CONWUTATIONALECONONUCSYSTEMS
Advances in Computational Economics
VOLUMES
SERIES EDITORS
Hans Amman, University ofAmsterdam, Amsterdam, The Netherlands
Anna Nagumey, University of Massachusetts at Amherst, USA
EDITORIAL BOARD
Anantha K. Duraiappah, European University Institute
John Geweke, University of Minnesota
Manfred Gilli, University ofGeneva
Kenneth L. Judd, Stanford University
David Kendriek, University ofTexas at Austin
Daniel MeFadden, University ofCali/ornia at Berkeley
Ellen MeGrattan, Duke University
Reinhard Neck, Universität Bielefeld
Adrian R. Pagan, Australian National University
John Rust, University ofWisconsin
Bere Rustem, University ofLondon
HaI R. Varian, University of Michigan
The titles published in this series are listed at the end of this volume.
Computational
Economic Systems
Models, Methods & Econometrics
edited by
Manfred Gilli
University ofGeneva, Switzerland
SPRINGERSCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 9789048146550
ISBN 9789401587433 (eBook)
DOI 10.1007/9789401587433
Printed on acidfree paper
All Rilzhts Reserved
© 1996 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1996
Softcover reprint ofthe hardcover 1st edition 1996
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
TABLE
OF CONTENTS
Preface
List of Contributors
vii
IX
Part One:
Modeling Computational Economic Systems
Evolutionary Games and Genetic Algorithms
Christopher R. Birchenhall
The Emergency and Evolution of SelfOrganized Coalitions
Arthur de Vany
Smart Systems and Simple Agents: Industry Pricing by Parallel and
Genetic Strategies
Raymond Board and Peter A. Tinsley
A Distributed Parallel Genetic Algorithm: An Application from
Economic Dynamics
Paul M. Beaumont and Pa trick T. Bradshaw
MultiItem Stochastic Inventory Models with Constraints and their
Parallel Computation
Yuan Wang and Yuafan Deng
Building and Solving Multicriteria Models Involving Logical Conditions
R.L. V. Pinto and Berc Rustem
3
25
51
81
103
123
Part Two:
Computational Methods in Econometrics
Wavelets in Econometrics: An Application to Outlier Testing
Seth A. Greenblatt
Linear Versus Nonlinear Information Processing: A Look at Neutral
Networks
Emilio Barucci, Giampiero M. Gallo and Leonardo Landi
Solving Triangular Seemingly Unrelated Regression Equations Models
on Massively Parallel Systems
Erricos J. Kontoghiorghes and E. Dinenis
Maximum Likelihood Estimation of Nonlinear Rational Expectations
Models by Orthogonal Polynomial Projection Methods
Mario J. Miranda
139
161
191
203
vi
Structural Breaks and GAReR Modelling
Stephen G. Hall and Martin Sola
Block Distributed Methods for Solving Multicountry Econometric
Models
ion Faust and Ralph Tryon
Efficient Solution of Linear Equations Arising in a Nonlinear
Economic Model
Are Magnus Bruaset
Solving Pathdependent Rational Expectations Models Using the
FairTaylor Method
F.J. Henk Don and Rudy M.G. van Stratum
Author Index
Subject Index
217
229
243
257
271
277
Preface
The approach to many problems in economic analysis has drastically
changed with the development and dissemination of new and more efficient computational techniques. The present volume constitutes a selection
of papers presented at the IFACMeeting on 'Computational Methods in
Economics and Finance', organized by Hans Amman and Ben; Rustem in
June 1994 in Amsterdam. The selected contributions illustrate the use of
new computatiollal methods and computing teehniques, such as parallel
proeessing, to solve eeonomic problems.
Part I of the volume is dedieated to modelling eomputational economic
systems. The eontributions present eomputational methods to investigate
the evolution of the behaviour of eeonomic agents. Christopher Birehenhall
diseusses the applieation of various forms of genetic algorithms to simple games and eompares the out comes with theory and experimental evidence. Arthur de Vany analyzes sequential and Edgeworth recontracting
eore formation with imprecise information, using the Boltzmann maehine as
a model to study the evolution of eoalitions. Board and Tinsley foeus on the
organization of interindustry communications for adjustment of producer
prices using parallel Jacobi iterations and genetic algorithms. Beaumont
and Bradshaw also explore the use of genetic algorithms in computational
economics. In particular, they present a distributed parallel genetic algorithm whieh is quite effective at solving complex optimization problems as it
avoids converging on suboptimal solutions. Wang and Deng develop a multiitem, multiperiod, doublerandom inventory model and demonstrate the
application of distributed memory MIMD parallel computers to solve such
models. Following the prineiples of behavioral realism, Pinto and Rustem
present a multiple eriteria decision support system for the construction and
solution of multicriteria models involving logical conditions.
