
An urn contains one white ball and one gray ball. Each period a ball is
drawn randomly and returned to the urn along with an additional ball of
the same color as the one drawn. The color of the ball drawn denotes the
outcome.
The Polya process captures a variety of social and economic phenomena.
A person’s choice of whether to learn to play tennis or racquetball could
depend on the choices of others. A person might be more likely to choose
tennis if more of her friends also choose tennis, as it increases her chances of
finding a game. Similarly, a person’s decision about what type of software to
buy, language to learn, or smart-phone to buy could also depend on earlier
choices by friends. Similar logic also applies to choices by firms over which
technological standards to adopt. They may base their choices on the actions
of others.
The model captures these social influences by changing the distribution of
balls. If gray balls represent people who choose tennis and white balls
represent people who choose racquetball, then as more people choose tennis,
the urn contains more gray balls, causing subsequent people to be more likely
to choose tennis as well. This increasing pull toward the outcome that more
people choose creates path dependence.
image
Figure 14.1: Outcomes Consisting of Two White Balls and One Gray Ball
We can derive two unexpected properties of the Polya process. First, any
sequence with the same number of white outcomes occurs with equal
probability. Second, every distribution of white and gray balls occurs with
equal probability. The second property implies extreme path dependence.
Anything can happen. Everything is equally likely. After 1,000 periods, the
probability that the urn contains 40% white balls equals the probability that it
contains 2% white balls.
To see why, consider all possible sequences of outcomes in the first three
periods. The first period outcome is gray with probability . If so, we add a
gray ball, increasing the probability that the second outcome will be gray to
. If that outcome is also gray, we add a third gray ball, increasing the
probability that the third outcome is gray to
. It follows that the total
probability of three gray balls (or three white balls) equals
times
times
, which equals .
The three sequences in which the first three outcomes consist of two white
balls and one gray ball are shown in figure 14.1. In the top row, the order of
the outcomes is gray, white, and then white. The probability of this sequence
is , as is the probability of other sequences. It follows that the probability of
getting one of the three sequences equals . By symmetry, the probability of
choosing two gray balls and one white ball also equals . Therefore, each set
of outcomes—three white, three gray, two white and one gray, and two gray
and one white—occurs with the same probability of . Moreover, sequences
of two white and one gray also occurs with equal probability. Similar results
can be shown for any number of periods.5
If we extend the Polya process to add balls of additional colors, extensions
of both regularity properties still hold. Any proportion of the colors can arise
and is equally probable. These results create a conundrum for producers of
consumer products. Long-run consumer preferences for some product
attributes may be random. Knowledge that an outcome cannot be predicted
can still inform action. Ford would not want to build 40,000 yellow pickup
trucks and later find that red emerged as the favorite color from a pathdependent process. The potential for unsold inventory in unwanted colors
points to two potential actions. A company could construct its supply chain so
that color choices come last; for example, a clothing company might wait to
dye sweaters until popular colors become clearer. Or a company could choose
to not give people a choice. Henry Ford offered his customers any color
Model T they desired, so long as it was black. Apple did the same when it
rolled out the first iPhone: you could get black, or, for the same price, you
could get black.
The Balancing Process
Our second model, the balancing process, makes the opposite assumption of
the Polya process. After drawing a ball of one color, we add a ball of the
opposite color. If we draw white balls in the first two periods, the urn will
contain three gray balls and only one white ball, resulting in a
probability
of drawing a gray ball. This process produces path-dependent outcomes, in
that the likelihood of an outcome in any period depends on the history of past
outcomes. However, it does not produce path-dependent equilibria. In the long
run, the urn converges to equal proportions of each color ball.6
The Balancing Process
An urn contains one white ball and one gray ball. Each period a ball is
drawn randomly and returned to the urn along with an additional ball of
the color opposite to the color drawn. The color of the ball denotes the
outcome.
