The Self Organized Criticality of Efficient Markets
(previously presented under the title
“On the Possibility of Informationally Efficient Markets”)
David Goldbaum*
Rutgers University
Abstract
In a dynamic model with learning, financial markets are shown to produce self-organized
criticality. The price contains private information that traders learn to extract and employ to
forecast future value. Since the price reflects the beliefs of the traders, the learning process is
self referencing. The sensitivity of the price to small errors in information extraction increases as
the market approaches full market efficiency due to the traders choice to deemphasize private
information as the price becomes a better source of information.
JEL codes: G14, C62, D82
Keywords: Efficient Markets, Learning, Dynamics, Computational Economics
*
Department of Economics, Rutgers University Newark, 360 Dr. Martin Luther King, Jr. Blvd., Newark, NJ 07302,
[email protected]. The author is grateful for comments received at the Conference on Computing
in Economics and Finance in Amsterdam. Financial support was provided by the Rutgers University Research
Council.
1. Introduction
When Grossman and Stiglitz (1980) find that no equilibrium exists able to produce efficient
markets under rational expectations, they create one by introducing a second source of noise that
hampers information extraction. The equilibrium is created at the expense of market efficiency.
Finding the conditions sufficient to create an equilibrium is, of course, the traditional thrust of
economic research. This paper explores learning starting from a point of disequilibrium in a
setting reflecting the Grossman Stiglitz paradox. The result is self-organized criticality. The
point of attraction is market efficiency, but it represents a point of phase transition as the market
clearing condition produces a discrete jump in the model coefficients, should it be reached.
A fully revealing efficient price is impossible in a Rational Expectations Equilibrium.
Grossman and Stiglitz (1980) (hereafter GS) reach this conclusion based on a market in which
fundamental information is costly to obtain. They find that traders who are informed with the
private information have an impact on the price while uninformed traders use the price to
costlessly extract the private information. Subsequent papers reach the same conclusion after
relaxing the assumption of full rationality, for example by replacing rationality with dynamic
learning on the part of the traders. This paper further generalizes GS by examining dynamic
populations in concert with a dynamic learning process. Different evolutionary processes are
examined to explore how the nature of the evolution in the population has an impact on the
asymptotic behavior of the market.
Bray (1982) introduced learning through repeated realizations of the GS single period
model. In each period a new risky asset is introduced to mature at the end of the period. Bray
provides support for the GS assumption of full rationality by finding that learning converges to
2
the rational expectations equilibrium for any fixed population proportion of uninformed to
informed traders.
Hussman (1992) and Timmermann (1996) also examine learning in a financial market
setting. Like Bray, they have a static population of traders, but with a multi-period asset paying
a dividend that follows an AR(1) process.
In order to create a rational expectations equilibrium that includes an equilibrium for the
population process Grossman and Stiglitz found it necessary to introduced an exogenous noise.
The population of “noise” traders disrupts the rational expectations equilibrium in the price,
hampering information extraction. Routledge (1999) examines a dynamic population in a noisy
GS model. With two types of traders, a dynamic population process is introduced by allowing
traders to switch strategies based on imitating more successful traders they encounter. Thus, the
single act of imitation embodies the two dynamic processes of learning and population evolution.
Modeled with a random supply of the risky asset, the exogenous noise ensures the existence of a
stable REE. The process generally converges to this REE. Goldbaum (forthcoming) separates
the learning process from the population process in an examination of a market in which
dividends are driven by a random walk.
The asymptotic properties of the market are examined under two types of population
processes. A population process in which the relative performance measure determines the
innovation of the popularity of the strategies is represented by the Replicator Dynamics process.
Replicator Dynamics (RD) have been used extensively in examining the evolution of strategies
in a repeated game setting, but have also been used to govern population dynamics by Sethi and
Franke (1995) among others. It is a process by which superior strategies attract population from
inferior strategies.
3
A population process in which the relative performance measures determine the level of
popularity of a strategy is represented by the discrete choice dynamics introduced by Brock and
Hommes (1997), the Adaptive Rational Equilibrium Dynamics (ARED). The dynamic process
was used in a financial market setting by Brock and LeBaron (1996) and Brock and Hommes
(1998). The model is based on the assumption of heterogeneity in the population that can
produce diversity in actions.
2. Model
The market setting is borrowed from Goldbaum (forthcoming) with the modification that
the dividend process here is an AR(1) process rather than a random walk. The following
provides a basic description of the environment and solutions based on the stationary dividend
process.