Papers in Part Ir concern new eomputational approaches to econometrie problems. Seth Greenblatt gives an application of wavelets to outlier
detection. Barucci, Gallo and Landi compare the information processing
capabilities of different architectures of neural networks to those of standard linear techniques. Kontoghiorghes and Dinensis propose an efficient
parallel iterative algorithm, based on orthogonal transformations, to solve
triangular seemingly unrelated regression equation models on massively
parallel computers. Mario Miranda presents a nested fixedpoint algorithm
for computing the full information maximum likelihood estimators of a nonlinear rational expectations model using orthogonal polynomial projection
methods. Hall and Sola propose a generalization of the standard GARCH
model which allows diserete switching. Follows a set ofpapers which diseuss
viii
approaches for the solution of nonlinear rational expectation models. Faust
and Tryon present variations on the FairTaylor algorithm exploiting the
block structure of multicountry, rational expectations macroeconometric
models to solve them in a distributed processing environment. Are Magnus Bruaset discusses efficient methods for solving certain linear systems
which arise in the solution process of nonlinear economic models. Don and
Stratum present a way to solve pathdependent rational expectations systems numerically using FairTaylor's method. If the stationary state of the
system is pathdependent FairTaylor's method may fail to converge to the
correct solution. They avoid this problem by rewriting the original model
in terms of scaled variables.
We are grateful to Hans Amman and Ber~ Rustem, the organizers of
the Amsterdam conference, for having provided researchers in the computational economics area with this excellent opportunity to exchange their
experiences. In particular this conference provided the basis for continuing
intellectual exchanges in this field with the setting up of the foundation
of the Society of Computational Economics. The Amsterdam conference
certainly significantly contributed to assessing computational economics as
a now established field in economics.
Manfred Gilli
Department of Econometrics,
University of Geneva,
Geneva, Switzerland
List of Contributors
Emilio Barucci
DIMADEFAS, University of Florence, Italy
Paul M. Beaumont
Supercomputer Computations Research Institute, Florida State University,
Tallahassee, FL, USA
Christopher R. Birchenhall
School of Economic Studies, University of Manchester, United Kingdom
Raymond Board
Federal Reserve Board, Washington, DC, USA
Patrick T. Bradshaw
Supercomputer Computations Research Institute, Florida State University,
Tallahassee, FL, USA
Are Magnus Bruaset
SINTEF Applied Mathematics, Oslo, Norway
Yuafan Deng
Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong
E. Dinenis
Centre for Insurance and Investment Studies, City University Business
School, London, Uni ted Kingdom
F. J. Henk Don
Central Planning Bureau, The Hague, and University of Amsterdam, The
Netherlands
Jon Faust
Federal Reserve Board, Washington, DC, USA
Giampiero M. Gallo
Department of Statistics, University of Florence, Italy
Seth A. Greenblatt
Cent re for Quantitative Economics and Computing, Department of Economics, University of Reading, United Kingdom
x
Stephen G. Hall
Centre for Economic Forecasting, London Business School, United Kingdom
Erricos J. Kontoghiorghes
Centre for Insurance and Investment Studies, City University Business
School, London, United Kingdom
Leonardo Landi
Department of Information Systems, University of Florence, Italy
Mario J. Miranda
Department of Agricultural Economics and Rural Sociology, Ohio State
University, Columbus, OH, USA
Rodrigo L. V. Pinto
Department of Computing, Imperial College, London, United Kingdom
Ben; Rustem
Department of Computing, Imperial College, London, United Kingdom
Martin Sola
Birkbeck College, London, United Kingdom
Rudy M. G. van Stratum
Central Planning Bureau, The Hague, The Netherlands
Peter A. Tinsley
Federal Reserve Board, Washington, DC, USA
Ralph Tryon
Federal Reserve Board, Washington, DC, USA
Arthur de Vany
Institute for Mathematical Behavioral Sciences, University of California,
Irvine, CA, USA
Yuan Wang
Center for Scientific Computing, State University of New York, Stony Brook,
NY, USA
PART ONE
Modeling Computational Economic Systems
EVOLUTIONARY GAMES AND GENETIC ALGORITHMS
Christopher R. Birchenhall
Abstract. While the use of GAs for optimization has been studied intensively, their
use to simulate populations of human agents is relatively underdeveloped. Much of the
paper discusses the application of vanous forms of GAs to simple ga.mes and compares
the outcomes with theory and experimental evidence. Despite the reported successes, the
paper concludes that much more research is required to understand both the experimental
evidence and the formulation of population models using GAs.
1. Artificial Evolutionary Modelling
An evolutionary model usually has two key elements, selection and mutation. Selection involves some concept of fitness, such that a variant (phenotype) with higher fitness has a higher probability of survival. Mutation
generates new variants. As in the biologieal world, it is to be expected that
the most interesting evolutionary economies models will involve the coevolution of two or more interacting populations or subgroups; see Maynard
Smith's book (1982).
