Computational economic systems models, methods econometrics

T ABLE OF CONTENTS Preface List of Contributors Part One: Modeling Computational Economic Systems Evolutionary Games and Genetic Algorithms A Distributed Parallel Genetic Algorithm: An

CONWUTATIONALECONONUCSYSTEMS

VOLUMES

SERIES EDITORS

Hans Amman, University of Amsterdam, 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

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V

ISBN 978-90-481-4655-0 ISBN 978-94-015-8743-3 (eBook)

DOI 10.1007/978-94-015-8743-3

Printed on acid-free 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

T ABLE OF CONTENTS

Preface

List of Contributors

Part One:

Modeling Computational Economic Systems

Evolutionary Games and Genetic Algorithms

A Distributed Parallel Genetic Algorithm: An Application from

Economic Dynamics

Multi-Item Stochastic Inventory Models with Constraints and their

Parallel Computation

Building and Solving Multicriteria Models Involving Logical Conditions

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

139

Solving Triangular Seemingly Unrelated Regression Equations Models

on Massively Parallel Systems

Maximum Likelihood Estimation of Nonlinear Rational Expectations

Models by Orthogonal Polynomial Projection Methods

Structural Breaks and GAReR Modelling

Block Distributed Methods for Solving Multicountry Econometric

Models

Efficient Solution of Linear Equations Arising in a Nonlinear

Economic Model

Solving Path-dependent Rational Expectations Models Using the

Fair-Taylor Method

Preface

The approach to many problems in economic analysis has drastically changed with the development and dissemination of new and more effi-cient computational techniques The present volume constitutes a selection

of papers presented at the IFAC-Meeting 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 sim-ple games and eompares the out comes with theory and experimental ev-idence 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 algo-rithm whieh is quite effective at solving complex optimization problems as it avoids converging on suboptimal solutions Wang and Deng develop a multi-item, multi-period, double-random 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 rie 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 stan-dard 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 fixed-point algorithm for computing the full information maximum likelihood estimators of a non-linear 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

economet-approaches for the solution of nonlinear rational expectation models Faust and Tryon present variations on the Fair-Taylor algorithm exploiting the block structure of multi-country, rational expectations macroeconometric models to solve them in a distributed processing environment Are Mag-nus 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 path-dependent rational expectations sys-tems numerically using Fair-Taylor's method If the stationary state of the system is path-dependent Fair-Taylor'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 compu-tational 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

Are Magnus Bruaset

SINTEF Applied Mathematics, Oslo, Norway

Birkbeck College, London, United Kingdom

Rudy M G van Stratum

Central Planning Bureau, The Hague, The Netherlands

PART ONE

Modeling Computational Economic Systems

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

An evolutionary model usually has two key elements, selection and tion Selection involves some concept of fitness, such that a variant (phe-notype) 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 coevo-lution of two or more interacting populations or subgroups; see Maynard Smith's book (1982)

muta-A computational model is a model that can be implemented on a puter and whose structure and behaviour can be interpreted as represent-ing some aspects of areal world situation Once implemented, these models could be the basis for experimentation; such computer experiments could be

com-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 ana-lytie 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 els 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

mod-as reality are mod-as useless mod-as a one-to-one road map Just mod-as simple games are used to illustrate a partieular point, so our computational models can

3

M Gilli (ed.), Computational EcOMmiC Systems, 3-23

© 1996 Kluwer Academic Publishers

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 self-contained

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

non-negative 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 ing of strings from an old population, the bias favoring the fittest strings in the old population In this way selection emulates the

draw-"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-

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 transfor-mation 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

viee-versa 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

P = "'~ LI.=1 2 n+1- i s· ••

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 2 n 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:

whieh are mapped onto [0, 1] as

This eoding gives m X 2 n - 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

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 U nless 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 lows 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

al-If fitness has to be calculated between selections, i.e in the mented forms of crossover and mutation, this is done with respect

aug-to the population immediately after the latest selection For ample, in assessing a mutation the fitness of the mutant is calcu-lated as if no mutations had taken place since the last selection Equally, in evaluating the effect of a crossover, fitness is calcu-lated as if no mutations nor crossovers had occurred since the latest selection During augmented mutation and crossover, it is

ex-as if the background population wex-as 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,