2.1 The market
A large but finite number of agents, indexed by i = 1, ..., N, trade a risky asset and a riskfree bond. The risk-free bond, with a price of one, pays R. The risky asset is purchased at the
market determined price, pt, in period t. In t+1, it pays a stochastic dividend dt+1, and sells for the
market determined price pt+1. The market participants are aware that the stochastic dividend is
based on an AR(1) process with mean d0:
d t  d 0  t ,
with t  t 1   t ,  t ~ IIDN (0,  2 ) .
(1)
In each period, each myopic trader maximizes a negative exponential utility function on
one period ahead wealth conditional on his individual information set (to be developed below).
This produces the demand for the risky asset,
qit ( pt ) = it ( Eit ( z t 1 )  Rp t )
4
(2)
with z t  pt  d t ,  it = 1/  it2 ,  it2 = varit ( z t 1 ) indicating the conditional variance, and γ is the
coefficient of absolute risk aversion.
Assume K strategies for estimating payoffs. In a Walrasian equilibrium, the market price
equates supply and demand for the asset. Supply is fixed to avoid the exogenous introduction of
noise. Without loss of generality, set fixed net supply of the risky asset to zero. Let Nk be the
total number of traders employing information I k . Let q tk be the per capita demand for the
1
risky security among group k traders, q ( pt ) 
Nk
k
t
Nk
q
i 1
it
proportion of the trader population employing strategy k,
( pt ) .
Let n k , 0  n k  1 be the
n
 1 . The price pt clears the
k
market by solving
K
0   n k qtk ( pt ) .
(3)
k 1
2.2 Information
2.2.1 The Fundamental Trader
The estimate of the future payoff is in the nature of Hellwig (1980). Trader i’s private
research indicates the fundamental value of the risky security captured by a signal based on dt+1
that is subject to idiosyncratic error. From the signal, the traders is able to extract a noisy
indicator of the dividend deviation from the long run mean,
sit  d 0  t 1  eit , with eit ~ IIDN (0,  e2 ) .
(4)
A linear projection of ηt+1 onto the information set produces the fundamental investor's
mean squared error minimizing forecast
Ei (t 1 | t , sit ) = (1  )t  si ,t
(5)
where the weight β is known based on the traders' knowledge of the dividend and information
5
processes,
 2
.
  cov( t 1 , sit ) / var( sit )  2
    e2
The "fundamental" price prevails in a market populated exclusively by fundamental
investors. Derive the fundamental price by using the estimate (5) in (2),
ptF  b0  b1F t  b2F t 1   t ,
t 
 1
R N
e
it
.
(6)
Price is a function of the current private and public information.
i
Advancing (6) one period, substituting it into the demand (2) and using the market clearing
condition (3), the price coefficients solve to b0  d 0 /( R  1) , b1F  (1  ) /( R  ) and
b2F   /( R  ) .
For large N, the impact of the idiosyncratic signal noise on the price is negligible. Assume
a sufficiently large N such that the  t term can be dropped.1
Fundamental traders rely on (6) in forming demand. Plug (6) back into (2) to solve for the
average demand of the group of fundamental traders,
qtF =  tF ((
Rd 0
R

((1  )t  t 1 )  Rp t ) .
R 1 R  
(7)
2.2.2 Regression traders
The regression traders model the relationship between the payoff, z t 1  pt 1  d t 1 , and
current market observables. The traders appropriately estimate
z t  c0  c1 pt 1  c2 t 1   t .
(8)
The traders employ least-squares learning to update the parameters of their model. The
1
Formally, νt is o(1).
6
learning process is self-referential with an endogenous state variable, pt-1, included as a
regressor. Let xt' = [1 pt ηt]. The regression traders update the coefficients, ct = [c0t c1t c2t], using
the standard recursive updating algorithm for least-squares learning of Marcet and Sargent
(1989a, 1989b):
Ct  Ct 1  (Qt1 xt 2 ( z t 1  Ct 1 xt 2 ))' / t ,
(9)
Qt  Qt 1  ( xt 1 xt' 1  Qt 1 ) / t ,
given (c0, Q0). The regression traders all rely on the same public information, and thus all
employ the same forecast, E ( z t 1 | xt ) . Per capita demand among regression traders is thus
qtR =  tR (c t xt  Rp t ) .