A computational model is a model that can be implemented on a computer and whose structure and behaviour can be interpreted as representing some aspects of areal world situation. Once implemented, these models
could be the basis for experimentation; such computer experiments could be
used to test our theories andjor suggest new questions. One of the reasons
for turning to computational models of evolution is the expectation that
their behaviour will be complex and not immediately susceptible to analytie methods. Experimentation may provide some insight into their nature;
subsequent analysis would need to substantiate these findings.
We need to develop an understanding on how these computational models are to be constructed. It has to be expected that in the early stages these
models will be relatively simple, just as simple games are important to the
development of game theory. Computable representations that are as rieh
as reality are as useless as a onetoone road map. Just as simple games
are used to illustrate a partieular point, so our computational models can
3
M. Gilli (ed.), Computational EcOMmiC Systems, 323.
© 1996 Kluwer Academic Publishers.
4
C.R. BIRCHENHALL
be designed to investigate specific types of model and susceptible to experimentation.
So wh at will be involved in constructing a computational, evolutionary
model? We need algorithms that suitably emulate the processes of selection and mutation. In this paper we report on our initial use of genetic
algorithms in building a computable evolutionary process.
2. Genetic Algorithms
In this section we report some results on the use of genetic algorithms
as the basis of computational, evolutionary models of behaviour in the
playing of simple games. Our immediate aim is to assess how weil such
models perform in the context of simple games. Insofar as the performance
of GAs is acceptable in this context, we can be more confident in their use
for more complex games. In assessing these computer models we compare
their performance against theoretical and experimental benchmarks. It is
to be noted that theory and experiments do not always coincide; in this
situation our primary interest will be in the ability of models to emulate
actual, rather than, theoretical behaviour.
2.1. SOME TERMINOLOGY
Goldberg (1989) offers an exceilent introduction to GAs. The foilowing
notes are not intended to be selfcontained.
1. GAs work with a population of strings; typicaily these strings are bit
strings, i.e. each string is a sequence of O's and 1'so In setting up a GA
the user must define a fitness function that maps strings onto some
nonnegative measure of the string's fitness; with the understanding
that string with greater fitness is more likely to propagated by the
algorithm. Typically, the user will specify a mapping from the set of
strings into an appropriate space e.g. a mapping of astring into one
or more numbers representing a point in the domain of the problem.
The fitness function then maps from the domain of the problem to the
realline.
2. The genetic algorithm involves three steps: selection, crossover and
mutation.
 Selection generates a new population of strings by a biased drawing of strings from an old population, the bias favoring the fittest
strings in the old population. In this way selection emulates the
"survival of the fittest". The results of this paper are largely based
on a GA using a form ofthe "roulette wheel", see Goldberg (1989):
the probability of astring being chosen is proportional to the fit
GAMES AND GENETIC ALGORITHMS
5
ness of that string. Repeated drawing, with replacement, is used
until the new population is the same size as the old population.
One of the dangers of GAs is premature convergencej following
Goldberg (1989) the fitness values are scaled by a linear transformation. This scaling is governed by a sealing factor Sj if possible
the linear transform makes the highest fitness value equal to s
times the average.
In crossover the population is viewed as a set of parental pairs of
strings, the parents in each pair are randomly crossed with each
other to form two children. In the standard GA, in the manner of
Goldberg (1989), these children replace the parents. In this paper
we make heavy use of Arifovie's augmented form of crossover, in
which a child replaces a parent to the extent that the child is
bett er than the parent, see Arifovie (1994). Crossover is governed
by the erossover probability Px, such that for each pair of parents
the probability ofthem being crossed is Px. In all the runs reported
he re Px = 0.6.
During mutation each string is subject to random "switching"
of its individual elements; in switching a '1' be comes a '0' and
vieeversa. This process is governed by the mutation probability
Pm, such that the prob ability of an element being "switched" is
Pm. In this paper we use the augmented version of mutation such
that a mutated string replaces the original only if the mutation
increases fitness.
In all runs reported here the strings were initialized bit by bit,
such that the prob ability of an individual bit being a '0' is Pi.
Hereafter this is prob ability is called the bias. As will be seen
from the reported results the choiee of this bias is important.
 The interpretation given to the strings can have an important
influence on the workings of the GA. Even the mapping from bit
strings to real numbers has several important variations. Three
will be briefly discussed here: geometrie, arithmetie and mixed.
Given a bit string s, we will use Si to denote the value of the i th
bit; all Si have values 0 or 1. Let n denote the length of the bit
string, so that i varies over the set {1, ... , n} .
