Complexity and the Economy
Page 14
θ
2
−
+
+ +
+
−
+ +
)] (A.1)
1
[(
1
1 )
(
t, i, j
et , i, j
pt
dt
Et, i, j pt 1 d
t+1
with θ = 1/75 in the complex regime, and 1/150 in the rational-expectations regime.
This accuracy is used in three places. First, if multiple predictors are active, only the most accurate is used. Second, it is part of the fitness measure for
selecting predictors for recombination in the genetic algorithm. This fitness
measure is defined as
f
=
− 2 −
(A.2)
t, i , j
M et, i, j Cs
where M is a constant; s is specificity, the number of bits that are set (not #) in the predictor’s condition array; and C = 0.005 is a cost levied for specificity.
The value of M is irrelevant, given tournament rankings.
Third, agents use the error variance of their current predictor for the
forecast variance in the demand Eq. (5). (We keep this latter variance fixed
between genetic algorithm implementations, updating it to its current value
in Eq. (A.1) at each invocation.)
Initial Expectations. We initialize agents’ expectations in both regimes by drawing the forecasting parameters from a uniform distribution of values centered upon the h.r.e.e. ones. We select a to be uniform (0.7, 1.2) and b to be uniform (−10, 19.002). The variance of all new predictors is initialized in all cases to the h.r.e.e. value of 4.0.
The genetic algorithm. New predictors are generated by updating each agent’s predictor set at random intervals, on average every 250 periods or 1,000
periods, depending on the regime, asynchronously across agents. The worst
a sse t Pr icing under endogenous exPectat ion s [ 63 ]
performing (least accurate) 20% of the agent’s 100 predictors are dropped, and are replaced by new ones, using uniform crossover and mutation. The
agents are initialized by seeding them with random predictors: condition bits
are set to 0 or 1 with probability 0.1, otherwise to #. This avoids bias in choosing predictors at the outset, and allows intelligent behavior to bootstrap itself up as the artificial agents generate predictive models that perform better. For the bitstrings, these procedures are standard genetic algorithm procedures for mutation and crossover (uniform crossover is used, which chooses a bit at
random from each of the two parents). The forecasting parameter vectors are
mutated by adding random variables to each individual component. And they
are crossed over component-wise, or by taking linear combinations of the two
vectors, or by selecting one or the other complete vector. Each of these pro-
cedures is performed with equal probability. Crossover on a predictor is per-
formed with probability 0.3 or 0.1 in the rational-expectations and complex
regimes, respectively. Individual bits are mutated with probability 0.03. New
predictors are brought into the predictor set with variance set to the average of their parents. If a bit has been changed, the new predictor’s variance is set to the average of that of all predictors. If this new variance is lower than the variance of the current default predictor less an absolute deviation, its variance is set to the median of the predictors’ variance. This procedure gives new predictors a reasonable chance of becoming used.
Market Clearing. The price is adjusted each period by directly solving Eqs.
(5) and (6) for p, which entails passing agents’ forecasting parameters to the clearing equation. In actual markets, of course, the price is adjusted by a specialist who may not have access to agents’ demand functions. But we note that
actual specialists, either from experience or from their “books,” have a keen
feel for the demand function in their markets, and use little inventory to balance day-to-day demand. Alternatively, our market-clearing mechanism simu-
lates an auction in which the specialist declares different prices and agents
continually resubmit bids until a price is reached that clears the market.
Calculation of the Homogeneous Rational-Expectations Equilibrium. We calculate the homogeneous r.e.e. for the case where the market price is a linear function of the dividend p = f d + g which corresponds to the structure of our forecasts. We t
t
can then calculate f and g from the market conditions at equilibrium. A homogenous equilibrium demands that all agents hold 1 share, so that, from Eq. (5)
E (
+
− +
=
2
1
λσ
(A.3)
+ 1
+ 1 )
(
)
t pt
dt
r pt
p + d
From the dividend process Eq. (4) and the linear form for the price, we can
calculate σ 2
= (1 +
2
+
)
and E ( p + d ) as
f
p
d
σe
t
t+1
t+1
E (
+
= 1 +
1 − ρ
+ ρ
+
1
] +
+ 1 )
(
) [(
)
t pt
dt
f
d
dt
g
[ 64 ] Complexity and the Economy
Noting that the right side of Eq. (A.3) is constant, we can then solve for f and g as
ρ
f = 1+ r − ρ
(1 + f )[(1 − ρ) d
− λσ 2 ]
g
e
=
r
Therefore, the expression:
λ(2 + r) 2
σ
E
e
(
+
= 1 +
+
+ 1
+ 1 )
(
)
t pt
dt
r pt
1 + − ρ
(A.4)
r
is the h.r.e.e. forecast we seek.
