i
t+k
t
And homogeneity allows us to drop the agent subscript and set the weights to
1/ N. It is then a standard exercise (see Diba and Grossman, 1988) to show that by setting up the arbitrage, Eq. (1), for future times t + k, taking expectations across it, and substituting backward repeatedly for E[ p | I ], agents can iterat+k
t
tively solve for the current price as6
∞
p
k
= ∑ β [
| ]
t
E dt + k It
(2)
k = 1
If the dividend expectations are unbiased, dividend forecasts will be
upheld on average by the market and, therefore, the price sequence will be in
rational-expectations equilibrium. Thus, the price fluctuates as the information
{ I } fluctuates over time, and it reflects “correct” or “fundamental” value, so that t
speculative profits are not consistently available. Of course, rational-expectations models in the literature are typically more elaborate than this. But the point so far is that if we are willing to adopt the above assumptions—which depend heavily on homogeneity—asset pricing becomes deductively determinate, in the
sense that agents can, in principle at least, logically derive the current price.
Assume now, more realistically, that traders are intelligent but heteroge-
neous—each may differ from the others. Now, the available shared information
6. The second, constant-exponential-growth solution is normally ruled out by an appropriate transversality condition.
[ 44 ] Complexity and the Economy
I consists of past prices, past dividends, trading volumes, economic indicat
tors, rumors, news, and the like. These are merely qualitative information plus data sequences, and there may be many different, perfectly defensible statistical ways, based on different assumptions and different error criteria, to use them to predict future dividends (see Arthur, 1992, 1995; Kurz, 1994, 1995). Thus,
there is no objectively laid down, expectational model that differing agents
can coordinate upon, and so there is no objective means for one agent to know
other agents’ expectations of future dividends. This is sufficient to bring indeterminacy to the asset price in Eq. (1). But worse, the heterogeneous price
expectations E [ p | I ] are also indeterminate. For suppose agent i attempts i
t+1
t
rationally to deduce this expectation, he may take expectations across the
market clearing Eq. (1) for time t + 1:
E [
= β
(3)
1
∑
+
+ | ]
{ , +1( [ +2| ]
[ +2| ])}|
i pt
It
Ei
wj t
Ej dt
It
Ej pt
It
It
j
This requires that agent i, in forming his expectation of price, take into account his expectations of others’ expectations of dividends and price (and relative market weights) two periods hence. To eliminate, in like manner, the
price expectation E [ p | I ] requires a further iteration. But this leads agents j
t+2
t
into taking into account their expectations of others’ expectations of others’
expectations of future dividends and prices at period t + 3—literally, as in Keynes’ (1936) phrase, taking into account “what average opinion expects the
average opinion to be.”
Now, under homogeneity these expectations of others’ expectations col-
lapsed into single, shared, objectively determined expectations. Under het-
erogeneity, however, not only is there no objective means by which others’
dividend expectations can be known, but attempts to eliminate the other
unknowns, the price expectations, merely lead to the repeated iteration of
subjective expectations of subjective expectations (or, equivalently, subjec-
tive priors on others’ subjective priors)—an infinite regress in subjectivity.
Further, this regress may lead to instability. If investor i believes that others believe future prices will increase, he may revise his expectations to expect
upward-moving prices. If he believes that others believe a reversion to lower
values is likely, he may revise his expectations to expect a reversion. We can, therefore, easily imagine swings and swift transitions in investors’ beliefs,
based on little more than ephemera—hints and perceived hints of others’
beliefs about others’ beliefs.
Under heterogeneity then, deductive logic leads to expectations that are not
determinable. Notice the argument here depends in no way on agents having
limits to their reasoning powers. It merely says that given differences in agent expectations, there is no logical means by which to arrive at expectations. And a sse t Pr icing under endogenous exPectat ion s [ 45 ]
so, perfect rationality in the market cannot be well defined. Infinitely intelligent agents cannot form expectations in a determinate way.
Forming Expectations by Inductive Reasoning
If heterogeneous agents cannot deduce their expectations, how then do they
form expectations? They may observe market data, they may contemplate the
nature of the market and of their fellow investors. They may derive expecta-
tional models by sophisticated, subjective reasoning. But in the end all such
models will be—can only be—hypotheses. There is no objective way to verify
them, except by observing their performance in practice. Thus, agents, in facing the problem of choosing appropriate predictive models, face the same problem
that statisticians face when choosing appropriate predictive models given a
specific data set, but no objective means by which to choose a functional form.