The primary driving forces in these GAs are selection and crossover; tion 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 par-ents that are the basis of the parents' fitness A proper understanding of crossover requires a study of schemata or similarity templates (see Gold-berg (1989) chap 2) Essentially the relationship between strings and fitness must allow "good" strings to be generated from "good" substrings

muta-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 hill-climbing or, in the case of population modelling, the replieator namie Depending on the problem and the coding scheme, a crossover can induce a highly nonlinear step by 'mixing' good substrings

dy-2.2 THE POWER 10 FUNCTION

Here we consider the function f( x) = x10 whose maximization Goldberg (1989) suggests is GA-hard An initial "population" of 20 strings was gen-erated 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 tation has a significant impact on the behaviour of the GA, the augmented version manages to induce all strings onto the optimum, while a standard

augmen-GA typically maintains a dispersed population This is an important ference when using aGA to simulate population behaviour When using a

dif-GA to optimize a function we judge performance by the fitness of the best string, not by the average fitness across the whole population

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 Right

Up a a b c

Down c b d d

non-negative The following observations can be made

Generic games have at least one pure strategy N ash 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 U p equal to

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

matri-ces are symmetrie across players, these games have asymmetrie libria

equi If a > c and b > d then Up and Left are dominant and (Up, Left) is the

unique N ash equilibrium Equally if a < c and b < d then Down and

Right are dominant and (Down, Right) is the unique N ash equilibrium

- If a + b > c + d then Up and Left are risk-dominant If a + b < c + d

then Down and Right are risk-dominant

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

opponents are drawn at random from a large population Insofar as the composition of a small population is subject to change in a random fash-ion, 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 sponse to the experience of its members This is not to suggest that the population is aiming to maximize any sodal welfare, nor that the popula-tion 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

re-Consider the case where players make pure choices In each period we assurne each player meets an opponent chosen at random from the pop-ulation 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 propor-

tion of players that are U p /Left players The expected payoff from playing Down/Right is er+d(1-r) 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 U p /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 U p /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 risk-dominant In these

circumstances, an initial T = 0.5 would lead to a tendency to Up/Left Equally, if Down/Right were risk-dominant then a random initial dis-tribution, 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 gests the population will organize itself into Up/Left and Down/Right players The mixed N ash p* can be interpreted as a stable equilibrium

sug-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 pro-posed dynamic, namely relatively successful strategies increase as a pro-portion 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 ful agents More generally we allow agents to learn from the behaviour of others While this process is analogous to biological processes - survival

success-of the fittest - it is not inconsistent with intelligent, if relatively phisticated, pursuit of self-interest The "genetic codes", Le the tendency

unso-to choose one play or another, can be seen unso-to be self-modifying, 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-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 mu-tation 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 em-ulation is a form of innovation for the individual Furt her more , a good deal

of innovation involves finding new mixtures of current ideas, different ponents 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)

com-3.3 GA MODELS

Below we report on GA models of simple 2 X 2 games

U nless 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

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 N ash 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 N ash 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 pseudo-random 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 0.1 0.2 0.3 004 0.5 0.6 0.7 0.8 0.9

Initial po 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

sym-metrie 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 offs for Down/Left and dis the common payoff for Down/Right For each

pay-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 non-striet Nash equilibrium, but it is unstable There is every reason to expect

(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 U p /Left converges to 1 as expected

and (Down, Right) are strict N ash equilibria The mixed N ash 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 inant strategies and (Down, Right) is the only pure N ash equilibrium The GA results gave Po = ° for an values of the bias, i.e p = ° is globally stable

dom-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 dom-inant strategies in coordination games This conclusion has to be qualifi.ed given his results for Gs, where the strength of the Pareto dominant combi-

nation 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 N ash equilibria The mixed strategy equilibrium has p* = 0.2 The risk dominant combination is (U p, Left) Straub observed 60% playing U p /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 U p /Left

In Straub Game G 2 = (35,35,25,55) (Up, Left) and (Down, Right) are pure N ash 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

as pure N ash 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

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,

- 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 dency for the players to divide themselves up into BIue and Red players This is tempered by the pull of risk dominant strategies

ten-One of the motivations behind this experiment was an attempt at a crude form of generalization The previous experiments involved the pop-ulation learning to play a particular game To make significant progress toward the modelling of real agents, our artificial agents will need the abil-ity to generalize from their specific experience It is possible to imagine a crude generalization process based on the outcome of the current exper-iment; e.g given a new game g, not in the original set S, treat g as if

it were that element g' E S which is dosest to g Generalization involves agents having access to some analysis of the games it faces, possibily based

on some concept of model In (Birchenhall, 1995a) a situation is discussed where a population of models coevolve with a population of actions It is to

be expected that significant progress toward generalization will require the adoption of Holland's dassifier systems (see Holland (1989), and/or Koza's genetic programming (Koza, 1992)) Currently, the author more readily perceives how the latter could generate readily interpretable mIes of play This possibility will be pursued in future work

3.6 CONCLUDING REMARKS ON GA AND SYMMETRIC GAMES

The behaviour of the GAs as a whole is consistent with the replicator story The convergence results dearly demonstrate that the initialization of the

GA is important This raises a major issue for the use of GAs as the basis

of computer experiments What is to be the basis for this initialization?