a sse t Pr icing under endogenous exPectat ion s [ 65 ]
APPENDIX B
The Santa Fe Artificial Stock Market
The Santa Fe Artificial Stock Market has existed since 1989 in various designs (see Palmer et al. (1994) for a description of an earlier version). Since then a number of other artificial markets have appeared: e.g., Beltratti and Margarita (1992), Marengo and Tordjman (1995), and Rieck (1994). The Santa Fe Market
is a computer-based model that can be altered, experimented with, and stud-
ied in a rigorously controlled way. Most of the artificial market’s features are malleable and can be changed to carry out different experiments. Thus, the
artificial market is a framework or template that can be specialized to focus
on particular questions of interest in finance: for example, the effects of different agents having access to different information sets or predictive behav-
iors; or of a transaction tax on trading volume; or of different market-making mechanisms.
The framework allows other classes of utility functions, such as constant
relative risk aversion. It allows a specialist or market maker, with temporary imbalances in fulfilled bids and offers, made up by changes in an inventory
held by the specialist. It allows a n
umber of alternative random processes for
{ d }. And it allows for the evolutionary selection of agents via wealth.
t The market runs on a NeXTStep computational platform, but is currently being ported to the Swarm platform. For availability of code, and for further
information, readers should contact Blake LeBaron or Richard Palmer.
[ 66 ] Complexity and the Economy
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[ 68 ] Complexity and the Economy
CHAPTER 4
Competing Technologies,
Increasing Returns, and Lock-In
by Historical Events
W. BRIAN ARTHUR*
This paper introduced many of the concepts of increasing returns economics: competing technologies, lock-in by small historical events, and the possibility of non-predictable, path-dependent outcomes. It was known beforehand that increasing returns could lead to multiple equilibria, not all of which could be optimal. What wasn’t known was how one particular outcome out of the many possible would be selected; from a static viewpoint, which outcome would be “chosen” was indeterminate. This paper p
ut forward a solution to the selection problem by redefining increasing returns problems as dynamic problems subject to random events. In a given realization under a particular set of small random events one outcome would be selected, in another realization under a different set of small random events a different outcome would be selected. Increasing returns problems should therefore be studied as probabilistic systems that unfold over time, an approach that has since become the accepted one.
The paper first appeared in September 1983 as Working Paper WP-83-90 at the
International Institute for Applied Systems Analysis. But with its emphasis on possible lock-in to non-predicable, inferior outcomes, it proved difficult to publish. The paper underwent six years of submissions before this version finally appeared in the Economic Journal, 99, 116–131, 1989. Since then it has become one of the most heavily cited papers in economics.1 I included it in my previous collected papers volume, Increasing
* I thank Robin Cowan, Paul David, Joseph Farrell, Ward Hanson, Charles
Kindleberger, Richard Nelson, Nathan Rosenberg, Paul Samuelson, Martin Shubik, and Gavin Wright for useful suggestions and criticisms. An earlier version of part of this paper appeared in 1983 as Working Paper 83-90 at the International Institute for Applied Systems Analysis, Laxenburg, Austria. Support from the Centre for Economic Policy Research, Stanford, and from the Guggenheim Foundation is acknowledged.
1. See H. Kim, A. Morse, and L. Zingales. “What Has Mattered to Economics since 1970.” J. Econ. Perspectives 20, 4, 2006.
Returns and Path Dependence in the Economy, University of Michigan Press, 1994, but because it is considered a foundational paper in complexity economics, I’ve included it here too.
This paper explores the dynamics of allocation under increasing returns
in a context where increasing returns arise naturally: agents choosing
between technologies competing for adoption.
Modern, complex technologies often display increasing returns to adop-
tion in that the more they are adopted, the more experience is gained with
them, and the more they are improved.2 When two or more increasing-return
technologies “compete” then, for a “market” of potential adopters, insignifi-