CONTRARY to popular belief, the most watched sporting event on Earth is not
the Olympic Games, but football’s World Cup. When the US hosted this global
spectacle in 1994, the final between Brazil and Italy was witnessed live
by more
than 100 000 people in the Rose Bowl in Pasadena, California, and by at least a
billion on television the world over. The two teams struggled through 90
minutes
of stirring but scoreless play, and 30 minutes of equally barren extra time, so
the game had to be decided on penalties. Brazil won the shoot-out 3-2, to
become
world champions.
With two such evenly matched teams, I think many fans must have wondered
along with me how Brazil and Italy would have fared had they played each other
again—or even 10 or 100 times more? Would the Brazilians have emerged as
the superior team? Or would they have battled on a more-or-less even
basis, just
as they did in Pasadena? I guess we will never really know, but one way to at
least imagine how this question might be answered would be to create
simulations
of the two teams. We could then let them fight it out on a theoretical pitch
instead of the grass of the Rose Bowl. To be convincing, such a simulation
would
have to take into account the playing characteristics of every player and the
strategies of the Brazilian and Italian coaches, not to mention the state
of the
pitch, wind speeds, home advantage, crowd noise, and all the other
minutiae that
help to determine the outcome of sporting events.
It may seem like fantasy that such a simulation could mimic real-world
phenomena involving human decision makers—and do so in enough detail to
accurately predict the future. But it is not. Let’s see why.
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El Farol is a bar on Canyon Road in Santa Fe, New Mexico, at which a band
used to play Irish music on Thursday evenings. Having been born and raised in
Belfast, W. Brian Arthur, an economist at the Santa Fe Institute, enjoyed going
to hear this band perform. But he was less than enthusiastic about going if he
thought that too many drunken louts would be there too. Trying to decide
whether
or not to go to El Farol on Thursday nights led Arthur to formulate what he has
termed the “El Farol problem”. It has all the essential characteristics of what
is called a “complex, adaptive system”. Here’s how it goes.
Assume there are 100 people in Santa Fe who, like Arthur, enjoy going to El
Farol to listen to the music. However, none of them wants to go if the bar is
going to be too crowded. Suppose also that all 100 people know how many people
were in the bar over the past several weeks. For example, the number of people
turning up over the past 10 weeks might be: 15, 67, 84, 34, 45, 76, 40, 56, 23
and 35. Each person then independently employs a prediction procedure to
estimate how many people will appear at the bar in the coming week. Typical
predictors might be the same number as last week (35), a mirror image around 50
of last week’s attendance or a rounded-up average of attendances over the past
four weeks (39).
Suppose that each person decides independently to go to the bar only if his
or her prediction is that less than 60 people will attend. Each person has
a set
of several different predictors, but acts on the result of the one that has
proved most accurate over preceding weeks. Because different people have
different predictors in their sets, some will turn up at the bar, while others
will expect it to be too crowded and stay at home. The new attendance figure is
published next day in the newspaper, and from this everyone updates the
accuracies of all their predictors. Things are then ready for another
round.
Fashionable predictions
This process creates what might be termed an “ecology” of predictors, in the
sense that at any given time a subset of all possible predictors are “live”,
which means that they are currently being used by at least one person,
while all
other predictors are “dead”. As time goes on, however, some predictors come to
life and others die. It is of some interest to know if the music fans will
ultimately settle on a few “immortal” predictors, or if good predictors
come and
go like fashions in other areas of life.
The problem faced by each person is to forecast the attendance as accurately
as possible knowing that the actual attendance will be determined by the
forecasts of others. Suppose, for example, that someone is convinced that 87
people will attend. If this person assumes others are equally smart, then it’s
natural for him or her to assume that they too will see 87 as a good forecast.
But then they will all stay at home, destroying the accuracy of that forecast.
So no common forecast can possibly be a good one. In short, deductive logic
fails.
From a scientific point of view, the problem is how to create a theory that
will describe how people decide whether or not to turn up at El Farol on
Thursday evenings, and the dynamics that these decisions create. It didn’t take
Arthur long, however, to discover that it is very difficult even to formulate a
useful model of the decision-making processes in conventional mathematical
terms. So he decided instead to create the “would-be” world of El Farol on his
computer. Into this world he lets loose agents—electronic Irish men and
women—and studies how they act.