A number of theoretical papers are putting emphasis on risk dominance, and Straub provides some evidence that this may be important in practice This may suggest biasing the GA toward risk dominant strategies But this clearly needs to be qualified; for example Straub's game 8 shows that Pareto dominance has some influence

At the end of the day we may have to adopt some rules of initialization that refiect our understanding of behaviour We need much more evidence

on how players actually behave

Theoretical guidance on the convergence properties of GAs in these game situations would be most helpful

Equally, we need to develop more understanding of the rate of gence of GAs

conver-4 Sefton-Yavaq Games

In this section we briefiy discuss a model of a Sefton-Yavaq game Sefton and Yavaq (1993) report on experiments to test the effectiveness of Abreu-Matsushima mechanisms (see Abreu and Matsushima (1992; 1993)) The basic game underlying the Sefton-Yavaq experiments is given in table 3 Here there is a Pareto dominant equilibrium (Red, Red) The mechanism

18 C.R BIRCHENHALL

to in du ce (BIue, BIue) proposed by Abreu and Matsushima is to replace the game by aTstage game, where the payoff in each stage is the payoff

in table 3 divided by T, together with a a fine (negative payoff) against

the first player to play Red In Sefton (1993) they consider the cases where

TABLE 3 Basic game In

game of this form, there is a unique rationalizable equilibrium wher.e both players choose to stay with BIue through all stages In the Sefton-Yavaq

experiments the players switch to Red after about 4 stages

A large number of computer experiments were carried out to emulate the results reported by Sefton and Yavaq In most of these models the

population converged onto the unique rationalizable equilibrium, Le play all BIue This should be of no surprise given that GAs have a reputation for finding global maxima Below we discuss an experiment in which this convergence is not apparent But we do not put a great deal of emphasis on this specific case It is not clear how we should interpret the evidence from the Sefton-Yavaq experiments While a player believes there is a reasonable chance that his opponent will choose to switch to Red then it is rational for the player to plan to switch as wen It might be the case that with sufficient experience with the game, actual players would converge onto the rationalable equilibrium From this perspective, the Sefton-Yavaq evidence would be consistent with a learning story which eventually leads to the theoretical equilibrium

The specific model we discuss here has each player choosing a probability

p of switehing from Blue to Red, with the assumption that before the play begins each player is BIue At each stage of the game, including the first,

if the player is BIue then a coin, with bias p, is flippedj if the eoin turns

up positive the player switches to Red Onee the player is Red there is no switehing back to Blue in this model The rationalizable equilibrium would correspond to p = O If p -:J 0 then the expected number of BIues is

1 - (1 _ p?+l

P

Note EN ~ (l/p) - 1 when R = (1 - p)T+l/p is small and EN ~ T as

p ~ O For p = 0.2, the expected number of blues is approximately 4 Figures 1 and 2 illustrate the outcome of GA models where the individ-ual strings represent values for p The figures demonstrate that the values

of p did not converge, but exhibited both short and long swings

Prob Max Avg Mio Prob

To calculate payoffs in each round of the GA the strings are converted

to probabilities; these probabilities are then used to generate a particular realization, i.e a sequenee of Blues and Reds Onee these realizations of the current population are formed, each existing or modified string is converted

to a realization and played against the current population of realized quences This use of realizations can be seen to reflect the limited ability of agents to calculate the expeeted payoffs given the distribution of p's across the population Rather than the calculations the actual outcomes of plays are used to control the imitation process

se-For the models illustrated we used 20 strings with 32 bits, mixed coding was used, with 16 bits for both parts of the code The crossover probability was 0.6, the mutation probability was 0.04 and the bias was 0.8 It is to

be noted that the use of pure geometrie coding led to significantly different results; there was a" mueh stronger convergence to p = O The outcomes

in the GA In this example, the use of realizations and mixed coding has significantly tempered the GA tendency to converge to the equilibrium While learning models may have boundedly rational agents converging to

