Springer artificial economics agent based methods in finance game theory and their applicationsspringer(2006)

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Preface

Agent-based Computational Methods applied to the fields of Economics, Management Sciences, Game Theory or Finance^ have received a great deal of academic
interest these past years, in relation with the Complex System approaches. Those
fields deal with the computational study of economies (at large), as complex adaptive systems, implying interacting agents with cognitive skills. One of the first use
of agent based models has been popularized by Axelrod, [1], in his theory of evolution of cooperation. In this early work, he used extensively computational simulations and methods in order to study strategic behaviour in the Iterated Prisoner's
Dilemma. This work is still influencing many researches in various scientific fields.
It has for instance been the foundations of a new approach of Game Theory, based
on computational ideas.
In the mid eighties, under the impulsion of the Santa-Fe Institute, and especially

Christopher Langton, [3], a new field of research, called Artificial Life (AL), has
emerged. The idea of AL was to mimic real life under its various aspects to understand the basic principles of life. This has lead to encompass wider ideas such as complexity, evolution, auto-organisation and emergence. All concepts induced by these
approaches have influenced social scientists among others. Following these initial
attempts to mix computational approaches and social sciences, for instance among
the pioneering works using Agent-based Computational Economics in finance, one
can refer to the Artificial Stock Market, [4]. This model, based on bounded rationality and inductive reasoning, [5], is one of the first allowing correct simulations of
real world stock market dynamics. This work has been done by people coming fi-om
various scientific fields (Economics, Game Theory, Computer Science and Finance).
All these approaches intensively use computer simulation as well as artificial intelligence concepts mostly based on multi-agents systems. In this context, some of
the most used models come from Game Theory. Therefore, Agent-based Computational Simulations is more and more an important methodology in many SocialSciences. It becomes now widely used to test theoretical models or to investigate
their properties when analytical solutions are not possible.
Le. ACE, ABMS, COT, ACF


VI

Preface

Artificial Economics'2005 is one attempt to gather scientists from various horizons that directly contribute to these fields. The book you have in hands reproduces
all the papers that have been selected by the programme committee. AE'2005 aims
and scopes were to present computer-science based multi-agent methodologies and
tools with their applications to social-scientists (mainly people fi'om economics and
the management sciences,) as well as to present uses and needs of multi-agent based
models and their constraints, as used by these social scientists, to computer scientists. Additionally, it has been a great occasion to favor the meeting of people and
ideas of these two communities, in order to be able to construct a much structured
multi-disciplinary approach.
For its first edition. Artificial Economics has presentend recent scientific advances in the fields of ACE, ABMS, CGT, ACF, but has also been widely open to
methodological surveys. Two prestigious invited-speakers have proposed analysis
and surveys on major issues related to Artificial Economics topics.
Cristiano Castelfranchi, from the Institute of Cognitive Sciences and Technologies, University of Siena, has developed a talk on "The Invisible (Left) Hand" . For

Pr. Catsellfanchi, Agent-based Social Simulation will be crucial for the solution of
one of the most hard problems of economic theory : the spontaneous organization of
a dynamic social order that cannot be planned but emerges out of intentional planning agents guided by their own choices. This is the problem that Hayek assumes
to be the real reason for the existence of the Social Sciences. In his talk, Pr Castellfranchi has examined the crucial relationships between the intentional nature of the
agents * actions and their explicit goals and preferences, and the possibly unintended
finality or function of their behavior [He] argues in favor of cognitive architectures
in computer simulations and proposes some solutions about the theoretical and functional relationships between agents' intentions and non-intentional purposes of their
actions.
[For him,] social order is not necessarily a real order or something good and desirable for the involved agents; nor necessarily the best possible solution. It can be bad
for the social actors against their intentions and welfare although emerging from
their choices and being stable and self-maintaining. Hayek's theory of spontaneous
social order andElster's opposition between intentional explanation and functional
[are also] criticized.
Robert Axtell, from The Brookings Institution, Washington DC and the Santa Fe
Institute, has emphasized a very stimulating reflection on "Very Large-Scale MultiAgent Systems and Emergent Macroeconomics". For Dr. Axtell, the relatively few
applications of agent-based computing to macroeconomics retain much of the representative agent character of conventional macro. [Robert Axtell] points out that
hardware developments will soon make possible the construction of very large scale
(one million to 100 million agent) models that obviate the need for representative
agents -either representative consumers, investors or single-agent firms.
[He also argues] that the main impediment to creating empirically-relevant artificial
agent economies on this scale is our current lack of understanding of realistic behavior of agents and institutions. [He] claims that this software bottleneck-what rules to
write for our agents ?-is the primary challenge facing our research community.