• Geometrie coding of numbers is the usual co ding of unsigned
integers, i.e. given string s the associated integer value is
"'~ 2n +1  i s·••
P = LI.=1
To map sinto the real unit interval, [0,1], we can use
6
C.R. BIRCHENHALL
r = p/(2 n

1).
This eoding gives a range with 2n different values .
• Arithmetie coding is based on a simple sum of the bit values
e.g. 8 is mapped onto the integer
and onto the unit interval with
r = q/n.
This has a range with only n different values .
• Mixed eoding uses a mixt ure of geometrie and arithmetie
eoding. The string is divided into two parts; the first m ::; n
bits are used in an arithmetie eode and the remaining k =
n  m bits are used in a geometrie eoding. Two integers are
formed so:
_
q
~m.
LJi=18\
_
P
~n
LJi=m+1
2n+1i.
S\,
whieh are mapped onto [0, 1] as
This eoding gives m
X
2n 
m
different values.
On the faee of it arithmetie eoding is very inefficient use of a bit
string; furthermore, unless a very long string is used the seareh
spaee is broken up to very few equivalenee classes. Its attraetion
eomes from the behaviour of GAs based on this coding. Consider
mapping a n bit string onto [0,1]; for the n different values q/n,
where q = 0, ... , n  1, there are C~ different related strings.
That is to say there are normally several different ways in whieh
a partieular end value may emerge from the GA. Equally, and
more importantly, there will be a large number of paths whieh
will lead the GA to the same outcome. In geometrie coding the
value of the most signifieant bits are very important, e.g. to get
into the upper half of the range the most signifieant bit must be
set to 1. It is suggested that arithmetie eoding is more "robust".
Mixed eoding aims to eombine the benefits ofboth basie methods;
the arithmetie eoding gives the rieher set of "broad band" seareh
GAMES AND GENETIC ALGORITHMS
7
paths, while the geometrie co ding can be used to fine tune the
final value. Many of the runs reported below have used mixed
coding.
It is to be remembered that our aim in this discussion is to use
GAs as a means of modelling the behaviour of groups of economie
agents. In judging coding schemes, as with other aspects of GAs,
our concern is with the GAs ability to emulate behaviour, not
with their efficiency in finding optima. From this point of view
the mixed coding scheme has intuitive appeal, at least to the
author. People tend to use rough and ready methods in early
stages of searches and concentrate on minor adjustments at the
latter stages; mixed co ding tends to work in this manner.
3. When using a GA to model a population it is important to be clear how
and when fitness is calculated; the problem arises from the fact that the
fitness of an individual depends on the composition of the population.
There is no fixed function that defines fitness. Selection, crossover and
mutation all change the population and thus the fitness function. This
can lead to some instability in the process. Unless otherwise stated the
results reported use the following procedure.
 Fitness of the initial population is calculated before the first round
of selection. Fitness values are recalculated immediately after each
selection. In the augmented GA it is these values that have to be
bettered by potential mutations and crossovers.
The population undergoes mutation and then crossover. This allows the mutations to propagate themselves; this is partieularly
valuable with the augmented GA, where mutations and crossovers
are implemented only if they are judged to improve performance.
If fitness has to be calculated between selections, i.e. in the augmented forms of crossover and mutation, this is done with respect
to the population immediately after the latest selection. For example, in assessing a mutation the fitness of the mutant is calculated as if no mutations had taken place since the last selection.
Equally, in evaluating the effect of a crossover, fitness is calculated as if no mutations nor crossovers had occurred since the
latest selection. During augmented mutation and crossover, it is
as if the background population was frozen in its state after the
latest selection.
Having assessed all mutations and crossovers, the fitness of the
ensuing population, Le. the population after all mutations and
crossovers, is recalculated before the next selection process. Clearly,
8
C.R. BIRCHENHALL
at this juncture some of the mutations and crossovers may prove
to be less desirable.
Our experience is that the GA behaves in a more predietable manner
with this process, when compared with a process where the implicit
fitness function is continuously updated.
The primary driving forces in these GAs are selection and crossover; mutation is seen as a safety valve to counter the effects of an inadequate initial
population and to prevent stagnation. Selection, in favouring the fittest,
spreads the infiuence of the fit individuals across the population. Crossover
improves the population insofar as it can mix those substrings of the parents that are the basis of the parents' fitness. A proper understanding of
crossover requires a study of schemata or similarity templates (see Goldberg (1989) chap. 2). Essentially the relationship between strings and fitness
must allow "good" strings to be generated from "good" substrings.
Crossovers give GAs their distinctive character; selection and mutation
appear in other evolutionary algorithms, e.g. evolutionary programming of
Fogei, Owens and Walsh (1966). In the augmented form, where mutations
and crossovers are fitness improving, the GA looks very much like aversion
of hillclimbing or, in the case of population modelling, the replieator dynamie. Depending on the problem and the coding scheme, a crossover can
induce a highly nonlinear step by 'mixing' good substrings.