(10)
2.3 Price Formation
With K = 2, let nt = ntF , and thus (1-nt) = ntR . From (3),
0 = nt qtF  (1  nt )qtR
(11)
Use (7), (10), and (11) to solve for the market clearing price. A consistent price function takes
the form
pt = b0  b1t t  b2t t 1
(12)
with
b0  d 0 /( R  1) ,
b1t  t1 (nt tF
R
R 
(1  )  (1  nt )c2 tR ) ,
b2t  t1 (nt tF
R
R 
) ,
t  nt R tF  (1  nt )( R  c1 ) tR .
7
(13)
3. Analysis and simulation
3.1 Learning under fixed n
As with Bray (1982), consider a learning process with nt = n to determine whether a REE
exists and if so whether the system will converge such that the traders can correctly extract dt+1
from the price.
3.1.1 A fixed point solution
Three equations describe the dynamic processes under a fixed n. Equation (1) is the
exogenous dividend process. Equation (12) is the endogenous price equation. The coefficients
of (12) depend on the beliefs of the regression traders as captured by (8) that evolve according to
(9).
A partial equilibrium solution for the fixed point has, for 0 < n ≤ 0:
n F (1  )
n F   (1  n) R
b1 =
, b2 =
,
( R  )( n F  (1  n) R )
( R  )( n F  (1  n) R )
c1 
c2 

R 
R
R(n F  (1  n) R )
=
,
( R  )b2
nF  (1  n) R
(14)
(15)
n(1  )F
,
( R  c1 ) = 
( R  )( nF  (1  n) R )
 2F  ((1  )( R /( R  )) 2  b22 ) 2 , and
(16)
 2R  b22  2 ,
and for n = 0:
c1 = c2 = 0, b1 =  /(R-  ), and b2 = b3 = b4 = 0.
The solution is partial since the conditional variance terms which are in the solution for b2,
remain a function of b2.
The numerically derived general fixed point solutions for the
endogenous parameters are plotted in Figure 1.
8
[Figure 1 about here]
At the fixed point, the price is fully revealing of the private information dt+1 and the
regression traders correctly extract this information to make an accurate forecast of the time t+1
value. The price itself does not fully reflect the time t+1 value since the fundamental traders
hedge their position to account for the noise in their individual signals. This act affects the price.
As n declines, pt becomes an increasingly accurate reflection of the t+1 value.
Let the measure of profits earned by each information source be the excess return realized
for the risky asset multiplied by the group average demand:
 tk  qtk ( pt 1  d t 1  Rp t ) , k = F, R.
(17)
Based on the fixed point solution,
 2
  
 
 for 0 < n  1 and
2
2



2


R
n
(
1


)
  2 
  F
E ( R )   F  R 
R 
 R     n  (1  n) 

(18)
E ( F )  β/(1-β), E(R) = 0 at n = 0.
(19)
 R 

E ( )    
 R 
F
F2
R
2
 n(1  n)(1  ) 2

F
R 2
 (n  (1  n) )
Thus, for all values of n > 0, the fixed point expected profits are positive for the regression
traders and weakly negative for fundamental traders. The regression trader profits result from
correctly learning to extract ηt+1 from the price. The performance of the fundamental traders
suffers as a result of the idiosyncratic noise associated with their private research. A discrete
jump in profits occurs at n = 0 since knowing ηt+1, even with noise, is useful if the price only
reflects the public time t information.
The discontinuity in the fixed point solution at n = 0 is similar in nature to that found by
GS. With no value of n that produces equal expected profits for the two groups, the setting also
produces the same paradox. GS specify that a long-run equilibrium requires that the two groups
9
of traders must earn equal expected returns in order to have a stable population; otherwise traders
would switch from the inferior to the superior strategy.
3.2 Evolution in the population
Allow for a dynamic population proportion. Two processes are considered, the ARED of
Brock and Hommes (1997) and Replicator Dynamics. Under both dynamic processes, the
traders rely on observed profits to indicate the expected profits for each strategy. Let  tFe and
 tRe be the traders’ performance measures of fundamental and market-based approaches,
respectively. They are updated according to the process
 tke   tke1   t ( tk1   tke1 ) ,
(20)
 0ke  0 , k = F, R, 0   t  1 .
