Complexity and the Economy
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co-create the world our forecasts are attempting to predict. Without know-
ing how others might determine their forecasts, mine are indeterminate.
There are some cases in economics where it is pretty obvious that everyone
can figure out what to do, where something like the above given scheme does
work. Otherwise the problem is fundamental. The agents in the economy are
in a Magritte world. When our ideas and preferences co-create the world they
are trying to forecast, self-reference renders the problem indeterminate. The
idea that we can separate the subjects of the economy—the agents who form
it—from the object, the economy, is in trouble. Pockets of indeterminism are
present everywhere in the economy. And the High Modern form of economic
determinism fails.
2. Marshal Canrobert’s remark on the charge of the Light Brigade at Balaclava.
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There are two questions we want to ask. One question is: Does it matter?
Maybe all of this happens on a set of measure zero, maybe this difficulty is
confined to some trivial examples in economics. The second question is: If
there is a real difficulty, how should we proceed?
I want to show you an argument taken from the field of capital markets,
from asset pricing theory. And I want to show you this theory lands in the
same trouble as the theory that explained the airlines’ choices.
The only difference is that this is a theory that matters.
In 1991, I was hired by Citibank in Hong Kong as a consultant to develop
sophisticated neural-network models to predict prices in foreign exchange
markets. My initial reaction as an economist was skepticism. I believed the
standard theory, and one of its implications is that there is no way to pre-
dict the financial markets. But soon I discovered that traders in the foreign
exchange market disagreed. They believe they can predict price movements—
at least to the degree they can make money. But first let me quickly outline
the standard theory. The standard efficient markets theory says that all information coming in will be used by speculators and investors and anything in
that information hinting about the future changes of the price will be used.
In other words, by an argument very much like the airline argument, that
I will show you in a moment, each stock’s price is bid to a unique level that
depends on the information currently available. Using past patterns of prices
to forecast future prices (technical trading), in this view, cannot lead to further profits. Otherwise the information inherent in past prices could be used
to make further profits, and by assumption investors have already discounted
all information into current prices. So the standard theory says investors use all information available to form expectations. These will determine stocks’
prices, which on average will uphold these same expectations. Rational expec-
tations again. Thus there is no way to make any money, and the market is
efficient. Traders, on the other hand believe that the market is forecastible.
They believe they can spot patterns in past prices helpful in prediction—they
believe in technical trading. They believe the market is anthropomorphic, that it has a psychology, that it has motives. “The market was nervous yesterday.
But it shrugged off the bad news and went on to quiet down.” Economists are
skeptical of this. I remember hearing one famous economist remark: “If tech-
nical trading could make money, there would be a lot of companies and banks
getting rich.” This puzzled me. Because there are a lot of companies and banks getting rich using many forms of technical trading.
The standard theory is wonderfully successful. It has its own logic. This
logic is complete and has desirable properties like mathematical unique-
ness. But the standard theory must face some unexplained phenomena. It
calls these empirical anomalies. (The basic notion is that there is something
wrong with these phenomena because they don’t fit the theory, rather than
that there is something strange with the theory because it doesn’t explain
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these phenomena.) So if there is a crash in the October 1987 stock market and the market loses 23% of its value, this is called a “correction.” Yet there is no news in October ’87 that calls for this crash. Another anomaly is “bubbles,”
like the famous Dutch tulip bubble where the prices stay high without any
apparent reason. Additionally the volume of market trades is orders of mag-
nitude higher than theory predicts. Several economists (Brock, Lakonishok,
and Le Baron, notably) have shown that technical trading is indeed profitable
statistically. Another puzzle is so-called GARCH behavior, (GARCH means
Generalized Auto Regressive Conditional Heteroscedasticity), which means
there are periods of high volatility in stock prices interspersed randomly with periods of quiescence.
In sum, there are at least half a dozen major statistical anomalies that
are not explained in the standard theory. This has led to a great deal of more modern and ingenious theorizing, some using ad-hoc behavioral observation,
some more sophisticated theorizing.
Let me show you now, as in the airline problem, how the standard theory
breaks down and leads to pockets of indeterminacy. Suppose investors can
put some portion of their money in a single stock that pays a dividend every
time period (a day, a year, say) that investors cannot perfectly predict. The
investors are buying the stock for this dividend plus any capital appreciation (tomorrow’s price), and they face the problem of forecasting these. To make
the standard solution work, we assume homogeneous, identical investors—
clones—who have identical forecasts of the dividend at the end of the period
and identical forecasts about the stock’s price in the future. Forecasts that
are on average unbiased and are therefore rational expectations. A little eco-
nomic reasoning then shows today’s price is equal to the common expecta-
tion of tomorrow’s price plus dividend (suitably discounted and weighted).