2.2. THE POWER 10 FUNCTION
Here we consider the function f( x) = x 10 whose maximization Goldberg
(1989) suggests is GAhard. An initial "population" of 20 strings was generated randomly and passed through fifty "generations" or iterations of
various forms of the GA. Each string contained 32 binary characters. The
prob ability of mutation was 0.033 and the prob ability of crossover was 0.6.
Mixed co ding was used, 16 bits being used for the arithmetic and geometrie
elements. The bias was 0.5.
The important lesson illustrated by this simple example is that augmentation has a significant impact on the behaviour of the GA, the augmented
version manages to induce all strings onto the optimum, while a standard
GA typically maintains a dispersed population. This is an important difference when using aGA to simulate population behaviour. When using a
GA to optimize a function we judge performance by the fitness of the best
string, not by the average fitness across the whole population.
GAMES AND GENETIC ALGORITHMS
9
3. Games and GAs
In this section we investigate coordination issues in the context of simple
2 X 2 games. In particular, GAs are used as the basis for artificial models
of populations learning to play various games. At the same time we aim to
compare the outcome of these computational models with evidence from
experiments. The form of the GA depends on the assumptions made about
the decision process of the agents in the population. This fact, together with
the dependency on the values of the standard GA parameters, implies that
a certain amount of experiment at ion is needed to find satisfactory models.
At this juncture, we can do little more than attempt to assess whether
the GA based models are reasonable. In the longer run, we might hope to
obtain sufficient understanding of human decision making and GAs to allow
us to investigate novel situations with some confidence. The author feels the
greatest problem is with our inadequate understanding of human decision
processes. We hope the following discussion assists in the appreciation of
GAs.
3.1. THEORETICAL NOTES ON 2 x 2 GAMES
Asymmetrie 2 x 2 game will have a payoff matrix of the form shown in
table 1. In this discussion we will assurne that all entries a, b, c and d are
TABLE 1. Payoff Matrix for
symmetrie game
Left
Up
Down
a
c
Right
abc
b
d
d
nonnegative. The following observations can be made.
Generic games have at least one pure strategy Nash equilibrium. A
game is generic if a 1= c and b 1= d.
Proof: If a > ethen (Up, Left) is Nash; if d > b then (Down, Right) is
Nash. If a < c and d < b then (Up, Right) and (Down, Left) are both
Nash.
Note with generic games the equilibria are strict, i.e. both players have
a positive incentive not to change their strategy, given the strategy of
the other player.
If a  c and d  b have the same sign then there is a mixed strategy
equilibrium wit h probability of playing Up equal to
10
C.R. BIRCHENHALL
p* =
db
,
ac+db
and prob ability of playing Left with
ac
ac+db
q*=If a > c and d > b then there are three equilibria with mixed strategy
probabilities (p,q) = (0,0), (p*,q*), (1, 1).
If a < c and d < b then there are three equilibria with mixed strategy
probabilities (p, q) = (0,1), (p*, q*), (1, 0). Although the payoff matrices are symmetrie across players, these games have asymmetrie equilibria.
 If a > c and b > d then Up and Left are dominant and (Up, Left) is the
unique Nash equilibrium. Equally if a < c and b < d then Down and
Right are dominant and (Down, Right) is the unique Nash equilibrium.
 If a + b > c + d then Up and Left are riskdominant. If a + b < c + d
then Down and Right are riskdominant.
3.2. AN EVOLUTIONARY STORY
Evolutionary game theory has recently attracted attention, see for example
the special issues of the Journal of Economie Theory(1992) and Games and
Economic Behaviour(1993), chapter 9 in Binmore (1992), as weIl as the
artieies by Friedman (1991), Kandori et al. (1993). Many of the concepts
in this area trace their origins to the work of Maynard Smith (1982). The
folIowing simple story will suffice for us to make a few salient points; the
reader should look to the aforementioned literat ure for a fulIer exposition
of evolutionary games.
Consider a population of unsophistieated and myopie players, that is to
say a set of players who, within a given time period have a given tendency
to play our game in a partieular way. Here tendency may be pure, i.e. the
player may be an UpjLeft or a DownjRight player, or the tendency may
be mixed, i.e. the player will choose UpjLeft with probability p. (Mixed
strategies can be given a much more appealing interpretation when discussing populations, see below.) These players are naive in the sense that
their choices are not necessarily the outcome of an analysis of the game;
these choiees are not necessarily optimal, though, as we shall see, they may
reflect experience. They are myopie in that they do not attempt to antieipate the impact of their current decisions on the future plays of the
game. The favoured justification for ignoring repeated game effects is that
GAMES AND GENETIC ALGORITHMS
11
opponents are drawn at random from a large population. Insofar as the
composition of a small population is subject to change in a random fashion, Le. through selection and mutation processes, then myopie decisions
may be perfectly reasonable, albeit boundedly rational.