3.2.1 ARED
Based on the randomized discrete choice model of Manski and McFadden (1981), the
action of the individual is partially determined by a random process. Assume a sufficiently large
population such that the nt reflects the probability that an individual chooses to employ the
fundamental analysis. According to the randomized discrete choice model,
nt  Pr( I it  I F ) 
exp( tFe )
.2
Fe
Re
exp( t )  exp( t )
(21)
The  parameter is an inverse function of the variance associated with the unmodeled individual
characteristics affecting the distribution of the agents’ choice. It is commonly thought of as an
“intensity of choice” parameter.
The greater  the less unmodeled aspects play a role in
2
Formally, as a population proportion, nt is restricted to be an element of the set of rational numbers. Allow that
(21) produces a near approximation of nt.
10
determining choice and thus the more sensitive traders are to the model's measured differences in
the expected profit. This increases the likelihood of each trader choosing the analysis technique
with the greater measured expected profit. In the model, the greater the value of , the more
extreme the value of nt associated with a given Fe - Re.
3.2.2 Replicator dynamics
The Replicator Dynamic produces an evolutionary dynamic population in which the
dominant strategy attracts converts from the inferior strategy. The two choice version of the
more general K choice replicator dynamic model found in Branch and McGough (2003) results
in the transition equation
nt  r ( tFe   tRe )(1  nt ) for  tFe   tRe
nt 1  
.
nt  r ( tFe   tRe )nt for  tFe   tRe
(22)
A number of different functional forms for r exist in the literature. The simulations that follow
are based upon
r ( x)  tanh( x / 2)) ,
(23)
a choice that ensures 0 < nt < 1 for bounded  tFe   tRe .3 The parameter δ determines how
responsive the population is to differences in expected profits, thus playing a role similar to ρ in
the ARED.
3.2.3 Comparison of population dynamics
Assume rational expectations by the regression traders so that based on nt they employ the
correct values of ct = c(nt). The analysis suggests that for all values of nt > 0, expected profits
earned by the regression traders exceeds the profits earned by the fundamental traders.
3
This is consistent with how GS envisions evolution in the population. "If the [expected utility of the informed] is
greater than the [expected utility of the uninformed], some individuals switch from being uninformed to being
informed (and conversely). An overall equilibrium requires the two to have the same expected utility." p394.
11
Under the ARED, the difference in the performance determines the level of nt. The
realization of  tFe   tRe = 0 produces nt+1 = 1/2, while  tFe   tRe > 0 produces nt+1 > 1/2 and
 tFe   tRe < 0 produces nt+1 < 1/2. It is possible under ARED for the population process to have
an interior fixed point value of nt = n* at which  tFe   tRe ≠ 0. At the fixed point the magnitude
of  tFe   tRe is just sufficient to sustain the bias in the population towards greater use of the
superior strategy. The value of the fixed point is a function of  as plotted in Figure 2. For
adaptive expectations, μt = 1, a bifurcation into a two-cycle occurs as  is increased. For
  1 / t , this bifurcation does not occur since the long memory serves to stabilize the
performance measures and thus stabilizes nt. That  tFe   tRe < 0 for all values of nt > 0 ensures
that the fixed point n* is between 0 and 1/2.
[Figure 2 about here]
Under the RD, the difference in the performance determines the innovation of nt. The
realization of  tFe   tRe = 0 produces nt+1 = nt ,while  tFe   tRe > 0 produces nt+1 > nt and
 tFe   tRe < 0 produces nt+1 < nt. Thus, the only fixed point to the population process occurs at
 tFe   tRe = 0. Since no value of nt produces equal profits, no fixed point exists under the RD.
The RD is thus true to the GS notion of the population dynamics both in concept and outcome.
Without the assumption of full rationality, the learning and the population processes must
evolve together. In the phase space plotted in Figure 3, the learning process moves the model
vertically while the population process moves the model horizontally. The curve c1 is the fixed
point for the learning process at which the regression traders have correctly estimated the
parameters of eqn. (8) for the given nt value and at which they correctly extract dt+1 from the
price. If nt is sufficiently stable to provide consistent parameter target for the learning process,
12
the market converges towards the curve marked c1.
The regression traders earn positive
expected profits when the market is located between the curves c1 and c1 . Under the RD, given
convergence in the learning process, the population process produces perpetual movement
towards the left axis as nt  0. Under the ARED, given convergence in learning there is a fixed
point to the population process on c1. The location of the fixed point on c1 depends on .