Preface

VII

Artificial Economics 2005 has been a two-days symposium. 20 papers have been
selected among roughly 40 submitted extended abstracts. The reviewing process has

been blind, and each paper has been reviewed by three referees. Space and time limitations are the reasons why no more papers have been accepted, although many of
the rejected submissions were really interesting. Nevertheless, the choice of avoiding parallel sessions has been made to favor interactions between participants. The
contributions have been gathered in six sessions, each of them devoted to one of
the following topics: Artificial Stock Markets, Learning in models, Case-Studies and
Applications, Bottom-Up approaches. Methodological issues and Market Dynamics.
This book is organized according to the same logic.
Artificial Economics'2005 as well as this book is the result of the combinatory
efforts of:
Frederic AMBLARD - Universite de Toulouse 1, France
Gerard BALLOT - ERMES, Universite de Paris 2, France
Bruno BEAUFILS - LIFE, USTL, France
Paul BOURGINE - CREA, Ecole Polytechnique, France
Olivier BRANDOUY - CLAREE, USTL, France
Charlotte BRUUN - Aalborg University, Danemark
Jose Maria CASTRO CALDAS - ISCTE, DINAMIA, Portugal
Christophe DEISSENBERG - GREQAM, France
Jean-Paul DELAHAYE - LIFE, USTL, France
Jacques FERBER - LIRMM, Universite de Montpellier II, France
Bernard FORGUES - CLAREE, USTL, France
Wander JAGER - University of Groningen, The Netherlands
Marco JANSSEN - CIPEC, Indiana University, USA
Alan KIRMAN - GREQAM, France
Philippe LAMARRE - LINA, Universite de Nantes, France
Luigi MARENGO - DSGSS, Universita di Teramo, Italy
Philippe MATHIEU - LIFE, USTL, France
Denis PHAN - Universite de Rennes I, France
Juliette ROUCHIER - GREQAM, France
Elpida TZAFESTAS - National Technical University of Athens, Greece
Nicolaas VRIEND - Queen Mary University of London, United Kingdom
Bernard WALLISER - CERAS, ENPC, France

Murat YILDIZOGLU - IFREDE-E3I, Universite Montesquieu Bordeaux IV, France
We also want to thank Rene Mandiau from the Universite of Valenciennes, as
he has been a very precious additional referee. Let all of them be thanked for their
participation in this scientific event, that surely appeals for fixrther similar manifestations.

Villeneuve d'ascq,
June 2005

Philippe Mathieu
Bruno Beaufils
Olivier Brandouy


VIII

Preface

References
1. R. Axelrod and W.D. Hamilton (1981), The evolution of cooperation. Science, pp. 13901396
2. R. Axelrod (1984), The evolution of cooperation, Basic Books
3. C. Langton (1995), Artificial Life, an overview. The MIT Press
4. R.G. Palmer and W.B.Arthur and J.H. Holland and B. LeBaron and R Tayler (1994),
Artificial Economic Life : A Simple Model of a Stockmarket, Physica D, vol 75, pp. 264274
5. B. Arthur (1994), Inductive Reasoning and Bounded Rationality: the El-Farol Problem,
American Economic Review, vol 84, pp. 406-417


Contents

Artificial Stock Markets

Time Series Properties from an Artificial Stock Market with a Walrasian
Auctioneer
Thomas Stumpert, Detlef Seese, Make Sunderkotter
Market Dynamics and Agents Behaviors: a Computational Approach
Julien Derveeuw
Traders Imprint Themselves by Adaptively Updating their Own Avatar
Gilles Daniel, Lev Muchnik, Sorin Solomon

3
15
27

Learning in Models
Learning in Continuous Double Auction Market
Marta Posada, Cesdreo Hernandez, Adolfo Lopez-Paredes
Firms Adaptation in Dynamic Economic Systems
Lilia Rejeb, Zahia Guessoum
Firm Size Dynamics in a Cournot Computational Model
Francesco Saraceno, Jason Barr

41
53
65

Case-Studies and Applications
Emergence of a Self-Organized Dynamic Fishery Sector: Application to
Simulation of the Small-Scale Fresh Fish Supply Chain in Senegal
Jean Le Fur
Multi-Agent Model of Trust in a Human Game
Catholijn M. Jonker, Sebastiaan Meijer, Dmytro Tykhonov, Tim Verwaart

79
91


X

Contents

A Counterexample for the Bullwhip Effect in a Supply Chain
Toshiji Kawagoe, Shihomi Wada