"optima" or equilibria, the speed of convergence is important In practice, they may never get dose to the equilibria, particularly in a complex and rapidly changing world

5 Ultimatum Game

In the ultimatum game two players are to divide a sum of money One player proposes a division of the money; if the second player (receiver) accepts the proposal then the proposed division determines the outcome; if the receiver rejects the proposal then both players get nothing Subgame perfect equilibria of these games has the proposer getting all of the money - maybe not the last "penny" Many laboratory experiments have been undertaken

to test this theoretical outcome and invariably in these experiments the receiver obtains a significant proportion of the money (see for example Roth et al (1991) and Thaler (1988)) Our interest in this game arose from Ken Binmore's paper at the RES Conference in 1994

The overall result of many simulations of this game, the situation was similar to that found for the Sefton-Yavq games Standard applications

of the GAs lead to the population converging onto the subgame perfect equilibrium Below we report on a variation where this was not so apparent Again we do not place a significant emphasis on this specific case It does raise issues similar about those discussed above, i.e speed of convergence and out of equilibrium behaviour of a proposer facing a receiver who might have a significant reservation price Furthermore, a number of writers feel that we need to introduce concepts of fairness to handle games, such as the ultimatum game, where the players may quest ion the "property rights" implicitly given to the proposer

The model reported here has two GAs, one GA for a population of posers and one GA for a population of receivers Proposers choose offers, while receivers choose a reservation level When a proposer and an receiver play they both receive nothing if the offer is less than the receiver's reser-vation level, otherwise the funds are divided as proposed In a round of the GA all proposers play all receivers and the fitness is the average payoff over all plays The GAs used 20 strings with 32 bits, crossover and mu-tation probabilities were 0.6 and 0.33, both GAs used a bias of 0.5 and a scaling factor of 2, the players were dividing 40 units Note the high level

pro-of mutation The previous discussion pro-of mixed coding in the context pro-of the Sefton-Yavaq games is applicable to this case

6 Conclusion

While the above GA models have proved to be more or less consistent with experimental evidence, we have not explored all variations in all games Where the experimental evidence causes traditional theory some difficulty, e.g the Sefton-Yavaq and Ultimatum Games, we have found the results are sensitive to the precise formulation of the GA model In particular, mixed coding, the mutation rates and the use of realizations have proved

to be important There is a need to undertake a much more systematic study of the variations Any such study will need to be clear what criteria

is to be applied in choosing good variations; consistency with experimental evidence should be prominent At the same time, furt her discussion of the interpretation of selection, mutation and crossover in these models is highly desirable An important problem in this area is going to be the speed of con-vergence; in practice, slow convergence will look much like no convergence and force to think more carefully about out of equilibrium behaviour

In our discussion of the simultaneous games model, we proposed the move to classifier systems and genetic programming Agents in the models presented above are simple; they are Httle more than strategies and do

22 C.R BIRCHENHALL

not undertake an analysis of the game before them In search for artificial models that will adequately simulate real agents, this is undoubtedly a weakness of the above models; humans do have an ability to generalize from their experience Some move to rule learning models such as classifiers and GPs will reduce the force of the simplicity criticisms Nevertheless, both

of these alternatives build on genetic algorithms and it is likely that a proper understanding of their use in modelling behaviour will require a commensurate understanding of GAs

Having acknowledged that our artificial agents need to be more cated if they are to be convincing, we would like to add the following caveats

sophisti-to that praposal Moving sophisti-to the use of classifiers and GPs would not change the emphasis on populations of boundedly rational agents learning through

a process of betterment To understand how modern economies more or less succeed to solve the highly complex coordination problems underlying modern technology, requires us to reduce the emphasis on the unboundedly sophisticated agent and try to understand how populations of relatively naive agents could succeed In a complex, fast moving world it is unlikely agents will have complete or precise models of that warld; success is likely

to be based on robustness as much as absolute correctness Experience, i.e learned behaviour, may be as potent as sophisticated modelling

Acknowledgements

This work has arisen from discussions with Stan Metcalfe and Nikos trinos Ericq Horler has read a previous draft of this paper; I am grateful for his corrections and suggested improvements I do wish to attribute any blame far any remaining errar or omission I am grateful for comments received at presentations to the U niversity of Siena, May 1994, and the IFAC Workshop on Computing in Economics and Finance, Amsterdam, June 1994