[Figure 3 about here]
An equilibrium in which one group of traders persistently outperforms the other may seem
undesirable from the GS perspective, but it is fully consistent with the ARED dynamics. Each
trader evaluates the two options differently. The center of this distribution is  tFe   tRe . Of the
full population of traders, n* is the proportion who are in the tail with  itFe   itRe > 0. The
persistence in the use of the fundamental analysis despite its lesser performance is simply the
result of heterogeneity in beliefs with the interpretation that there are always some traders who
think that the fundamental analysis will do better than the regression analysis.
3.3 Simulations
Simulations are necessary to determine convergence properties when allowing for the
interaction between the learning and population processes.
Monte Carlo experiments are
conducted based on 100 iterations of each market condition. The model is examined under both
ARED and RD and settings of μt = 1/t and μt = 1.
3.3.1 ARED
Setting μt = 1/t creates an environment favorable to stability and convergence. As the
history length grows, the traders' estimates of the expected profits become less affected by
individual realizations, expanding the conditions that produce stability in nt. Market 1 and
Market 2 highlight the issues that arise. At the extreme with ρ = 0 nt is fixed, nt = n* = 0.5. As
13
has already been established, the learning process converges under a fixed n. In Market 1 ρ = 6
produces n* = 0.40 while Market 2 sets  = 500 for n* = 0.05.
Let ni indicate the mean value of nt for the last 500,000 periods of the 2.5 million period
simulation i, ni 
t1 2.510^ 6
1
 n i,t . The simulations and the sample are long because of the
t1  t 0  1 t 02.010^ 6
substantial level of persistence observed in the endogenous parameters. Table 1 reports statistics
from the Monte Carlo experiments, including n , the mean of the ni estimates. Reported in the
parenthesis is the standard deviation for the n statistic. The table also includes the coefficient
on the mean and the standard deviation for of the coefficient on the time variable in a simple
regression of the magnitude of the deviation from the fixed point, | nit  n* | , on time.
Table 1: Monte Carlo results on convergence of nt


n*
n
n -n*
T coef
Discrete Choice Dynamics
Market 1
Market 2
Market 3
1/t
1/t
1 
6
500
10 
0.3979
0.0500
0.3572 n*
0.3979
0.0514
0.4522 n
1.960E-05
1.423E-03 *
9.496E-02 * n -n*
1.916E-05
2.891E-05
1.358E-04
-3.535E-11 -5.505E-10 * -1.033E-09 * T coef
2.487E-11
5.768E-11
3.465E-10
Replicator Dynamics
Complete Information
Market 4
Market 5
Market 6
1/t
1/t
--0.01
1
--0
0 c 3*
0
0.0028
0.0094 c
0.0033
2.751E-03 *
9.423E-03 * c
3.294E-03 *
6.355E-07
7.170E-05
8.845E-04
-5.741E-10 * -9.748E-10 T coef
-1.136E-09 *
1.138E-12
6.956E-11
3.804E-10
n is the estimate of the average value of nt evaluated in the last 500,000 periods of a 2.5 million
period simulation. T-coef is the coefficient on the regression | nit  n* | a  b * time  eit .
Standard errors reported in parenthesis. In Markets 4 and 5 n* is an attractor, but not a fixed
point.
* indicates a statistically significance.
In both markets, the estimate of n is greater than the fixed point, n* and the regression of
nt against the time trend produces an average coefficient that is negative. In the case of  = 6, n
is insignificantly different from n* and the time coefficient is insignificantly statistically different
from zero. These results indicate that a sufficiently long sample run produces a statistically
insignificant bias in the estimate of n* as nt continues to converge towards n*.
14
In the case of  = 500 the estimate n is statistically different from n*. Of the sample of
100 iterations, none produced an estimate of ni < n* whereas 42 of the 100 simulations produced
an estimate of ni < n* for  = 6. The statistical significance of the n and the time coefficient
suggest that even a long sample run of 2.5 million periods produces a biased estimate of n* as nt
continues to converge towards n*.
The larger  creates a greater challenge to convergence to the fixed point. The cause is the
greater need for accuracy in the regression parameter when the proportion of traders relying on
regression analysis is high. From Figure 3 as nt approaches zero the range between c1 and c1
that produces regression trader profits narrows. Small coefficient errors that were acceptable for
larger values of nt cause substantial pricing error as nt nears zero. The pricing error caused by
incorrect regression coefficients when nt is low leads to profits for the fundamental traders that
lead to an increase in nt. The increasing need for accuracy as nt  0 explains the persistence of
pricing errors even as the parameters converge. This feature is seen in all six of the examined
markets. The interaction of the two processes hampers convergence. Increasing ρ has the dual
affect of increasing the sensitivity of nt to the magnitude of  tFe   tRe realizations and of
lowering n* so that even small coefficient errors in the learning process can lead to substantial
fundamental trader profits, increasing the variance of  tFe   tRe .