(Of course, the situation here is made more difficult by the fact that the expectational models investors choose affect the price sequence, so that our statisticians’ very choices of model affect their data and so their choices of model).
In what follows then, we assume that each agent acts as a “market statis-
tician.”7 Each continually creates multiple “market hypotheses”—subjective,
expectational models—of what moves the market price and dividend. And
each simultaneously tests several such models. Some of these will perform
well in predicting market movements. These will gain the agent’s confidence
and be retained and acted upon in buying and selling decisions. Others will
perform badly. They will be dropped. Still others will be generated from time
to time and tested for accuracy in the market. As it becomes clear which expectational models predict well, and as poorly predicting ones are replaced by
better ones, the agent learns and adapts. This type of behavior—coming up
with appropriate hypothetical models to act upon, strengthening confidence
in those that are validated, and discarding those that are not—is called inductive reasoning.8 It makes excellent sense where problems are ill defined. It is, in microscale, the scientific method. Agents who act by using inductive reasoning we will call inductively rational.9
7. The phrase is Tom Sargent’s (1993). Sargent argues similarly, within a macroeconomic context, that to form expectations agents need to act as market statisticians.
8. For earlier versions of induction applied to asset pricing and to decision problems, see Arthur (1992, 1994, 1995) (e.g., the El
Farol problem), and Sargent (1993). For accounts of inductive reasoning in the psychological and adaptation literature, see Holland et al., 1986), Rumelhart (1977) and Schank and Abelson (1977).
9. In the sense that they use available market data to learn—and switch among—
appropriate expectational models. Perfect inductive rationality, of course, is indeterminate. Learning agents can be arbitrarily intelligent, but without knowing others’
learning methods cannot tell a priori that their learning methods are maximally efficient. They can only discover the efficacy of their methods by testing them against data.
[ 46 ] Complexity and the Economy
Each inductively rational agent generates multiple expectational models that “compete” for use within his or her mind, and survive or are changed on
the basis of their predictive ability. The agents’ hypotheses and expectations adapt to the current pattern of prices and dividends; and the pattern of prices changes to reflect the current hypotheses and expectations of the agents. We
see immediately that the market possesses a psychology. We define this as the collection of market hypotheses, or expectational models or mental beliefs,
that are being acted upon at a given time.
If there were some attractor inherent in the price-and-expectation-formation
process, this market psychology might converge to a stable unchanging set of
heterogeneous (or homogeneous) beliefs. Such a set would be statistically validated, and would, therefore, constitute a rational expectations equilibrium.
We investigate whether the market converges to such an equilibrium below.
3. A MARKET WITH INDUCED EXPECTATIONS
The Model
We now set up a simple model of an asset market along the lines of Bray (1982) or Grossman and Stiglitz (1980). The model will be neoclassical in structure,
but will depart from standard models by assuming heterogeneous agents who
form their expectations inductively by the process outlined above.
Consider a market in which N heterogeneous agents decide on their desired asset composition between a risky stock paying a stochastic dividend, and a
risk-free bond. These agents formulate their expectations separately, but are
identical in other respects. They possess a constant absolute risk aversion
(CARA) utility function, U(c) = − exp (− λc). They communicate neither their expectations nor their buying or selling intentions to each other. Time is discrete and is indexed by t; the horizon is indefinite. The risk-free bond is in infinite supply and pays a constant interest rate r. The stock is issued in N units, and pays a dividend, d , which follows a given exogenous stochastic process t
{ d } not known to the agents.
t The dividend process, thus far, is arbitrary. In the experiments we carry out below, we specialize it to an AR (1) process
d = + ρ (
(4)
− −
1
) +
t
d
dt
d
e
t
where e is gaussian, i.i.d., and has zero mean, and variance σ 2.
t
e
Each agent attempts, at each period, to optimize his allocation between
the risk-free asset and the stock. Assume for the moment that agent i’s predictions at time t of the next period’s price and dividend are normally distributed with (conditional) mean and variance, E [ p + d ], and σ 2 + . (We say i, t
t + 1
t + 1
t, i, p d
a sse t Pr icing under endogenous exPectat ion s [ 47 ]
presently how such expectations are arrived at.) It is well known that under CARA utility and gaussian distributions for forecasts, agent i’s demand, x , i, t
for holding shares of the risky asset is given by:
E ( + +
1
1
+ −
1
( + ))
i, t pt
dt
p
r
x =
(5)
i, t
2
λσ
i, t, p + d
where p is the price of the risky asset at t, and λ is the degree of relative risk t
aversion.