Eco-Binmore, K., 1992, Fun and Games, Lexington: Heath & Co

Binmore, K., 1994, 'Learning to be Imperfect: The Ultimatum Game', Paper presented

to the RES Conference Exeter, forthcoming in Games and Economic Behaviour Birchenhall, C R., 1995a, 'Modular Technical Change and Genetic Algorithms', Compu- tational Economics 8(4), 233-253

Birchenhall, C R., 1995b, Toward a Computational, Evolutionary Model of Technical Change, School of Economics Discussion Paper, Manchester University

Bullard, J and J Duffy, 1994, A Model of Learning and Emulation with Artificial tive Agents, mimeo

Adap-Fogei, 1 J., Owens, A J., and M J Walsh, 1966, Artificial Intelligence through Simulated Evolution, New York: Wiley

Friedman, D., 1991, 'Evoltionary Games in Economics', Econometrica 59, 637-666 Games and Economic Behaviour 5(3-4), 1993, special issues on Adaptive Dynamics

Glazer, J and R W Rosenthai, 1993, 'A Note on Abreu-Mechanisms', Econometrica

Pro-Holland, J H and J H Miller, 1991, 'Artificial Adaptive Agents in Economic Theory',

AER Papers and Proceedings, May 1991, 365-370

Journal 0/ Economic Theory 57(2), 1992, special issue on Evolutionary Game Theory

Kandori, M., G J Mailath, and R Rob, 1993, 'Learning, Mutation, and Long Run Equilibria in Games', Econometrica 61(1), 29-56

Koza, J R., 1992, Genetic Programming: On the Programming 0/ Computers by Means

0/ Natural Selection, Bradford, MIT Press

Lane, D A., 1993, 'Artificial worlds and economics, Part 1', Journal 0/ Evolutionary Economics 3, 89-107

Marimom, R., McGrattan, E., and T J Sargent, 1990, 'Money as a Medium ofExchange

in an Economy with Artificially Intelligent Agents', Journal 0/ Economic Dynamics and Control14, 329-373

Maynard Smith, J., 1982, Evolution and the Theory 0/ Games, Cambridge University

Press

Roth, A R., Prasnikar, V., Okuna-Fujiwara, M., and S Zamir, 1991, 'Bargaining and Market behaviour in Jerusalern, Ljubljana, Pittsburgh, and Tokyo: An Experimental Study', American Economic Review 81(5), 1068-1095

Sefton, M and A Yavas, 1993, Abreu-Matsushima Mechanisms: Experimental Evidence, mimeo, University of Manchester

Straub, P., 1993, Risk Dominance and Coordination Failures in Static Games, mimeo,

Winston, P H., 1992, Artificial Intelligence, Third Edition, Addison-Wesley

Young, H P., 1993, 'The Evolution of Conventions', Econometrica 61(1),57-84

THE EMERGENCE AND EVOLUTION OF SELF-ORGANIZED COALITIONS

Arthur de Vany

Abstract This is a study of emergent economic order - order that is the result of human action but not human design A coordination garne is studied in whieh loeally eonneeted agents act without deliberation Their loeally optimal actions propagate through neigh- bors to others, and eoalitions form adaptively The garne is mapped onto a hypereube and a eonnectionist model is developed Simulation results show that the proeess is self- organizing and evolves to optimal or near-optimal equilibriaj the agent network eomputes the co re of the garne Its equilibria are path-dependent and the dynamie may become trapped on loeal optima: broken symmetry and noise promote evolution to global optima

1 Introduction

When economic institutions and the actions of agents are well-adapted,

a coherence and regularity is revealed in behavior at a macroscopic level

of observation and we say there is economic order Order is an emergent property of behavior when it cannot be deduced from the behavior of the agents alone We can describe such order and characterize it, but where does

it come from? If it is not cent rally designed, it must come from the way the agents interact with one another and their environment That is, order must come from sornething that was not evident in OUf understanding of how the agents operate As Hayek (1988) might say, emergent or spontaneous order is not the intent of any agent, it is '" " the result of human action but not human design."l Because it emerges from the way the agents act rat her than how they think and deliberate, emergent order is "boundedly rational"

One way to think about the problem of emergent order is to find ways

of characterizing the institutions and behavior that arise from the way the agents go about their business-this is the economics of self-organization

An economics of self-organization would study emergent order among agents

Ir am grateful to Dan Klein for pointing out that Adam Smith's eolleague Adam Ferguson, (1767, p 187), put it that way