Figures 4 and 5 display the typical evolution of endogenous parameters produced in Market
1 and Market 2. The data are from the last first 10,000 periods after dropping the first 500
periods. The horizontal lines represent the fixed point solution for the market.
For Market 3, set µ = 1 for adaptive expectations in the computation of the expected
profits. The population proportion remains responsive to the latest realization of profits making
15
the model less stable. Even settings of  less than the bifurcation value result in a model that is
unable to settle. This can be seen in Figure 6 for which  = 10 suggests a failure of the
parameters to converge. From Table 1, n is substantially biased upwards from n* and while the
convergence coefficient is negative and statistically significant, the magnitude is small given the
magnitude of the bias.
3.3.2 Replicator Dynamics
The parameter δ determines the rate of adjustment in the population towards the superior
choice. This is a pure learning model with no population process since all traders are employing
the same strategy. Two patterns emerge from the simulations. If δ is set sufficiently low, as in
Market 4, then the evolution towards nt = 0 is slow, but also smooth and continuous. The
evolution in the population does not outpace the evolution in the coefficient values. This can be
seen in Figure 7 with δ = 0.01.
If δ is set high, then the evolution in nt proceeds in declining cycles. This results from the
evolution in nt outpacing the evolution in the regression coefficients. When nt becomes too low
for the current parameters, the pricing error allows the fundamental traders to profit, temporarily
reversing the progress in nt. The jagged progress of nt in Market 5 can be seen in Figure 8 with δ
= 1.
For both δ = 0.01 and δ = 1 the system appears to be in the process of convergence towards
nt = 0. The system itself is bounded away from nt = 0 by the chosen r (x ) function of the RD
process since it requires  tFe   tRe   in order to produce nt = 0. The diminishing  tFe   tRe ,
which converges towards 0 as n  0, also diminishes the incentive for the remaining
fundamental investors to switch. Statistics on the convergence of the RD based markets are
included in Table 1.
16
4. Complete Information
4.1 Learning
In this section traders are allowed to employ both the public price and the private signal in
determining a forecast of the value of the risky security. Traders are modeled as learning agents
individually estimating the relationship
zt  c0  c1it pt 1  c2it t 1  c3it sit  it
(24)
based on knowledge of d0. Let cit = [c0it c1it c2it c3it] be the individual regression coefficients,
where xt’ = [1 pt t (t 1  ei ,t ) ]. Individual demand is
qit  (c it xt  Rp t ) / it2 .
The equilibrium price solution that sets
q
it
(25)
= 0 takes the same linear structure as in (12),
i
pt = b0  b1t t  b2t t 1 ,
(26)
b1t  t1   it c2it , b2t  t1   it c3it ,
(27)
but with
i
i
with
t    it ( R  c1it ) .
i
All traders share the same time consistent beliefs at the fixed point. The fixed point for the
learning process produces the regression coefficients
c1  R , c2  c3  0 ,
while the price and the price coefficients are indeterminate,
b1  c2 /( R  c1 ) , b2  c3 /( R  c1 ) .
As the system approaches the fixed point, traders are able to rely heavily on the price for
17
information on the value of dt+1. They place almost no weight on their noisy private signal since
the price is an almost perfect indicator of dt+1.
At the fixed point, the both price and private information are removed from the agents’
demand function and demand for the risky asset becomes zero. The No Trade solution is
attained. The price is removed because the traders believe the market has converged to the point
at which pt fully reveals dt+1, removing the ability of the price to reveal profitable trades. At the
same time, the noisy private information is completely dominated by pt, and thus receives zero
weight. In a GS type paradox, the reality of the fixed point is just the opposite of the traders’
belief.
The price contains no information, both because no one trades based on the dt+1
information and because the price can take any value to clear the market. Unlike GS, the fixed
point is stable because once attained, future prices also fail to reflect fundamental values and thus
the private signal contains no profitable information.