An alternative interpretation of mixed strategies arises from the idea
that each agent is really a representative of a 'large' subpopulation. Each
agent in the subpopulation uses pure strategies and the mixed strategy
probability associated with the representative agent is the proportion of the
subpopulation choosing the first pure strategy. We can consider a situation
where an individual is drawn from the subpopulation at random for each
play of the game.
There is systematie learning in our storYj but this learning takes place
at the level of the population. The population as a whole changes in response to the experience of its members. This is not to suggest that the
population is aiming to maximize any sodal welfare, nor that the population is an active agent. Rather we are suggesting that learning is not an
internal matter for individual agentsj rat her agents behaviour responds to
the experiences of other agents as much as their own. We return to these
issues below.
Consider the case where players make pure choices. In each period we
assurne each player meets an opponent chosen at random from the population. If the player chooses Up/Left then his expected payoff will be
ar + b(l  r), where r is the prob ability that his opponent is an Up/Left
playerj if the population is large then r can be interpreted as the proportion of players that are Up /Left players. The expected payoff from playing
Down/Right is er+d(1r). The Up/Left players will do better, on average,
than Down/Right players if ar + b(l  r) > er + d(l  r)j this inequality
can be written as r > p* if a  c + d  b > 0 or as r < p* if a  c + d  b < O.
Let us now assurne that, at the end of each period, information about
payoffs is disseminated through the population. If the Up /Left players tend
to do better, we conjecture a net switch from Down/Right to Up/Leftj
conversely, we expect a net move to Down/Right if they do better.
 If a > c and d > b, so that both (U p, Left) and (Down, Right) are
Nash, then a  c + d  b > 0, and the proportion of Up/Left players,
r, will rise or fall as r > p* or r < p* respectively. With an initial
r > p*, then r will tend to rise toward 1, i.e. the population will tend
to converge on a common Up /Left choiee. If initially r < p* then there
is a tendency to a uniform Down/Right population, with r falling to
O. This has the implication that the "emergent" equilibrium depends
on the initial distribution of players; the outcome is path dependent.
Both pure equilibria are locally stable.
12
C.R. BIRCHENHALL
If p* < 0.5 then d  b < 0.5(a  c + d  b) and 0.5(d  b) < 0.5(a  c)
and 0.5(c + d) < 0.5(a + b), Le. Up/Left is riskdominant. In these
circumstances, an initial T = 0.5 would lead to a tendency to Up/Left.
Equally, if Down/Right were riskdominant then a random initial distribution, T = 0.5, would tend to Down/Right in the limit. That is
to say, that if the initial population is "unbiased", Po = 0.5, then the
population tends to the risk dominant equilibrium.
 If a < c and d < b, so that (Down, Left) and (Up, Right) are Nash,
then a  c + d  b < 0, and the proportion of Up/Left players, T,
will rise or fall as T < p* or T > p* respectively. There is a tendency
for T to converge to p*. Note weIl: despite the symmetry in the game
and the lack of differential treatment of the players, this analysis suggests the population will organize itself into Up/Left and Down/Right
players. The mixed Nash p* can be interpreted as a stable equilibrium
distribution for the population.
This is not a rigorous analysis, but illustrates the basic idea behind
many of the evolutionary stories in game theory. The process is very much
in the nature of the replicator dynamic, see for example Binmore (1992,
chapter 9).
Critics suggest these stories treat the agents as unthinking  they are,
to borrow a term from Paul Geroski, "monkeys". While this is a valid
interpretation of many of the models in this paper, this reaction misses the
essential purpose of the exercise. Recall our initial premise that knowledge
is embodied in populations and that individuals have limited understanding
of the world. In order to place emphasis on distributed knowledge systems it
is useful to simplify the stories and leave the individuals unsophisticated. To
the extent it is necessary to make the individuals more or less sophisticated
the models can be developed furt her. We return to this point latter.
With this in mind it is useful to consider the interpretation of the proposed dynamic, namely relatively successful strategies increase as a proportion of the population. Consider the following approach: the individual
agents obtain information on the strategies of other players and their payoffs
and are able to modify their own strategies. Our dynamic can be viewed as
an assumption that agents tend to emulate the behaviour of other successful agents. More generally we allow agents to learn from the behaviour of
others. While this process is analogous to biological processes  survival
of the fittest  it is not inconsistent with intelligent, if relatively unsophisticated, pursuit of selfinterest. The "genetic codes", Le. the tendency
to choose one play or another, can be seen to be selfmodifying, rather
than driven by some external selection process. The choice of a player in
any given period is naive but will in general reflect the experience of the
population. It is suggested that in complex environments, this use of emu
GAMES AND GENETIC ALGORITHMS
13
lation  staying with the crowd  maybe more effective than attempting
a sophisticated modelling and optimization process. Often the latter option
may not be available; this is particularly true in coordination games where
pure analysis does not dearly identify outcomes.