As an economist, Arthur’s interest is in self-referential
problems—situations where the predictions made by economic agents act to
create the world they are trying to forecast. In such problems, if you let the
agents use different forecasting models, you quickly run into a morass of
conceptual and technical difficulties. So conventionally, economists
assume that
the agents are homogeneous: that they agree on the same model, and know
that others know that others know that…they are using this model. This view
then asks which forecasting model would be consistent, on average, with the
outcome it creates. But nobody asks how agents come up with this magical model.
Unfortunately, this approach leads to models and predictions about the workings
of economies that don’t match up to reality. Agents are not homogeneous in
their
attitudes to risk and they do not all share the same ideas and models of the
future.
In his virtual El Farol, Arthur has removed some of this homogeneity by
giving his agents different ways to predict the size of the crowd on Thursdays.
His experiments reveal two interesting patterns. First, if the predictors are
not too simplistic, then the number of people who attend fluctuates around an
average level of 60. In fact, whatever threshold level Arthur chooses, that
level seems to emerge as the long-term average of the number of people who go to
the bar. The second pattern is even more intriguing: the number attending week
on week looks as though it is random (see
Diagram above). This is despite the
fact that no inherently random factor dictates how many people actually
turn up.
The number of people going to the bar each week is a purely deterministic
function: it is dictated solely by the individual predictions which, in turn,
are deterministic functions of the number of attendees in past weeks.FIG-20384301.gif
Mathematically, these observations lead to a fairly definite and specific
conjecture: the average number of people who go to the bar converges on the
threshold value as the number of weeks becomes large. It might also be worth
pondering the related conjecture: the time-series of attendance levels is a
deterministically random process. In other words, it is “chaotic”.
Gamblers’ code
Anyone trying to refine or refute these conjectures runs into a huge
obstacle. The simple fact is that there are no formal mathematical structures
within which to even meaningfully phrase the questions that might help us to
study these conjectures. I’ve just described the problem in a few paragraphs of
everyday English, and Arthur created a world within which to explore these
conjectures empirically in a few lines of computer code. But mathematically
speaking, we are stuck. And this is symptomatic of the whole field. We have no
good mathematical framework within which to probe the properties of complex,
adaptive systems.
In this sense we face the same problem as gamblers back in the 17th century
who sought a rational way to divide the stakes in a game of dice that had to be
stopped prematurely by the police— or, perhaps, the gamblers’ wives. The
description and analysis of that very definite, real-world problem led Fermat
and Pascal to create the mathematical structure we now call probability theory.
Complex system theory still awaits its Pascal and Fermat.
So for now, the only way forward appears to be to use computers to create
laboratories for experimenting with these systems. The hope is that a deeper
understanding of how complex, adaptive systems work will suggest the right type
of mathematical structures leading to a decent theory of these processes. Let’s
have a look at one of these models in action.
A few years ago, the US Environmental Protection Agency specified
environmental impact standards for just about any change that anyone might want
to make to anything involving the human habitat. In particular, these standards
apply to proposed modifications to road traffic systems—changes such as
the building of a bridge or freeway. Unfortunately, there is no known way to
assess whether any such proposal really meets the standards laid down by the
EPA.
In 1991, Chris Barrett, a researcher at the Los Alamos National Laboratory,
New Mexico, had the bright idea of building an electronic counterpart of the
nearby city of Albuquerque inside his computer. This electronic world, which he
called TRANSIMS, would come complete with every street, house, car and
traveller
in the real city. Barrett thought that it would be possible to couple this
silicon city to an air-pollution model and calculate directly the environmental
impact of any proposed change to the road network. Happily, a few visionary
thinkers at the EPA and the US Department of Transportation provided the money
needed to turn Barrett’s fantasy into reality.
Albuquerque is a city of about half a million people. It sits on a high
desert plain, with the Sandia Mountains towering over the city to the east and
the Rio Grande river running through the middle. The road network is
distinguished by two freeways that intersect in the centre of the city.
Interstate 25 runs north-south while Interstate 40 crosses it in the east-west
direction.
When travellers are dropped into the network, they begin making their way
from one place to another, and make decisions about such things as how fast to
drive, where to turn and which traffic lane to use. With TRANSIMS we can take a
God-like view of the system, zooming in wherever we wish to look at local
traffic patterns. It also allows us to examine the behaviour of individual
vehicles by freezing them at a particular moment in time. Information is
available about the car’s position, speed, acceleration, and (for the air
pollution model) fuel enrichment state.
Jammed up
As an illustration of the way TRANSIMS can be used, consider the build-up of
morning rush-hour traffic. The Figure (top left, next page) shows the traffic
density at 6.44 am. The lightest traffic density is shown in white, moving up
through green and orange to the densest traffic, which is shown in red and
purple. As the Figure shows, the traffic build-up starts on the north-south
freeway, the main roads in the east, and the northernmost bridge across the Rio
Grande. For the most part, this traffic represents movement from the
residential
suburbs in the south, east and north towards the central business districts.