103

Bottom-Up Approaches
Collective Efficiency in Two-Sided Matching
Tomoko Fuku, Akira Namatame, Taisei Kaizouji

115

Complex Dynamics, Financial Fragility and Stylized Facts
Domenico Delli Gatti, Edoardo Gaffeo, Mauro Gallegati, Gianfranco
Giulioni, Alan Kirman, Antonio Palestrini, Alberto Russo

127

Noisy Trading in the Large Market Limit
Mikhail Anufriev, Giulio Bottazzi

137

Emergence in Multi-Agent Systems: Cognitive Hierarchy, Detection, and
Complexity Reduction part I: Methodological Issues
Jean-Louis Dessalles, Denis Phan

147

Methodological Issues
The Implications of Case-Based Reasoning in Strategic Contexts
Luis R. Izquierdo, Nicholas M. Gotts

163

A Model of Myerson-Nash Equilibria in Networks
Paolo Pin

175

Market Dynamics
Stock Price Dynamics in Artificial Multi-Agent Stock Markets
A. O,I Hoffmann, S.A. Delre, J.H. von Eije, W, Jager

191

Market Failure Caused by Quality Uncertainty
Segismundo S. Izquierdo, Luis R. Izquierdo, Jose M. Galdn, Cesdreo IIerndndez203
Learning and the Price Dynamics of a Double-Auction Financial Market
with Portfolio Traders
Andrea Consiglio, Valerio Lacagnina, Annalisa Russino

215

How Do the Differences Among Order Distributions Affect the Rate of
Investment Returns and the Contract Rate
Shingo Yamamoto, Shihomi Wada, Toshiji Kawagoe

227


List of Contributors

Mikhail Anufriev
S.Anna School for Advanced Studies
Italy
Jason Barr
Rutgers University
USA
Giulio Bottazzi
S. Anna School for Advanced Studies
Italy
Andrea Consiglio
Universita degli Studi di Palermo
Italy
Gilles Daniel
University of Manchester
UK

Jean Louis Dessalles
ENST
France

J.H. von Eije
University of Groningen
The Netherlands
Tomoko Fuku
National Defense Academy
Japan
Edoardo Gaffeo
University of Trento
Italy

Domenico Delli Gatti
Catholic University of Milan
Universita Politecnica delle Marche
Italy

Jose M. Galan
University of Burgos
Spain

S.A. Delre
University of Groningen
The Netherlands

Mauro Gallegati
Universita Politecnica delle Marche
Italy

Julien Derveeuw
Universite des Sciences et Technologies
de Lille

France

Gianfranco Giulioni
Universita Politecnica delle Marche
Italy


XII

List of Contributors

Nicholas M. Gotts
The Macaulay Institute
UK

Valerio Lacagnina
Universita degli Studi di Palermo
Italy

Zahia Guessoum
Universite de Reims
Universite de Paris VI
France

Jean Le Fur
Institut de Recherche pour le
Developpement
France

Cesareo Hernandez

University of Valladolid
Spain

Adolfo Lopez-Paredes
University of Valladolid
Spain

A.O.I. Hoffmann
University of Groningen
The Netherlands

Sebastiaan Meijer
Wageningen UR
The Netherlands

Luis R. Izquierdo
The Macaulay Institute
UK

Lev Muchnik
Bar Ilan University
Israel

Segismundo S. Izquierdo
University of Valladolid
Spain

Akira Namatame
National Defense Academy
Japan

W. Jager
University of Groningen
The Netherlands

Antonio Palestrini
Universita di Teramo
Universita Politecnica delle Marche
Italy

Gatholijn M. Jonker
Radboud University Nijmegen
The Netherlands

Denis Phan
Universite de Rennes I
France

Taisei Kaizouji
International Christian University
Japan

Paolo Pin
Universita Ca' Foscari di Venezia
Italy

Toshiji Kawagoe
Future University-Hakodate
Japan

Marta Posada
University of Valladolid
Spain

Alan Kirman
EHESS
Universite d'Aix-Marseille III
France

Lilia Rejeb
Universite de Reims
Universite de Paris VI
France


List of Contributors
Annalisa Russino
Universita degli Studi di Palermo
Italy

Thomas Stiimpert
University of Karlsruhe
Germany

Alberto Russo
Universita Politecnica delle Marche
Italy

Malte Sunderkotter
University of Karlsruhe

Germany

Francesco Saraceno
Observatoire Fran^ais des Conjonctures
Economiques
France
Detlef Seese
University of Karlsruhe
Germany
Sorin Solomon
Hebrew University of Jerusalem
Israel
ISI Foundation
Italy