It is to be noted that our story to date has placed little emphasis on
innovation. In biological stories the focus at this point would turn to mutation. At the end of the day there is a useful mapping between human
innovation and mutation; while we may like to think of innovation as being
directed and the outcome of analytical reasoning, truly original ideas are not
predictable and must be more or less random. Having said that, it is worth
stressing that not all innovation involves totally original ideas. Indeed emulation is a form of innovation for the individual. Furt her more , a good deal
of innovation involves finding new mixtures of current ideas, different components of our activity can be measured against different "benchmarks".
This is the basis for an interpretation of crossover as a form of innovation
and imitation; we develop this theme in (Birehenhall, 1995a).
3.3. GA MODELS
Below we report on GA models of simple 2 X 2 games.
Unless otherwise stated, the strings in the GA are interpreted as the
mixed strategy probability of a player choosing UpjLeft, in each run of
the GA astring plays against all players (including itself), the bit string
coding is mixed and the GA uses protection and augmentation. Note there
is no distinction between strings playing column or row strategies. The
players do not know if they are column or row. The fitness of astring is the
average expected payoff given the probabilities represented by itself and by
its opponents.
We offer a few comments on the interpretation of the GA. Insofar as the
GA involves the development of a population and indudes a fitness based
selection it is reasonably called an evolutionary model; as indicated above
it is possible to view selection being driven by individuals modifying their
strings after observing the experience of the population as a whole. This
form of emulation involves some agents copying the complete strategy of
some other relatively successful agent. Crossover, at least in the augmented
form, can be viewed as a form of emulation that involves the copying of
part of some other player's strategy. This is of particular interest when the
knowledge embodied in the strings is decomposable or modular, i.e. where
there is an operational interpretation to Holland 's concept of schema. In the
current context, the importance of crossover is unclear, but it is retained
in anticipation of its value in more complex situations (see Birchenhall
(1995a) für a discussion of crossover in the context of technical change in
14
C.R. BIRCHENHALL
the context of modular technology). Mutation can be interpreted as players
experimenting with variants.
Note the augmented form assurnes players can calculate the payoff of
the crossed strategy. Essentially we are assuming all agents have detailed
information on the state of the population. This is questionable given the
basic flavour of our models Le. unsophistieated and boundedly rational
agents. A better approach requires agents to have a model of the world in
which they are acting. This in turn requires us to investigate the coevolution
of these models. This theme is discussed in (Birchenhall, 1995a).
We say a GA converges if all strings become, and remain, identieal.
It is suggested, without formal proof, that if the GA converges then the
common limiting value will be a Nash equilibriumj clearly this equilibrium
will be stable in the sense that the GA converges onto it. The informal
argument goes as follows. If the limiting value is not Nash then "eventually"
a mutation will find a better strategy and eventually selection and crossover
would spread this mutation through the population, Le. the limiting value
would be disturbed.
All the runs reported in this section on symmetrie 2 x 2 games use 20
strings each of length 32, and they use the same seeds for the pseudorandom number generator. Hence they share the same mapping from bias
to initial average mixed strategy probability, PO' Table 2 is the common
form of this mappingj the averages have been rounded to two significant
digits. As a rough rule ofthumb, the initial average prob ability is one minus
TABLE 2. Mapping of bias to initial average probability I
Bias
Initial po
0.1
0.2
0.3
004
0.5
0.6
0.7
0.8
0.9
0.90
0.80
0.71
0.61
0.51
0040
0.30
0.19
0.10
the bias. It is to be noted that this simple relation is largely due to the use
of mixed coding.
In reporting the results we describe the specific game in the form G =
(a, b, c, d) where the a, b, c and d match the values in the general symmetrie game in table 1, Le. a is the common payoff if the players choose
Up/Left, (b,c) are the payoffs if they play Up/Right, (c,b) are the payoffs for Down/Left and dis the common payoff for Down/Right. For each
game we summarize the mapping from the bias used to the limiting average
mixed strategy probability p.
The Corner game G = (80,0,0,0) has (Down, Right) as a nonstriet
Nash equilibrium, but it is unstable. There is every reason to expect
GAMES AND GENETIC ALGORITHMS
15
(Up, Left) to emerge from plays of this game. In the GA runs we
observed p = 1 for an values of the bias, i.e. in an cases the prob ability
of playing Up /Left converges to 1 as expected.