4.2 Simulations
Simulations Market 6 produce market behavior much like that produced by the Replicator
Dynamics. The positive value of the coefficient c3 reported in Table 1 indicates that the traders
choose to rely on the fundamental signal. The convergence of c3 towards zero suggests that the
traders are able to reduce their reliance on the private signal as the price becomes an increasingly
accurate indicator of dt+1. The greater reliance on the price is reflected in the increasing value of
c1 towards its fixed point value of R. The progress towards the fixed point is not perfectly
smooth as over reliance on the price in excess of its contemporaneous accuracy results in an
increase in the pricing errors. This, it turn, drives the traders to temporarily increase their weight
on their private signal. The pattern can be seen clearly in Figure 9. In a result that lends support
to the choice of a bounded r (x ) function for the Replicator Dynamics, it does not appear as
18
though the model is able to attain the discontinuity produced by the fixed point in finite time.
5. Conclusion
The analysis explores the convergence of an asset market towards and Efficient Markets
Equilibrium under three information and population environments. Two of the markets divide
the population into informed traders receiving a private signal and uninformed traders who learn
to rely on the price as a source of information. Replicator Dynamics serves as an evolutionary
population process consistent with dynamics envisioned by Grossman and Stiglitz (1980) in
which the superior strategy continues to attract adherents from the inferior strategy until all
traders have switched or until performance is equal. Simulations perform consistent with the
conclusions of Grossman and Stiglitz (1980). Both the learning process of the uninformed
traders and the population process converge towards the attractor of a market of no informed
traders as the uninformed continue to improve their extraction of the private information from
the price and the superior performance attracts trades away from engaging in fundamental
research. As a greater proportion of the market relies on the price for information and fewer use
the noisy signal, the price converges towards full reflection of the private information, but
without a fixed point in the population process, full market efficiency cannot be achieved.
The discrete choice dynamics of the ARED process serves as an evolutionary population
process that allows for an interior fixed point in the population proportion and the market as a
whole. At the fixed point, the price is perfectly revealing of the private information, but it does
not fully reflect it. The fixed point persists despite the superior performance of the nonfundamental approach due to trader heterogeneity. Differences in opinion or information lead
some of the traders select the seemingly inferior choice. At the fixed point, the price is fully
revealing of the private information, but it does not fully reflect it. As the intensity of choice is
19
increased towards infinity, the fixed point converges full market efficiency. Simulations indicate
that the market does converge to the fixed point, though increasing the intensity of choice also
increases the time required for convergence.
Convergence in both markets relies on the trader’s use of long history of past profits to
compute the performance measures.
Adaptive expectations produce high volatility in the
population proportion thus preventing convergence in the uninformed traders’ learning process.
The third market examined allows all of the traders to employ both the private signal and
the public price to forecast value. Like Grossman and Stiglitz (1980), the attractor represents a
discrete shift in the market as the fundamental information is ignored ensuring that the solution
fails to reflect full market efficiency. Unlike Grossman and Stiglitz (1980), the attractor is a
stable fixed point. In simulation, the market converges towards the fixed point without achieving
it in finite time.
In all variations of the markets considered, the magnitude of the markets’ pricing error
appears unaffected by convergence of the endogenous parameters towards those implying market
efficiency. This is because the more the market depends on the price as an information source
the greater the impact of small coefficient errors on the price. It remains unclear how cleanly the
price can reveal private information as the model continues to converge.
Note: The clustering of pricing errors is similar to that produced in Goldbaum
(forthcoming), indicating that the pattern is not exclusively linked to the unstable region that
exists in Goldbaum (forthcoming), but not here.
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20
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22
Figure 1 Fixed point values of endogenous parameters based on n
23
Figure 2: Based on the ARED dynamics, the fixed point for nt is decreasing as  increase. Under adaptive expectations, the fixed
point becomes unstable, bifurcating into a two cycle.
24
Figure 3: Phase Space in nt and c t1 .
25
Figure 4: Evolution of endogenous parameters. Market 1: ARED dynamics, =6, μt = 1/t.
26
Figure 5: Evolution of endogenous parameters. Market 2: ARED dynamics, =500, μt = 1/t.
27
Figure 6: Evolution of endogenous parameters. Market 3: ARED dynamics, =10, μ = 1.
28
Figure 7: Evolution of endogenous parameters. Market 4: RD dynamics, δ = 0.01, μ = 1/t.
29
Figure 8: Evolution of endogenous parameters. Market 5: RD dynamics, δ = 1, μ = 1/t.
Figure 9: Evolution of endogenous parameters. Market 6: Complete Information.
30
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