Dmytro Tykhonov
Radboud University Nijmegen
The Netherlands
Tim Verwaart
Wageningen UR
The Netherlands
Shingo Yamamoto
Future University-Hakodate
Japan
Shihomi Wada
Future University-Hakodate
Japan

XIII

Artificial Stocl< iVIaricets


Time Series Properties from an Artificial Stock
IVIarl^et with a Walrasian Auctioneer
Thomas Stumpert, Detlef Seese, and Malte Sunderkotter
Institute AIFB, University of Karlsruhe, Germany,
{stuempert|seese}@aifb.uni-karlsruhe.de
Summary. This paper presents the results from an agent-based stock market with a Walrasian
auctioneer (Walrasian adaptive simulation market, abbrev.: WASIM) based on the Santa Fe
artificial stock market (SF-ASM, see e.g. [1], [2],[3],[4],[5]). The model is purposely simple
in order to show that a parsimonious nonlinear framework with an equilibrium model can
replicate typical stock market phenomena including phases of speculative bubbles and market
crashes. As in the original SF-ASM, agents invest in a risky stock (with price pt and stochastic
dividend dt) or in a risk-free asset. One of the properties of SF-ASM is that the microscopic
wealth of the agents has no influence on the macroscopic price of the risky asset (see [5]).
Moreover, SF-ASM uses trading restrictions which can lead to a deviation from the underlying
equilibrium model. Our simulation market uses a Walrasian auctioneer to overcome these
shortcomings, i.e. the auctioneer builds a causality between wealth of each agent and the
arising price function of the risky asset, and the auctioneer iterates toward the equilibrium.
The Santa Fe artificial stock market has been criticized because the mutation operator for
producing new trading rules is not bit-neutral (see [6]). That means with the original SFASM mutation operator the trading rules are generalized, which also could be interpreted as a
special market design. However, using the original non bit-neutral mutation operator with fast
learning agents there is a causality between the used technical trading rules and a deviation
from an intrinsic value of the risky asset in SF-ASM. This causality gets lost when using a
bit-neutral mutation operator. WASIM uses this bit-neutral mutation operator and presents a
model in which highfluctuationsand deviations occur due to extreme wealth concentrations.
We introduce a Herfindahl index measuring these wealth concentrations and show reasons for
arising of market monopolies. Instabilities diminish with introducing a Tobin tax which avoids

that rich and influential agents emerge.

1 Introduction and Model Description
One main focus of agent-based financial markets is to find possible reasons to explain
phenomena observed on real-world markets v^hich cannot be explained with classical
equilibrium models (see e.g. [8]). Among numerous agent-based financial markets,
the Santa Fe artificial stock market is one of the pioneering and thus probably most
well-knovm market model. The Walrasian adaptive simulation market (WASIM) is


4

Thomas Stiimpert et al.

based on the Santa Fe artificial stock market with the purpose to improve it. The market design of SF-ASM allows no connection between the price function of the risky
asset and the wealth of the agents (see [5]), i.e. stock price dynamics are modelled to
be independent from the influence of each agent's wealth. In the Walras simulation
market we use a Walrasian auctioneer to build a causality between the equilibrium
model and the wealth of agents. Operating each period, the Walrasian auctioneer permits interaction of agents on a microscopic level that has influence on the stock price
on a macroscopic level. Agents try to forecastfixtureprices and dividends, and then
combine these forecasts with their own preferences for risk and return. The market
consists of stock shares and a risk-free asset. WASIM uses a bit-neutral mutation operator (see [6]) and presents a model in which highfluctuationsoccur due to extreme
wealth concentrations.
1.1 The Market and the Market Structure

The underlying basic model of SF-ASM and WASIM is an equilibrium model, i.e. a
model where supply and demand are balanced and the market is cleared each period.
The market consists of iV heterogeneous agents. Agents invest in a risky stock (with
price pt and dividend dt) or a. risk-fi*ee asset. The demand function of all agents is
equal to the sum of all stock shares on the market:

N

J2'^i,t = N

(1)

i=l

where xu denotes the number of shares agent i demands to possess in t and N
denotes the absolute number of shares on the market (which is equal to the number
of agents). In the equilibrium model agents can be short sellers, i.e. they can sell
more stock shares than they possess. Furthermore agents can buy more stocks than
they can afford, because the equilibrium model is independent of each agent's wealth
function. To satisfy the equilibrium condition, it is even possible that some agents
must buy a negative amount of shares of the risk-free asset. The risk-free asset is
supported infinitely and has afixedreturn r. The stock has a stochastic dividend dt,
dt = d + p'{dt-i