The balancing process captures sequences of decisions or actions that
include pressures toward equal allocation. Parents with two children may try
to give equal time to each. Spending an afternoon with one child creates a
desire to spend more time with the other child. The balancing process could
even model organizational efforts to achieve equity. The International
Olympic Committee (IOC) would like every region of the world to host
games. In 2013, the IOC announced that Tokyo had been selected as the host
city for the 2020 Summer Olympic and Paralympic Games. Two European
cities, Istanbul and Madrid, lost. Four years later, the IOC awarded Paris the
2024 games and a North American city, Los Angeles, the 2028 games. Tokyo
won the 2020 games in part on the strength of its proposal and in part because
the Summer Games had not been held in Japan since 1964. Geographic
fairness appears to exert sway. Europe, Asia and Oceania, and the Americas
have hosted the games approximately equal numbers of times in the period
following World War II. Europe has been awarded the games eight times, the
Americas six times, and Asia and Oceania seven times.
Path Dependence or Tipping Point
Path dependence, a gradual effect on outcomes, differs from a tipping point,
an abrupt change in outcomes. The growth of Microsoft provides a good
example of path dependence. Founded in 1975, Microsoft developed
interpreters for the BASIC computer language. In 1979, Microsoft inked a
deal with International Business Machines (IBM) to provide the operating
system for IBM’s personal computer. This deal set Microsoft on a path that
transformed a company with forty employees into one of the most valuable
companies in the world.
The IBM contract contributed to Microsoft’s upward path but did not
guarantee long-term success. At the time, the personal computer market was
small. The internet did not exist, nor did sophisticated word processing,
business software, or video games. Moreover, the success of the personal
computer depended in part on the DOS operating system that Microsoft
developed. As the personal computer market grew, other companies
developed software compatible with DOS, providing more positive feedbacks.
These events—the success of DOS, the growth of the personal computer
market, and the development of software running on the DOS platform—can
be thought of as one color of ball being consistently drawn from the urn. Each
outcome made the next more likely. The computer age may have been
inevitable, but Microsoft’s central role and the growth of the personal
computer represent one of many potential paths.
We can contrast the path dependence of Microsoft’s growth with the
assassination of Archduke Franz Ferdinand on June 28, 1914, which many see
as a tipping point that led to World War I. Six years prior to the assassination,
Austria-Hungary had annexed Bosnia and Herzegovina. Among the Serbians
unhappy with that development was Gavrilo Princip, who shot and killed
Franz Ferdinand and his wife, Sophie. Austria-Hungary blamed Serbia, a
near-inevitable reaction, and then turned to Germany’s Kaiser Wilhelm for
assurance as they prepared for war against Serbia. Tensions escalated. Serbia
had an alliance with Russia, which in turn had alliances with France and the
United Kingdom. By August 2, Germany had declared war on France. On
August 3, Belgium refused to grant Germany free passage into France, and
full-scale war began. This vastly simplified version of events suggests that
given the alliances, the killing of the archduke tipped the world toward war.
We can measure path dependence and tipping points through changes in
the probabilities of the possible outcomes.7 For the Polya process, the initial
probability distribution is uniform over all distributions in the urn. This is the
maximum entropy distribution. As events unfold, the distribution slowly
narrows, indicative of path dependence: what might happen changes as
outcomes occur. The reduction in entropy is gradual. At a tipping point, the
probability distribution changes abruptly. Entropy may fall quickly. Figure
14.2 demonstrates the difference in two processes, each with two possible
outcomes. After an event occurs—the contract for Microsoft or the killing of
the archduke—the probabilities of each change. Subsequent events also
change the probabilities. The process with the tipping point has a sharp kink.
The path-dependent process changes slowly.
image
Figure 14.2: A Tipping Point vs. Path Dependence
Further Applications
In real situations, path dependence may not be as extreme as in the Polya
process. Nevertheless, we can infer from the model that when behavior has a
large social component, almost anything can happen. On one college campus,
most students may wear black winter parkas; on another, they may wear blue
peacoats. Model thinking suggests that differences could be the result of social
influence as much as distinct underlying preferences. That holds in any
context where people choose among a fixed set of options and their choices
depend on the choices made earlier by others. Examples include democratic
elections, which movie to see, and which technology to purchase.