This yields a sequence of equations at each time, and with a pinch or two of
conditional-expectation algebra, we can solve these for the expectations of
future prices conditioned on current information, and wind up with today’s
price expressed as a function of expected future dividends. Problem solved.
But it is only solved, providing we assume “identical investors who have iden-
tical forecasts of the dividend at the end of the period and identical forecasts about the stock’s price in the future.” But what if we don’t? What if we assume investors differ?
Let us look at the same exercise assuming our investors agents are not
clones—not homogeneous. Note that the standard theory’s requirement of
identical “information” means not just the same data seen by everyone, but
the same interpretation of the data. But imagine yourself in a real setting,
like the Hong Kong foreign exchange market. Information then consists of
past prices and trading volumes, moves made by the central banks of New
Zealand or the bank of Singapore or the central bank of China, rumors, CNN,
news, what your friends are doing, what they are telling you by telephone,
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what somebody’s aunt thinks is happening to the market. All of these things comprise actual information and it is reasonable to assume that, even if everybody has identical access to all this information, they would treat this information as a Rorschach inkblot and would interpret it differently. Even if we
assume that the people interpreting this information are arbitrarily intelligent (they may be infinitely smart) and they are all perfectly trained in statistics, they will still interpret this data differently because there are many different ways to interpret the same data.
So there is no single expectational model. Each individual investor can still
come up with an individual forecast of the dividend. But tomorrow’s price is
determined by this investor’s and other investors’ individual forecasts of the dividend and of next period’s price. And there is no way for the individual
investor to fathom the forecasts of the others—to figure “what average opin-
ion expects the average opinion to be” (to use Keynes’ words). To do so brings on a logical regress. “I think that they might think, but realizing that I think that, they will think this.” Unless we assume identical investors, once again
our agents are trying to forecast an outcome (future price) that is a function of other agents’ forecasts. As before with the airlines problem there is no deductive closure. Expectations become indeterminate, and the theory fails.
Worse, expectations become unstable. Imagine a few people think that
prices on the market are going to go up. If I believe this and I believe that
others believe this, I will revise my expectations upward. But then I may pick up some negative rumor. I will reassess downward, but realizing that others
may reassess and that they too realize that others, I may further reassess.
Expectations become fugitive, rippling up or down whether trades are made or
not. Predictions become unstable. This is the way price bubbles start. If somehow people expect prices to go up, they will forecast that other people will
forecast that prices will go up. So they will buy in. A bubble starts. People can see prices go up and their expectations of upward motion fulfilled. Therefore
prices may continue to go up. The bubble is self-fulfilling.
Similar logic applies to “floors” and “ceilings.” If, for example, the price is 894, many investors believe that at 900 there is some sort of membrane, a
ceiling, and when the price reaches this ceiling it will bounce back down with a certain probability or it may “break through.” My first reaction to hearing
about floors and ceilings was one of disbelief. Then I started to realize that many investors may have sell orders at 900, simply because it is a round
number. So expectations that the price will fall if it hits 900 are likely to
be fulfilled. Ceilings and floors emerge as partially self-fulfilling prophecies, held in place by their being convenient sell and buy places. We are now a long way from homogeneous rational expectations. Under the realistic assumption that traders may interpret the same information differently, expecta-
tions become indeterminate and unstable. And they may become mutually
self-fulfilling.
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To summarize all this: if we look at a serious branch of economics, the theory of capital markets, we see the same indeterminacy as we saw in the airline problem. Agents need to form expectations of an outcome that is a function
of these expectations. With reasonable heterogeneity of interpretation of
“information,” there is no deductive closure. The formation of expectations is indeterminate.
And yet . . . and yet . . . in every market, in every day, people do form expectations. How do they do this? If they cannot do this deductively, then
should we model their behavior in this area? . . . In 1988, John Holland and
I decided that we would study situations like this by forming an artificial
stock market in the computer and giving the little agents—artificially intel-
ligent computer programs—some means by which they can do the reason-
ing that is required. This was one of the very earliest artificial, agent-based markets. Later we brought in Richard Palmer who is a physicist, Paul Tayler
who is a finance expert and Blake LeBaron who is a financial theorist in
economics. When we started, John Holland, the renowned computer sci-
entist who devised the genetic algorithm, could program only in BASIC.