 Coordination Game 1 G = (10,0,0,5) illustrates agame where (Up, Left)
and (Down, Right) are strict Nash equilibria. The mixed Nash has
p* = 1/3. The GA results largely conform with the simple replieator
story in section 3.2, i.e. with initial average probabilities Po greater
than p* the population converges onto Up/Left and with initial Po
below p* the population converges to Down/Right.
Coordination Game 2 G = (9,0,0,1) illustrates agame where (Up, Left)
and (Down, Right) are pure Nash equilibrium. The mixed Nash has
p* = 0.1. As with coordinate Game 1 the results from the GA are
consistent with the replicator story.
 The prisoner's dilemma G = (9,0,10,1) has Down and Right as dominant strategies and (Down, Right) is the only pure Nash equilibrium.
The GA results gave Po = for an values of the bias, i.e. p = is
globally stable.
°
°
3.4. THE STRAUB GAMES
Straub (1993) has described a set of experiments using the symmetrie and
asymmetrie games; we do not discuss the latter here. In all the symmetrie
games the (Down, Right) combination is Pareto dominant. Straub argues
that his observations support the proposition that players choose risk dominant strategies in coordination games. This conclusion has to be qualifi.ed
given his results for Gs , where the strength of the Pareto dominant combination clearly won over some of the players.
In each experiment there were 9 rounds of the game, with each player
meeting a different opponent in each round (see Straub 's paper for the
details of the arrangements made to remove repeated game effects).
Straub Game GI = (80,80,0,100) has (Up, Left) and (Down, Right)
as pure Nash equilibria. The mixed strategy equilibrium has p* = 0.2.
The risk dominant combination is (U p, Left). Straub observed 60%
playing Up /Left in round 1, with the proportion rising to 100% in
period 9. The GA results match the replicator story and are consistent
with Straub's results, i.e. with a bias of 0.4, giving initial Po ~ 0.6,
leads to all players choosing Up /Left.
In Straub Game G 2 = (35,35,25,55) (Up, Left) and (Down, Right)
are pure Nash equilibria. The mixed strategy equilibrium has p =
2/3 ~ 0.67. The risk dominant combination is (Down, Right). Straub
observed 10% playing Up/Left in round 1, with the proportion falling
to 0% in round 3. The GA results are consistent with the replicator
16
C.R. BIRCHENHALL
theory and Straub's results; with a bias of 0.9, giving Po ~ 0.1, all
players converge on Down/Right.
 Straub Game G 3 = (60,60,20,100) has (Up, Left) and (Down, Right)
as pure Nash equilibria. The mixed strategy equilibrium has p* = 0.5.
There is no risk dominant combination here, but (Down, Right) is
Pareto dominant. Straub observed 40% playing Up/Left in round 1,
with the proportion falling to 0% in round 5. The GA results are again
consistent with the replicator story and Straub's results; a bias of 0.6,
giving Po ~ 0.4, leads to all converging onto Down/Right.
 Straub Game G 4 = (80,80,20,100) has (Up, Left) and (Down, Right)
as pure Nash equilibria. The mixed strategy equilibrium has P = 0.25.
The risk dominant combination is (Up, Left). Straub observed 80%
playing Up/Left in round 1, with the proportion rising to 100% in
round 7. The GA results are consistent with the replicator story and
Straub's results; a bias of 0.2, giving Po ~ 0.8, converges to all playing
Up/Left.
Straub Game Gs = (30,70,10,80) has (Up, Left) and (Down, Right)
as pure Nash equilibria. The mixed strategy equilibrium has P = 1/3.
The risk dominant combination is (Up, Left). Straub observed 10%
playing Up or Left in period 1, with the proportion oscillating between
10% and 30%. The GA results are consistent with the replicator story.
In contrast to Straub's observations the GA converges; with a bias of
0.9, giving Po ~ 0.1 all players converged onto Down/Right.
3.5. SIMULTANEOUS GAMES
This section reports the results from a model where 30 agents are playing
125 simultaneous games. Each game has the form given in table 1 where
d = 0.5. The values of a, band c each ranged over the 5 values {O, 0.25,
0.5, 0.75, 1.0}.
The GA was based on 30 strings each of length 125. The i th bit in the
string was interpreted as specifying the choice of strategy in the i th game;
a bit value of 0 indicating that the player chooses Red and a value of 1
indicating the choice of BIue. The GA was run for 300 rounds.
The following summary comments are based on a full set of results
presented in (Birchenhall, 1995b):
 Risk dominance seems to have a strong, but not overriding influence
on the outcomes. As an approximate rule the choice between multiple
equilibria is governed by risk dominance.
 When the only equilibria are Red/BIue and BIue/Red there is a tendency for the players to divide themselves up into BIue and Red players.
This is tempered by the pull of risk dominant strategies.