The model can be extended to allow social influence to vary by the
alternative chosen. Vanilla ice cream may have a constant level of feedback.
The more exotic green tea ice cream may generate more variation in feedback:
a friend may not like it and discourage you from trying it, or a friend may love
it and encourage you to order it. It can be shown that less variation in
feedbacks increases the likelihood of choosing an outcome.8 The model can
also be changed so that people differ in their susceptibility to social influence;
some people give more or less weight to the balls added to the run.
In any variation of the model, we can measure (or estimate) the extent of
path dependence and compare it to other versions. If the assumptions we make
in constructing a model of new product introductions shows that outcomes
depend on the early part of the path, then entering, intervening, or subsidizing
early may be a good strategy. The model provides a logic for companies to
rush their products to market or offer steep discounts to generate early
adopters. Other assumptions may show that having the better product may
matter more than entering early and that the better strategy is to focus on
quality. By using models, we can identify the features relevant to a particular
situation—the relative importance of individual preferences and social effects,
the variation in feedbacks, and the relative differences in quality—and deploy
that knowledge to inform strategy and guide data collection.
On a final note, the Polya process shows how positive feedbacks can
produce path-dependent outcomes and equilibria. Path dependence arises in a
far broader set of contexts. Some degree of path dependence (in outcomes if
not equilibria) occurs whenever one action bumps into or interacts with future
actions. That is true when making decisions on large public projects.9 The
decision to build a park or a highway constrains future planning decisions.
The extent of that path dependence generally will depend on the size of the
project. Central Park has had a profound impact on how New York City has
developed. While the Polya process reveals the core idea that interactions
produce path dependence, we need more realistic models for that insight to
guide action.
Value at Risk and Volatility
We can interpret the standard deviation in a time series of data as
volatility. Investments in stocks, real estate, and privately held
businesses all exhibit volatility. Value at risk (VaR) measures the
probability of a loss of a given amount during a specific time period. An
investment with a one-year 5% VaR of $10,000 has a 5% probability of
losing more than $10,000 at the end of one year.10 Banks use VaR
calculations to determine the amount of assets that must be kept on hand
to avoid bankruptcy. For example, to secure an investment with a twoweek 40% VaR of $100,000, an investor may be asked to hold $100,000
in cash.
If an investment follows a simple random walk with an increase or
decrease of size M each period, then it has an N period 2.5% VaR of
.11 Thus, an investment that randomly goes up or down $1,000 each day
has a nine-day 2.5% VaR of $6,000, and a one-year 2.5% VaR of
$38,000. Notice that VaR increases linearly in the size of the steps but
that it increases like the square root of the number of periods. We can
use the formula for VAR to explain why the FDIC only requires that
banks hold around 2% of their assets in cash overnight, but banks
require that consumers put down 20% deposits on houses. The duration
on the overnight loans is one day. Home loans can last for over a decade.
The square root of three thousand and sixty-five days is approximately
sixty.
Here, we have assumed a normal random walk. Analysts calculating
VaR often consider the past empirical distribution of returns. If the
empirical distribution has a longer tail, that is, if it includes more large
events, then VAR would increase as large events are more likely.
Though VaR originated in finance, the idea can be applied broadly. A
nonprofit that operates a volunteer-led Saturday morning soup kitchen
that requires twenty-five volunteers might want to know the likelihood
of lacking sufficient volunteers. If the number of volunteers follows a
simple random walk that increases or decreases by 1 each week, then
using the formula for VaR above, and setting M = 1 and N = 52, we find
a one-year 2.5% VaR of 15, implying the nonprofit has a 2.5% chance of
a volunteer shortage.