-d)-{-£t

(2)

(with J = 10, p = 0.95, €t = A/'(0,cr^)). In each period agents evaluate their
portfolio and use a market structure vector to estimate how much stocks they want
to buy or sell in the next period. At the beginning of each period the market structure
vector Zt is calculated,
Zt : {pt,...,Pt-k^dt,...,dt-n)^

{0,1} X ... X {0,1},

with/c > 0, n > 0. (3)

The market structure vector Zt identifies the basic technical and fundamental market
state, e.g. Zt signals the relative strength (technical market state) and Zt signals
whether a stock is fundamentally overvalued resp. undervalued (fundamental market
state), seefigure1. In the following, we will call bits 1-6 of Zt fundamental bits, bits


Time Series Properties from an Artificial Stock Market with a Walrasian Auctioneer
Bit

Task

1-6
7
8
9
10
11
12

stock*retum/dividend > 0.25, 0.5,0.75, 0.875,1.0,1.125
pt > 5-period moving average
pt > 10-period moving average
Pt > 100-period moving average
Pt > 500-period moving average
control bit
control bit
Fig. 1. Market structure vector

7-10 technical bits and bits 11-12 test/control bits. Bits 11 and 12 are constantly
set on (1) respectively off (0). Bits 7 — 10 show the recent price trend. The market
structure vector builds the basis for the buy/sell orders of the agents. In order to
determine the attractiveness of the risky asset the return of the stock resulting from
the dividend process (dt/pt) is compared to the return of the risk-free asset r. This
leads to

i = !:^=p..r-M.

(4)

Pt

If the ratio is greater (less) than 1, the stock is overvalued (undervalued) for a riskneutral investor. Setting equation (4) equal to 1 and solving it for pt leads to the
intrinsic or fundamental value of the risky asset pt = p^ = ^ for a risk-neutral
investor. The dividend process follovv^s an autoregressive process of order 1, i.e. an
AR(l)-process. It can easily be shovm that the constant, d, absorbs the mean, i.e. the
expected value of dt+k given the set of information available at time t is d:
Et[dt-^i] = E t [ J + p{dt -d)f
st^i]
= J + pdt
Et[dt^2] =Et[d_+ p{dt+i -d)+ et+2]
= Et[d + p^dt + pst^i + et+2] = d + p'^dt
Et[dt+k] = --^

=d + p^dt''^^

^^^
d.

Equations (4) and (5) imply that in both WASIM and SF-ASM the expected intrinsic
value is constant, but not necessarily its second moment.
1.2 The Agents and their Prediction Rules
In this section we describe how an agent chooses between different rules to calculate
his demand on the basis of a mean-variance maximizer and the market structure vector. For calculation of the demand ftmction Xi^t of an agent i, we introduce forecasting rules. Each agent estimates the expected return of investing in the stock under risk
and makes his buy/sell order to a predefined price. The remaining money is invested
in the risk-free asset. A forecasting rule is a 3-tuple consisting of condition part, a
forecasting part and a fitness measure, the prediction rule r (r = 1,2,3,..., 100) of
agent i is defined as follows:


6

Thomas Stumpert et al.

PRi^r =

(rni,m2,...,mi2,a,hj)

where rrin € { 0 ; 1 ; # } with n = 1,2,3, ...,12 denotes the condition part, a G
[0.7,1.2] and b G [—10,19] denote the forecasting part and / denotes the fitness
measure. The forecasting part consists of m symbols (here 12), m is equal to the
number of market situation bits of Zf. Each m^, k G [1,12], has the values 0, 1
or #. The resulting pattern (mi, m2, ms,...., 77112) fi'om the condition part is compared to market situation bits, # represents the don't-care-symbol i.e. both bits 0 and
1 are possible (# generalizes the prediction rule). The market situation bits evaluate
both the attractiveness of the stock compared to the risk-free asset and the current
stock price compared to its moving averages. A rule is called active in case that the
comparison is true, i.e. if the symbols rrik fi*om the condition part match with all
market situation bits bitk. Forecasting is a linear combination of two randomly chosen parameters a, b, the current price pt and dividend dt. From a and b the price and
dividend is estimated as follows:

Et\ptA-i + dt+i] = a{pt + dt) + b

.

(6)

The fitness of a rule should be high, if the rule has many "#".
Now define the fitness of a forecasting rule,
eli^^ = (1 - ^)e2_i,i,r + 0[{pt+i + dt+i) - Et,i,r{Pw

+ dt+i)]^

.