25

M Gilli (ed.), Computational Economic Systems, 25-50

© 1996 Kluwer Academic Publishers

who act and adapt in a world they are always discovering; their knowledge

is "in the world, not their heads".2 This is very much different from an nomics based on agents whose knowledge is in their heads and who learn to model the world in order to find optimal solutions to the economic problems

eco-it presents Emergent order would come from the constraints of the world

in which the agents operate and from their exploratory and adaptive tions and the order they achieve might be far from optimal The quest ion examined here is: Is there emergent structure or order-for-free in a self-organized system? What are the properties of emergent behavior and what forms of organization can it produce? How closely related are patterns of self-organized order to patterns of optimal order? This paper is an attempt

ac-to engage these questions; it develops a model of self-organized coalitions

to investigate the emergence and evolution of economic organization The hard part in thinking about these questions is to find some way

of modelling agents and their interactions that doesn't give them all the information and foresight they need to solve the problem The problem

is to model agents who act on their own information and knowledge, but who are ignorant of the global picture There must be enough structure in the way the agents are interrelated to one another and the environment to lead them to a pattern of order that is recognizable to an observer of their actions on a macra scale Yet, the structure cannot be so strang as to make the solution evident or trivial for then we could immediately deduce it and there would be no sense of emergence in the behavior of the agents as a group

In this paper we develop a model of the emergence of self-organized coalitions among decentralized agents playing a network coordination game and use it to study the evolution of the coalitions on the path to equilibrium

2 The Coordination Problem

Consider a coordination problem where many agents seek their best choice

in an environment where the value of their choice depends on the choices made by some or all of the other agents Suppose there are two choices, say, technology a and technology band let the benefit to agent i, denoted

Vi( Cl, C2, ••• , C n ) depend on the choices of the other agents i =/;1, n If the

value function has a similar structure for all the agents and is maximized when they all make the same choice, this is strong form of network external-ity The externality creates an economy of scale which can be achieved by coordinating all the agents onto a common choice But, this can be difficult

to do (Katz and Shapiro, 1985)

2This is the research program earl Menger set out in his book on Problems in nomics and Sociology

Eco-SELF-ORGANIZED COALITIONS 27

Because agents act on their own, they may not internalize the efits of coordinating their actions Early choices may so condition later choices as to set the system onto a path leading to a local optimum of mixed choices rather than a global optimum of common choices (David, 1985) and (Arthur, 1989) Markets may evolve de facto standards over time or they may explicitly set standards through cooperative standard-setting bodies (Farrell and Saloner, 1986) A dominant firm (Bresnahan and Chopra, 1990), or a standard-setting authority (David and Greenstein, 1990) can help But, the solutions needn't be efficient if the dominant firm

ben-or authben-ority prben-ornotes technologies which turn out to be inferiben-or to others Consider a process where the agents set out to solve the problem in a de-liberate way Their task is to organize themselves optimally into coalitions

To do this, there are several difficulties that must be overcome The search space is large There are 2 n possible coalition structures when nagents make binary choices among two alternatives.3 When they are connected to kother agents and must coordinate with them, there are 2 k possible com-binations of signals among the agents and there are 2 2k possible decision rules (Kauffman, 1989)

The search space is complex Because returns to any one agent depend

on the choices of some or all of the other agents, the landscape of coalitional values is rugged and complex and contains many local optima The search process is path-dependent Because coalitions are recruited sequentially, they are constructed recursively and each later structure must build on the structure it inherits Earlier choices deform the value landscape and they condition and constrain later choices; thus, the coalition value landscape co-evolves with the coalitions as they form

To represent the problem, suppose there are three agents selecting one

of two equally effective choices 1 or O The agents do not uniformly prefer

1 or 0, but every agent uniquely prefers 1 if all the other players choose 1 and similarly for O The possible configurations of the choices are 23 = 8 coalition structures Let each agent's choice of A or B be represented by 1

or O The binary strings representing the possible out comes are:

Agent 1 Agent 2 Agent 3

3The situation is more complex if the agents are to form coalitions that are partitions

of the set of all n agents There are Bell number of these coalition structures, a number

far larger than in the binary case With 8 agents, there are 2 8 = 256 binary strings denoting coalitions of agents committed to 0 or 1, but there are 4041 coalition structures (De Vany, 1993)

In the first strueture, all the agents ehoose 0; in the last, all the agents choose 1; the other strings indicate mixed choices The first and last strings represent the globally optimal choices in which every agent's choice is com-patible with every other agent 's choice