And I could only program in BASIC. However, Richard was a sophisticated
programmer and we rapidly progressed. We designed our artificial stock
market within the machine (first on a Macintosh then a NeXT) and got it
working.
In this market there was no feed-in from the real stock market. It was an
artificial world going on inside the machine. The artificial agents, the little artificial investors, are all buying and selling a “stock” from one another. The computer could display the stock’s price and dividend, who is buying and selling, who is making money and who is not, who is in the market and who is
out, and so on. The price is formed within the machine by bids and offers.
Another little program—a specialist—sets the price to clear the market, as in
actual stock markets.
The modeling question was: If the agents cannot form their expectations
deductively, how are they going to form them? We decided to follow modern
cognitive theory about how actual human beings behave in such situations.
So we allowed our artificial agents looking at the recent history of the stock’s price to posit multiple, individual hypothetical models for forecasting and test these on a continual, ongoing basis. Each of these hypotheses has a prediction associated with it. At any stage each agent uses the most accurate of its hypotheses, and buys or sells accordingly. Our agents learn in two ways: they
learn which of their forecasting hypotheses are more accurate, and they con-
tinually toss out ones that don’t work and replace these using a genetic algo-
rithm. So they are learning to recognize patterns they are collectively creating, and this in turn collectively creates new patterns in the stock price, which they can form fresh hypotheses about. This kind of behavior—bringing in hypotheses, testing them, and occasionally replacing them—is called induction.
t He end of certain t y in economics [ 179 ]
Our agents use inductive rationality. And this is a much more realistic form of behavior.
Alright then. But now the key question is: Does our market converge to
the rational expectations equilibrium of the academic theory or does it show
some other behavior? What we found to our surprise was that two different
regimes emerged. One, which we called the rational expectations regime, held sway when we started our agents off with sets of predictive hypotheses close
to rational expectations. We could plot the parameters of all the predictive
hypotheses on a chart, and in this case, over time, we could watch them get-
ting gravitationally pulled into the orbit of the rational expectations solution, forming a “fuzz” around this point, as they made occasional predictive forays
away from rational expectations to test different ideas. It is not hard to see why rational expectations prevailed. If the overall mass of predictions is near rational expectations, the price sequence will be near rational expectations,
and non-rational expectations forecasts will be negated. So the academic the-
ory was validated.
But there was a second regime, which we called the complex regime, and it prevailed in a much wider set of circumstances. We found that if we started
our agents w
ith hypotheses a little removed from rational expectations, or
alternatively, if we allowed them to come up with hypotheses at a slightly
faster rate than before, the behavior of the market changed. Subsets of mutu-
ally reinforcing predictions emerged. Imagine we have 100 artificial agents
each using 60 different prediction formulas, so that there is a universe of some 6,000 predictors. Some of these predictors that emerge are mutually reinforcing, some are mutually negating. Suppose many predictors arise that say the
stock price cycles up and down over time. Such predictors would be mutu-
ally negating because they will cause agents to buy in at the bottom of the
cycle, and sell at the top of the cycle, mutually negating profits, and therefore eventually disappearing from the population of predictors. But if a subset of
predictors emerged by chance that said “the price will rise next period if it has risen in the last three periods,” and there were enough of these, they would
cause agents to buy, which on average would cause the price to rise, reinforcing such a subpopulation. Such subsets could then take off, and become embedded in the population of predictors. This was what indeed happened in the
complex regime, endowing it with much richer set of behaviors. Another way
to express this is that our artificial traders had discovered forms of technical trading that worked. They were using, with success, predictions based upon
past price patterns. And so technical trading was emergent in our artificial
stock market. This emergence of subsets of mutually reinforcing elements,
strangely enough, is reminiscent of the origin of life, where the emergence of subpopulations of RNA in correct combinations allows them to become mutually enforcing.
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Another property that emerged in the complex regime was GARCH behavior—periods of high volatility in the stock price followed by periods of qui-
escence—another property unexplained in the standard model. How did
GARCH become an emergent property? What happens in our artificial mar-
ket is that every so often some number of investors discover a new way to
do better in the market. These investors then change their buying and sell-
ing behavior. This causes the market to change, even if slightly, causing other investors in turn to change. Avalanches of change sweep through the market,