(7)

This leads to the fitness fixnction ft,i,r'
ft,i,r := M - eli^, - Cs,

(8)

where
/t,i,r
M
C
s
0
^t,i,r

= fitness of rule r of agent i in period t,

= constant scaling factor, e.g. 0,
= weight of s (here: C = 0.005),
= number of symbols G {0; 1} in the condition part, (i.e. unequal to # ) ,
= speed of change of the fitness function, between 0.. 1,
"" variance .

Then the expected price is calculated as follows:
Ei^t{Pt+l + dt+l) = Ciript + dt) + br

.

(9)

From all active rules the rule with the highest fitness is chosen for forecasting the
fixture stock price. After calculating the expected price, the agent puts his buy/sell
order:
^
^ _ %APt^i + dt^i) - (1 + r)pt
where xi^t denotes the number of shares the agent wants to possess. 7 is the global
risk aversion, which is constant for all agents (7 = 0.5) and df^ denotes the empirical


Time Series Propertiesfroman Artificial Stock Market with a Walrasian Auctioneer

7

variance of the forecast. The agent's new wealth Wi^t+i from investing in stock
shares or in the risk-free asset is:
Wi^t+i = Xi^t{pt+i + dt+i) + {1 + r){Wi,t -PtXi,t)'

(11)

Then the agent updates the fitness / of all active rules, i.e. agents buy or sell on
the basis of those rules who perform best, and confirm or discard rules according
to their performance. When some agents are bankrupt, they leave the market and
are replaced by new agents, who bring new money into the market. A genetic algorithm (see [9],[2] and [4]) enables agents who started with a rule set containing
bad performing rules to produce better performing ones (i.e. new a, b and / are generated). The genetic algorithm of an agent of WASIM is activated each k periods,
k e [200, ...,300]. Then 20% of the forecasting rules with the lowest fitness are replaced (for details of the genetic algorithm see [4]).
To overcome the limitation of the equilibrium model we introduce the concept of a
Walrasian auctioneer which produces a dependency between the basic market equilibrium model and the wealth of the agents.
1.3 The Walrasian Auctioneer
The Walrasian auctioneer needs the demand for shares of each agent to calculate the
stock price iteratively. The basic underlying model of SF-ASM is the equilibrium
model described in the previous section with additional trading restrictions, which
may lead to situations when the equilibrium assumption is not fulfilled. Expanding
the equilibrium model the auctioneer of WASIM both takes into account the wealth
of agents and trading restrictions (see algorithm below).
The Walrasian auctioneer is an iteration method towards an equilibrium point (see
[8]). Let p be the initial price for the auctioneer and let Cu := Wi^t-i — xi^t-i • Pt-i
be the free available cash of agent i, where Cu is a fiinction of the past paid dividends, the risk-free return r and the initial wealth {Wio = 100) of agent i. Then each
agent i uses Ei^t{pt-\-i + <^t+i) to calculate his demand xu for all possible prices.
The auctioneer executes the following algorithm and iterates the calculation of the
new price towards the equilibrium price up to the given resolution £ > 0 (in WASIM
s = 10-4):
1. Start at any price p > 0, e.g. the price of the previous period.
2. Calculate to price p the demand for shares Xi of agent i (see equation (10)),
where xa satisfies the trading restrictions:
• if (xit < 0), then xu = 0, i.e. do not allow short selling,
• if (xit > maxown)^ then xu = maxown, i-e. the maximal numbers of shares
an agent possesses is restricted to max own3. Calculate the number of shares to be traded.

8

Thomas Stumpert et al.
Axit = {xit - Xi^t-i),

4.
5.
6.
7.

where xu denotes the number of shares agent i demands to possess and Xi^t-i
is the number of shares the agent currently possesses.
The agent does not sell more shares than max trader
if Axit < -maxtrade^ then Axu = -maxtrade •
The agent does not buy more shares than he can afford:
if (Axit • p > Cit), then Axu = ^ .
The agent does not buy more shares than max trade,
if Axit > maxtrade, then Axu = maxtrade •
Excess supply (resp. excess demand) leads to a decreasing price p with the step
width Ap (resp. increasing). To improve the convergence speed, we rescaled
dynamically,