Total demand must equal the number of shares issued:
N
∑ x =
i, t
N
(6)
i = 1
which closes the model and determines the clearing price p—the current market price—in Eq. (5) above.
It is useful to be clear on timing in the market. At the start of time period t, the current dividend d is posted, and observed by all agents. Agents then use t
this information and general information on the state of the market (which
includes the historical dividend sequence { . . . d , d , d } and price sequence t−2
t−1
t
{ . . . p , p }) to form their expectations of the next period’s price and dividend t−2
t−1
E ( p + d ). They then calculate their desired holdings and pass their demand i, t
t+1
t+1
parameters to the specialist who declares a price p that clears the market. At t
the start of the next period the new dividend d is revealed, and the accuracies t+1
of the predictors active at time t are updated. The sequence repeats.
Modeling the Formation of Expectations
At this point we have a simple, neoclassical, two-asset market. We now break
from tradition by allowing our agents to form their expectations individu-
ally and inductively. One obvious way to do this would be to posit a set of
individual-agent expectational models which share the same functional form,
and whose parameters are updated differently by each agent (by least squares,
say) over time, starting from different priors. We reject this in favor of a different approach that better reflects the process of induction outlined in Section 2
above. We assume each agent, at any time, possesses a multiplicity of linear
forecasting models—hypotheses about the direction of the market, or “the-
ories of the market”—and uses those that are both best suited to the cur-
rent state of the market and have recently proved most reliable. Agents then
learn, not by updating parameters, but by discovering which of their hypoth-
eses “prove out” best, and by developing new ones from time to time, via the
genetic algorithm. This structure will offer several desirable properties: It will
[ 48 ] Complexity and the Economy
avoid biases introduced by a fixed, shared functional form. It will allow the individuality of expectations to emerge over time (rather than be built in only to a priori beliefs). And it will better mirror actual cognitive reasoning, in which different agents might well “cognize” different patterns and arrive at
different forecasts from the same market data.
In the expectational part of the model, at each period, the time series of
current and past prices and dividends are summarized by an array or informa-
tion set of J market descriptors. And agents’ subjective expectational models are represented by sets of predictors. Each predictor is a condition/forecast rule (similar to a Holland classifier which is a condition/action rule) that contains both a market condition that may at times be fulfilled by the current state of the market and a forecasting formula for next period’s price and dividend.
Each agent possesses M such individual predictors—holds M hypotheses of the market in mind simultaneously—and uses the most accurate of those that
are active (matched by the current state of the market). In this way, each agent has
the ability to “recognize” different sets of states of the market, and bring to bear appropriate forecasts, given these market patterns.
It may clarify matters to show briefly how we implement this expectational
system on the computer. (Further details are in Appendix A.) Suppose we
summarize the state of the market by J = 13 bits. The fifth bit might correspond to “the price has risen the last 3 periods,” and the tenth bit to “the price is larger than 16 times dividend divided by r,” with 1 signaling the occurrence of the described state, and 0 its absence or nonoccurrence. Now, the condition part of all predictors corresponds to these market descriptors, and thus, also consists of a 13-bit array, each position of which is filled with a 0, or 1, or #
(“don’t care”). A condition array matches or “recognizes” the current market
state if all its 0’s and l’s match the corresponding bits for the market state with the #’s matching either a 1 or a 0. Thus, the condition (####1########) “recognizes” market states in which the price has risen in the last 3 periods. The condition (#########0###) recognizes states where the current price is not
larger than 16 times dividend divided by r. The forecasting part of each predictor is an array of parameters that triggers a corresponding forecasting expression. In our experiments, all forecasts use a linear combination of price and
dividend, E(p + d ) = a ( p + d ) + b. Each predictor then stores specific val-t+1
t + 1
t
t
ues of a and b. Therefore, the full predictor (####1 ####0###) / (0.96, 0) can be interpreted as “if the price has risen in the last 3 periods, and if the price is not larger than 16 times dividend divided by r, then forecast next period’s price plus dividend as 96% of this period’s.” This predictor would recognize—would
be activated by—the market state (0110100100011) but would not respond
to the state (0110111011001).
Predictors that can recognize many states of the market have few 1’s and
0’s. Those more particularized have more 1’s and 0’s. In practice, we include
for each agent a default predictor consisting of all #’s. The genetic algorithm a sse t Pr icing under endogenous exPectat ion s [ 49 ]
Complexity and the Economy Page 11