Even at this early hour the freeway and a couple of the major roads
already have
perilously high traffic densities.
Two hours later, at the height of the rush hour, the system is completely
clogged as shown in the second Figure (top right). Now both freeways, all
bridges, and a number of the main roads are in the high-density orange, red and
purple zones. In addition, many of the secondary streets in residential areas
ringing the centre have moved into the medium-density green category. At this
point, fully developed rush-hour congestion has taken hold, and drivers are
stuck in traffic moving at less than walking pace. Albuquerque’s road traffic
network has failed to fulfil its purpose of delivering people from one part of
town to another.
These results suggest innumerable “what if” games that can be played with
TRANSIMS. Here’s one. A burning political question in Albuquerque recently has
been whether to build a bridge across the Rio Grande River at Montaño
Road, to relieve rush-hour traffic on the northernmost bridge. Opponents
say the
bridge will merely encourage developers to build more low-cost housing near the
bridge, which will generate more traffic to and from the central city. So what
effect would the new bridge across the Rio Grande have on rush-hour traffic? By
making different assumptions about the carrying capacity of the proposed bridge
and the kinds of housing developments that might take place, politicians and
city planners could use TRANSIMS to get valuable information upon which to base
their decision.
Beyond planning
The list of questions that one can envision is endless. These questions are
not new, but with laboratories such as TRANSIMS, we would not have to undertake
expensive building projects or do possibly dangerous tinkering with a system in
order to answer them.
And these systems are much more than just planning tools. They could
give us,
perhaps for the first time, laboratories in which we will be able to do
controlled, repeatable experiments with fully fledged complex, adaptive
systems.
The point of these experiments would be to gain the intuitive understanding
needed to create a formal mathematical structure within which to address
questions of the type I have posed. Only by seeing where interesting traffic
patterns—congestion, accidents or whatever—emerge can we hope to
invent mathematical frameworks that allow us to describe and predict such
patterns. These as yet unknown mathematical structures will then form the basis
for a proper theory of complex, adaptive systems.
Arguing again by analogy with dice-throwing and the creation of probability
theory, what was important to Fermat and Pascal was not the specific problem in
dice-throwing, but the overall issue of how to characterise mathematically a
general set of such problems. The same is true for the El Farol puzzle. Trivial
in itself, it is representative of a much broader class of problems that
permeate almost every nook and cranny of social and behavioural sciences.
Namely, how do the rules employed by individual agents to arrive at their
decisions combine to generate large-scale, global patterns such as traffic
congestion, a thrashing by a rival football team, stock market crashes, or the
disappearance of a species from an ecosystem? Answers to this deep-reaching
class of questions are still beyond us. A good formulation of the El Farol
problem will help to unlock them.
* * *
A little local intelligence goes a long way
The El Farol problem contains all the components that characterise what we
mean by a “complex, adaptive system”. The “fingerprints” of complexity are:
-
A medium number of agents
In the El Farol Problem we have postulated 100 music fans, each of whom acts
independently in deciding whether or not to go to the bar on Thursday evening.
Complex systems involve what one might call a medium-sized numbers of agents.
This sets them apart from the “simple” systems that people have studied in the
past several millennia—such as planetary systems, which tend to involve a
small number of interacting agents, or big systems like containers of ideal
gases, which have so many “agents” that we can use statistical means to study
their behaviour. What “medium” means can vary from case to case, but it usually
means a number too large for intuition and hand-calculation and too small for
statistical methods to give useful answers to the questions we want to
answer. -
Intelligent and adaptive agents
Not only are there a medium number of agents, these agents are intelligent
and adaptive. They make decisions and take actions on the basis of rules like
the predictors used by patrons of El Farol. Moreover, the agents are ready to
modify their rules on the basis of new information. Finally, they can generate
new rules that have never before been used, rather than being hemmed in by
predefined rules for action. -
Local information
No single agent has access to what all the other agents are doing. At most,
an agent gets information from a relatively small subset of the total
population. Each one then processes this “local” information in order to decide
what to do next. In the El Farol problem the information available to each
agent
is as local as information gets, since each person knows only what he or she is
doing. This is an extreme case, however, and in most complex, adaptive systems
the agents are more like drivers on a road network or traders in a speculative
market, each of whom has information about what at least a few other drivers or
traders are up to.