N

N

where ^ Xi^t is the aggregation of Axu over all agents i and represents the
excess demand (resp. supply). The higher the excess is the larger the step width

gets. This leads to a new price p = poid + ^P8. If the sum of supply and demand (i.e. negative supply) is under the threshold e,
then the equilibrium
up to the precision e, else go to step 2.
2 Specification
ofprice
Testis found
Criteria
To analyze the influence of heterogeneous agents to the occurrence of interesting
market structure, we will define in the next subsection a Herfindahl index that measures the wealth concentration. After that we divide a simulation run into equidistant
market phases, in order to analyze the arising of bubbles and market crashes resp.
price movements parallel to the intrinsic value of the stock.
2.1 Herfindahl Index for Measuring Market Microscopic Characteristics
In the equilibrium model the price results from the aggregated expectations of the
agents about the future price and dividend. Additionally, in WASIM the price of the
next period is calculated dependent on the wealth of the agents. For measuring wealth
concentrations, we define the wealth ratio p-u) as a Herfindahl index:

Pwit) = '-'

. _ 1

,

p^(i)e[0,i]

(13)

N
N

where W/^ = 2 Wj^t denotes the cumulative wealth of all agents in t, Wu is the
i=l

wealth of a single agent. Remember, that each agent chooses among 100 rules and


Time Series Properties from an Artificial Stock Market with a Walrasian Auctioneer

9

the investigation of microscopic influences on prices without the above ratio is like
looking for a needle in a haystack.
2.2 Dividing a Simulation Run into IVIarket Phases
A simulation run in WASIM consists of / = 150000 periods. To analyze market
phases of overvaluation and undervaluation, we segmented the simulation run in J
equidistant market phases of length n = 7500, where j e [1, J]. In each market
phase [tj,..., tj +n], we logged the following key numbers and metrics: The average
market price ptj = l/n Y^^^^. Pk (similarly the average fundamental price Ptj), the
lowest market (fundamental) price Pminj (respectively Pmin,j)> ^^^ highest market
(fundamental) price pmax,J (respectively pj^^^^). In order to easily evaluate the order
of market price deviations from the intrinsic value of the risky asset, we defined the
following relative volatility measure, Pa^j:
2
r<^^,J ~

2
2

'

where cr^(^) . = ;^^ Y^k^t^- (Pk ~Pt])'^ ^^ ^^e market (fundamental) price variance
in interval j . Furthermore, we computed the average wealth ratio py^^j for market
phase j and the respective maximum py^ rnax j 3^nd minimum pyj rnin j • Ari important
dimension to characterize a market situation in terms of the agent's risk preferences
and extreme market interactions is the deviation of the market price pt from the
known fundamental price p^ = ^ . We capture the degree of over-/ undervaluation
of the risky asset in a market phase by logging the relative frequency fij of market
price deviation in an interval / in market phase j . For instance, if /(o.85;0.9],j =
0.15, in 15% of all trading periods in market phase j the market price was greater
than 0.85 • Pi and less than or equal 0.9 • p^. Classifying market price deviation in
10 different classes, we get a histogram-like logging of the price deviation. Ntbuj,
Nfbitj, ^cbitj denote the aggregated number of technical bits, fundamental bits and
control (or test) bits which are marked as active in interval j (see section 1.2).

3 Empirical Results
We used two different scenarios and ran each scenario with 25 experiments. Each experiment leads to different trajectories of the dividend process and the price process.
Parameter configurations included a fast setting of the genetic algorithm (gamin =
200;
9^max — 300, for details see [4]), no trading restrictions (jnaxown —
maxtrade = N) and dividend processes with p = 0.95. All other parameters are set
as in the original Santa Fee artificial stock market (see [2]). The fast genetic algorithm guarantees that agents can easily adapt their strategy to a change in the market
structure. Prohibition of trading restrictions enables single agents to build up monopolies. In the following subsections k denotes the number of the simulation run and j
denotes the number of the market phase in a simulation run (j = 1,..., 20).


10

Thomas Stumpert et al.

3.1 Scenario 1: Market Characteristics without Taxes and Trading

Restrictions
For the first scenario we used the plain market setting without trading restrictions
{maxtrade = N^TTiaXown = N) and without taxation. There are three different market conditions which can be observed in different phases (see table 2 with
k — 1,2,..., 10): Overvaluation (with /(i.03,oo])» undervaluation (with /(q,o.97])> and
a synchronous run of the market price with the fundamental price (with /(o.97,1.03])The sum of all / is chosen to be equal or less than 1, e.g. /(o,o.85] = 0.137 means
that at least 13.7% of time periods the price pt of the risky asset deviates from p*
by at least -15%. Over all simulations, overvaluation occurs more rarely than undervaluation. The reason for this is that the fundamental price reflects the valuation of a
risk-neutral investor, whereas the market price is determined by supply and demand
of agents who use a risk-averse utility function. One exception of this market behavior can be observed, if supply and demand would result in a negative market price
(which is prohibited and set equals to zero). Figure 2 and figure 3 show the price
evolvement and the wealth ratio for the simulation run with k= 1 over 150000 periods. Highfluctuationsoccur with a wealth ratio higher than 0.7. As assumed before
a trend is visible that the wealth ratio increases within the simulation run because
the parameter setting empowers agents to build up monopolies. Table 1 shows the
calculated key ratios for the different market phases in this simulation run.
Table 1. Simulation run without taxes and trading restrictions, k = 1
3
1
2
3
4
5
6
7
8
9
10
11
12
13
14