Now consider a more interesting structure with interaction over a more complicated architecture Let each agent be connected to three neighbors and let there be eight agents in all The agents and their neighborhoods are shown in Figure 1 Each vertex represents an agent who must choose 1 or

O Agent 1 interacts with agents 2, 3, and 7 In turn, agent 2 interacts with agents 1, 4, and 8, and so on around the cube Even though agents 1 and

5 are not directly connected, they both interact with agents 3 and 7 and, hence, indirectly influence one another Thus, direct interactions between neighboring agents indirectly influence the choices made farther away; local interactions "chain" to extend their influence even though distant agents

do not interact directly

Figure 1 Three dimensional hypercube of eight agents each having three local neighbors

The three dimensional cube used here can be extended to larger games

on hypercubes By extending its dimension, the hypercube model can thus represent coordination among agents with network externalities of loeal or global scope.4 For example, if the values 0 and 1 represent dialects, then the distribution of Os and 1s over the vertices of the hypercube would show the dialects used by the agents in each region of the hypercube By altering the degree of the graph, it is possible to represent any kind of local or global interaction among the agents It can represent the compatibility of railroad gauges in a network of lines, apower network containing points using AC

or DC current, or traditions in the common law

4 Any tree can be imbedded in a hypercube (see Baldi and Baum (1986)), so the model can represent any game with arbitrary interaction among the agents

SELF-ORGANIZED COALITIONS 29

It is important to recognize the local structure of interaction between the 8 agents in the game considered here Every agent stands between three other agents, each of whom connects to two other agents, who connect with other agents, and so on When neighbors make compatible choices, their neighbors have an incentive to do so as wen and this spreads over the network This by no means insures compatibility throughout the space however, for two distant neighbors may make incompatible choices which their immediate neighbors propagate to other neighbors At some place in the space, these local choices will be incompatible When both choices are equally good, there are two compatible configurations of the cube; one in which all the choices are O's and one in which they are all 1 'so

3 Emergent Coordination

The study of emergent cooperation is in part inspired by the desire to count for empirical regularities of the equilibria observed and the paths leading there Brian Arthur (1989; 1990), has shown that adaptive pro-ces ses driven by increasing returns and non-linear feedback exhibit path dependence and sharp transitions to final states that may be non-optimal and he summarizes evidence of these characteristics in the adoption of tech-nologies The model developed here exhibits similar characteristics

ac-An emergent coalition structure forms through an undirected search over the complex space of possible coalitions It must form in a blind or non-directed way The structure must emerge from the process without be-ing its intention Forrest 's definition of emergent computation is a useful characterization of emergent economic order Forrest (1991) defines emer-gent computation in the following way:

a collection of agents who follow instruction;

interactions among the agents which form macroscopic patterns;

an ob server who interprets the patterns

Patterns that are interpretable as computations process information, which distinguishes them from noise or chaos An advantage of emergent compu-tation is its use oflow level agents (instructions) that are directly connected

to the problem domain An emergent computation bypasses the difficulty

of representing patterns at the emergent level These are the natural erties of an emergent economic order in which low level agents are coupled directly to their environment and interact to produce macroscopic patterns discernible to an observer-economist By studying economic order at the emergent level, we bypass the problem of representing how the macroscopic patterns are formed and we bypass the problem of how the agents learn and represent these patterns as part of their knowledge-their memory is in the

prop-world, and the intelligence they reveal is borne of their interactions at the emergent level, it is nowhere programmed in any agent

In our model, the agents are not learners, they are actors who learn the environment directly through the feedback they receivej they never learn the task by representing it internally as a problem to be solved Their col-lective "intelligence" emerges from the patterns that are produced through their interactions in the same way that order or computation emerges in a complex biological system (Kauffman, 1991), or in a computational system

in which the collective system is more computationally powerful than the elements (Forrest, 1991)

Since the agents do not deli berate and form their coalitions recursively, whatever patterns of order they achieve must come from the way they interact and from the dynamic of the process What properties of this process could so structure its dynamic as to produce order? There are several sources from which emergent order might arise

Symmetry breaking

It is understood that a model that frames the network coordination problem

as a symmetrie game may constrain the agents artificially There is great inertia against departing from a fully symmetrie state Several authors have noted how inertia may be overcome to promote evolution toward a solution

of the game (see Sugden (1986), Farrell and Saloner (1986), and Bresnahan and Chopra (1990)) Liebowitz and Margolis (1990) show that institutions, like patents, free training, rentals or discounts can overcome inertia by pro-moting early moves To generalize, the devices that overcome inertia break symmetry in some way We study four kinds of broken symmetry in our model that generalize the sources identified by these authors: asymmet-rie information, broken time symmetry, asymmetry of agent weights, and asymmetry in the form of bias

Adaptation and feedback

According to Day (1975), adaptive optimizing by boundedly rational agents

is a sequence of optimizations with feedback and a proper model of this cess would be a recursive model of adaptive man 's struggle with reality If the feedback is non-linear, then that may be a source of order In our model the agents build their coalitions recursively and are informed by feedback.5 The process is highly non-linear because there are thresholds for actions and non-linear responses It would ordinarily be argued by economists that thresholds limit response and would, therefore, be a limitation on coalition

pro-SIn a follow-on study, the agents will be capable of meta-adaptation (see Day (1975, p 9)) through refinement of the connections and strengths in their network of interactions

Noise

It is known that noise ean be a souree of order in physieal systems In the formation of erystals and in spin glass models, noise smooths the spaee of energy minima and inereases the number of paths to lower minima in the energy landseape Simulated annealing and genetie algorithms use noise to improve the seareh for optima On the other hand, eeonomic models tend to regard noise as a souree of ineffieieney Beeause noise inereases the number

of paths along whieh eoalitions evolve, it ean free them from loeal optima even while it opens the door to many more equilibria Noise might destroy uniqueness, but it might improve the efficieney of the adaptive se ar eh for effieient eoalitions

Reversals and partial commitments

Few deeisions are all or none ehoiees; most have degrees associated with them In the network coordination game, some agents eould use mixed strategies by committing partially to both teehnologies For example, in the evolution of eleetrie power there were multiphase systems and sys-tems that mixed AC and DC power (Hughes, 1983) Many companies have computer systems where more than one operating system is installed By making partial eommitments rat her than all-or-nothing ehoiees, the agents retain their adaptability and they ean respond to the evolution of the sys-tem more gracefully by ehanging only part of their system Leasing and renting equipment supports flexible eommitment We ean formalize partial commitments llsing Aubin's (1993) eoneept of a fuzzy eoalition

6This is analogous to a computer architecture, in this case, one that is organized in parallel Sah and Stigletz (1986) model the architecture of economic systems (Sah and Stiglitz, 1986)

4 A Connectionist Model of Coalitions

The problem of coordinating agents on a hypercube has a network sentation where each agent is a vertex in the hypercube and the weighted edges represent the existence and strength of the interactions between the agents The "architecture" of agent interaction, in the Sah and Stiglitz (1986) sense, or the size oftheir neighborhood in the sense ofEllison (1993),

repre-is given by the "wiring" of the network and the way actions flow over it Now, consider agents i and j who interact with one another They stand to gain when they coordinate their choices over the two alternatives We can express this as a constraint to be satisfied Wijaiaj, where Wij is a weight

If the weight is positive, then the agents maximize the constraint value when they both choose 1 When the weight is negative, they maximize by both choosing o By specifying the weights between the vertices, we have

a neural network model of economic interaction on the cube The weights represent the value of the economic interaction between vertices (agents) in the trading network Since agents do not trade with themselves, the weight

of the connection of i with itself is Wii = O Since agents trade with their immediate neighbors, these weights will be positive Since agents do not trade with one another if they are not neighbors on the cube, the weight between non-adjacent neighbors is zero

If we let each agent be a "neuron" in the network, then the action of the agent will induce astate in the neuron and that state will be communicated

to other neurons State information will flow to the other agents who are coupled with a particular agent and they will integrate this information with all the other information they receive to select an action The sequence of moves and signals will vary in the analysis; we will consider simultaneous moves and signals and we will consider sequential moves and signals In addition, we will consider a variation in the magnitude of the adaptive response to signals

Suppose each agent chooses an action ai E [0,1] i = 1, ,8 We shall consider situations where ai is (approximately) a boolean (0, 1) choice variable and situations where ai is a continuous variable in the unit interval

We shall also consider situations where ai is boolean, but in which the choice

of 0 or 1 is made with error or according to a prob ability distribution The first case, where ai is discrete, would represent an accurate choice between two alternatives that are exclusive In the second case, the level of

ai would represent i's partial commitment to 1 and of 1 - ai to O It could also represent the prob ability or proportion of time each of the choices are made The coalitions so formed are soft or, to use Aubin's (1993) term, they are "fuzzy" The third case would represent a choice ai made with a

"trembling hand" It also would represent tentative commitments later to