15
16
17
18
19
20

pt
99.86
84.31
99.43
105.17
87.92
98.87
98.70
103.28
96.38
99.75
100.43
95.02
94.31
100.22
96.12
98.78
99.89
103.56
102.69
97.61

Pmax

Pmin

Pt

105.92
105.91
266.42
214.35
183.10
148.83
116.98
196.98
144.40
137.38
132.07
139.31
168.51
149.16
140.78
137.46
135.42
155.02
161.37
141.82

66.64 100.15
21.77 100.28
0
100.16

0
100.28
0
99.99
0
100.08
85.65 99.71
0
99.88
0
99.56
0
99.77
72.13 100.27
0
100.00
0
100.08
11.04 100.08
0
100.30
85.16 100.06
80.15 99.93
72.65 99.87
0
100.15
0
99.73

P^ax

P'^in

P
P^

pw,max

108.16 91.06 -1.05 0.041 0.046
108.40 92.31 -70.83 0.141 0.350
108.62 92.81 -125.49 0.185 0.928
110.2193.57 -378.45 0.512 0.974
107.65 92.17 -188.10 0.749 0.998
108.89 92.07 -12.35 0.710 0.952
108.01 91.02 0.03 0.712 0.738
108.70 91.03 -95.85 0.703 0.960
107.76 91.57 -63.12 0.799 0.997
108.25 91.40 -57.52 0.804 0.977
108.43 92.81 -2.67 0.789 0.833
108.00 91.88 -30.98 0.853 0.988
109.27 91.13 -42.76 0.849 0.989
107.93 92.02 -28.93 0.833 0.987
108.73 92.93 -62.82 0.834 0.999
110.38 92.79 -0.99 0.817 0.867
109.37 89.57 -2.09 0.743 0.811
108.05 91.27 -11.10 0.725 0.771
110.41 91.35 -49.55 0.795 0.972
107.76 89.50 -44.84 0.893 0.992

piv,min
0.04
0.046
0.057
0.128
0.490
0.647
0.673
0.629
0.644
0.713
0.777
0.758
0.739
0.797
0.777
0.778
0.700
0.671
0.714
0.799

Ntbit

Nfbn

63650
77211
64999
66113

53634
59703
77173
54080
51209
50515
48603
45531
51851
56867
47887
37808
37322
38855
41407
58167

96524
74296
99280
106583
88247
76371
54297
84591
97735
79980
59933
72232
91738

101758
88656
56635
43796
29315
66268
81653

Ncbit
19700
21619
13894
19655
16595
10980
2786
3653
16928
9720
6375
11240
14967
10480
4295
6249
4542
5222
5034
13238

3.2 Scenario 2: Market Characteristics under Taxation
In order to reduce high fluctuations we introduced a Tobin tax (see table 2 with
k = 11, ...,20 and table 3), which significantly lead to avoidance of crashes, e.g.


Time Series Properties from an Artificial Stock Market with a Walrasian Auctioneer

11

Fig. 2. [Up] Price pt of the risky asset ioxk — I and j = 1,2,..., 20 without taxes

Fig. 3. [Down] Wealth ratio pw for /c = 1 and j = 1,2,..., 20 without taxes
reduction of price volatility and a price movement of the risky asset parallel to its
intrinsic value. Each agent has to pay taxes at a rate of 5% on his total wealth, i.e.
the sum of the stock value and the value of the risk-free asset. The tax is payable
every 100 trading periods into a tax pool. The funds are repayed in equal parts to
all agents in the period following after the tax payment. Thus, the tax functions as
a simple wealth redistribution system on the market. Prohibition of wealth concentrations reduces longer periods with high spreads between the market price and the
fundamental price, see p^2 in table 2. Single price peaks like those observable in the
scenario without taxation do not occur any longer, e.g. Pmin ^ 0. In order to verify
these assumptions statistically it is necessary to use a significance test. Before using
a F-test, we have to do some calculations. First of all, we want to consider the variance of the fundamental price. The conditional variance of the price process under
the condition that the process is known up to the previous period is can be computed
as follows: