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

Contents Artificial Stock Markets Time Series Properties from an Artificial Stock Market with a Walrasian Auctioneer Thomas Stumpert, Detlef Seese, Make Sunderkotter 3 Market Dynamic

Preface

Agent-based Computational Methods applied to the fields of Economics, ment 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 adap-tive systems, implying interacting agents with cognitive skills One of the first use

Manage-of agent based models has been popularized by Axelrod, [1], in his theory Manage-of lution of cooperation In this early work, he used extensively computational simu-lations 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

evo-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 under-stand the basic principles of life This has lead to encompass wider ideas such as com-plexity, 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 rational-ity 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 in-telligence concepts mostly based on multi-agents systems In this context, some of the most used models come from Game Theory Therefore, Agent-based Compu-tational Simulations is more and more an important methodology in many Social-Sciences 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 zons 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 scien-tists 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

hori-For its first edition Artificial Economics has presentend recent scientific vances 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

ad-Cristiano Castelfranchi, from the Institute of Cognitive Sciences and

Technolo-gies, 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 ning agents guided by their own choices This is the problem that Hayek assumes

plan-to be the real reason for the existence of the Social Sciences In his talk, Pr franchi 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

Castell-in computer simulations and proposes some solutions about the theoretical and tional relationships between agents' intentions and non-intentional purposes of their actions

func-[For him,] social order is not necessarily a real order or something good and able 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

desir-Robert Axtell, from The Brookings Institution, Washington DC and the Santa Fe

Institute, has emphasized a very stimulating reflection on "Very Large-Scale

Multi-Agent Systems and Emergent Macroeconomics" For Dr Axtell, the relatively few applications of agent-based computing to macroeconomics retain much of the rep- resentative 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 behav- ior 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 lim-itations are the reasons why no more papers have been accepted, although many of the rejected submissions were really interesting Nevertheless, the choice of avoid-ing 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 manifes-tations

Villeneuve d'ascq, Philippe Mathieu June 2005 Bruno Beaufils

Olivier Brandouy

VIII Preface

References

1 R Axelrod and W.D Hamilton (1981), The evolution of cooperation Science, pp

1390-1396

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

264-274

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 3

Market Dynamics and Agents Behaviors: a Computational Approach

Julien Derveeuw 15

Traders Imprint Themselves by Adaptively Updating their Own Avatar

Gilles Daniel, Lev Muchnik, Sorin Solomon 27

Learning in Models

Learning in Continuous Double Auction Market

Marta Posada, Cesdreo Hernandez, Adolfo Lopez-Paredes 41

Firms Adaptation in Dynamic Economic Systems

Lilia Rejeb, Zahia Guessoum 53

Firm Size Dynamics in a Cournot Computational Model

Francesco Saraceno, Jason Barr 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 79

Multi-Agent Model of Trust in a Human Game

Catholijn M Jonker, Sebastiaan Meijer, Dmytro Tykhonov, Tim Verwaart 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

Domenico Delli Gatti

Catholic University of Milan

Universita Politecnica delle Marche

J.H von Eije

University of Groningen The Netherlands

Jose M Galan

University of Burgos Spain

XII List of Contributors

Sebastiaan Meijer

Wageningen UR The Netherlands

Denis Phan

Universite de Rennes I France

Lilia Rejeb

Universite de Reims Universite de Paris VI France

List of Contributors XIII

Malte Sunderkotter

University of Karlsruhe Germany

Shingo Yamamoto

Future University-Hakodate Japan

Shihomi Wada

Future University-Hakodate Japan

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,

{ s t u e m p e r t | s e e s e } @ a i f b u n i - k a r l s r u h e d e

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 SF-ASM 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 high fluctuations and 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

mar-ket 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

per-mits interaction of agents on a microscopic level that has influence on the stock price

on a macroscopic level Agents try to forecast fixture prices 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

op-erator (see [6]) and presents a model in which high fluctuations occur 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 a fixed return 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), see figure 1 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 stock*retum/dividend > 0.25, 0.5,0.75, 0.875,1.0,1.125

7 pt > 5-period moving average

8 pt > 10-period moving average

9 Pt > 100-period moving average

10 Pt > 500-period moving average

11 control bit

12 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

risk-neutral 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

vec-tor For calculation of the demand ftmction Xi^t of an agent i, we introduce

forecast-ing rules Each agent estimates the expected return of investforecast-ing 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

com-pared 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

cho-sen 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 = fitness of rule r of agent i in period t,

M = constant scaling factor, e.g 0,

C = weight of s (here: C = 0.005),

s = number of symbols G {0; 1} in the condition part, (i.e unequal to # ) ,

0 = speed of change of the fitness function, between 0 1,

^t,i,r "" 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 Properties from an 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

al-gorithm (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

gen-erated) 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

re-placed (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

equi-librium 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

divi-dends, 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

own-3 Calculate the number of shares to be traded

8 Thomas Stumpert et al

Axit = {xit - Xi^t-i),

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

4 The agent does not sell more shares than max trader

if Axit < -maxtrade^ then Axu = -maxtrade •

5 The agent does not buy more shares than he can afford:

if (Axit • p > Cit), then Axu = ^

6 The agent does not buy more shares than max trade,

if Axit > maxtrade, then Axu = maxtrade •

7 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 +

^P-8 If the sum of supply and demand (i.e negative supply) is under the threshold e,

then the equilibrium price is found up to the precision e, else go to step 2

2 Specification of Test 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

mea-sures 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:

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 2

r<^^,J ~ 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)

algo-olies 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

differ-ent market conditions which can be observed in differdiffer-ent 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 The sum of all / is chosen to be equal or less than 1, e.g /(o,o.85] = 0.137 means

/(o.97,1.03])-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 valuation 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

under-of agents who use a risk-averse utility function One exception under-of this market ior 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

behav-evolvement and the wealth ratio for the simulation run with k= 1 over 150000

peri-ods High fluctuations occur 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 pt Pmax Pmin Pt P^ax P'^in P<T^ P^ pw,max piv,min Ntbit Nfbn Ncbit

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

-70.83 0.141 -125.49 0.185 -378.45 0.512 -188.10 0.749 -12.35 0.03 -95.85 -63.12 -57.52 -2.67 -30.98 -42.76 -28.93 -62.82 -0.99 -2.09 -11.10 -49.55 -44.84

0.710 0.712 0.703 0.799 0.804 0.789 0.853 0.849 0.833 0.834 0.817 0.743 0.725 0.795 0.893

0.350 0.928 0.974 0.998 0.952 0.738 0.960 0.997 0.977 0.833 0.988 0.989 0.987 0.999 0.867 0.811 0.771 0.972 0.992

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

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 trations 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

concen-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 ance 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

vari-as follows:

12 Thomas Stiimpert et al

Table 2 20 runs (without taxes for /cnotaa:=!, »10 and with taxes for ktax^"^ 1, ,20)

/(0.95;0.97]

0.056 0.073 0.061 0.076 0.088 0.104 0.057 0.111 0.116 0.085 0.115 0.091 0.102 0.099 0.064 0.072 0.098 0.092 0.062 0.048

/(0.97;1.0]

0.287 0.188 0.270 0.293 0.337 0.365 0.360 0.273 0.367 0.282 0.241 0.170 0.208 0.151 0.163 0.181 0.218 0.194 0.156 0.130

/(1.0;1.03]

0.220 0.160 0.191 0.216 0.219 0.175 0.254 0.158 0.143 0.192 0.135 0.083 0.077 0.062 0.106 0.081 0.122 0.094 0.089 0.079

/(1.03;1.05]

0.048 0.059 0.032 0.028 0.024 0.028 0.035 0.031 0.016 0.028 0.012 0.005 0.003 0.007 0.005 0.003 0.013 0.014 0.003 0.003

/(1.05;1.1]

0.055 0.084 0.044 0.028 0.037 0.032 0.039 0.042 0.020 0.037 0.005 0.003 0.001 0.005 0.000 0.001 0.007 0.007 0.000 0.000

/ ( l l , o o ] 0.110 0.159 0.100 0.084 0.099 0.061 0.059 0.071 0.045 0.162 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Pa2

-77.39 -68.30 -52.05 -66.77 -39.59 -44.33 -53.18 -39.31 -31.91 -62.11 -3.90 -4.76 -3.50 -4.33 -3.36 -4.21 -3.85 -5.08 -3.75 -3.08

vart((it+i) = v a r t ( 4 + p{dt - d) f St+i) = erf

vart(cZt+2) = vart(d + p{dt+i - d)-{- £^+2)

= Y2iYt{d + p^dt + pst^i + et-^2) = (1 + P^)cr'^

y Sir t{dt^k) = ( l + p 2 ^ ^ + p2(/c-i))a2

with limfc^oo vart{dt+k) = E ^ o P^^^e = r = ^ ^e = var(cft)

Straightforward, the variance of the fundamental price process p^ = dt/r computes

to

var(p*) = var(dt)/r2 =

and with the values used in our models we get var(p^) = 0.07429 • 0.1~^ • (1 —

O-OS"^)"^ = 76.19 = ap* Even sufficient for an interval length of n = 7500 trading periods, the empirically observed variances cr^* j match this value and therefore we assume var(p^) ^ var(p^^) Knowing the average relative volatility for each simula- tion run fi*om the empirical results, p^^j, we can compute the empirical market price variance as an average value over all intervals j , with j e [ 1 , , J ] , per simulation run k, with he [1, ,K] Since we defined

(15)

the average empiric market price variance per simulation run k is

^p,k = (1 - Pa^,k ' J) -CTp^ • (16)

We statistically test our assumption that market price volatility is significantly higher

in simulations runs without taxation compared to a market with Tobin tax with

Time Series Properties from an Artificial Stock Market with a Walrasian Auctioneer 13

the F-test^ The values of a^j^, provided for each simulation run in table 3 with

^«^(^p,tax,/c) = 7817.09, and mm(a2^

notax k) "" 25645.55, generate a significant

F-value of F = 25645.55 -r- 7817.09 = 3.28 (critical value at the 99%-confidence level is /(24;24;0.99) = 2.659) Thus, we can reject the h3^othesis that the tax sys-tem has no influence on the market price volatility Comparing the maximum of the average empiric market price variance of all simulation runs with taxation with the minimum empiric market price variance of all simulation runs without taxation, we make sure that the market setting with taxation has significantly lower volatility in all cases compared to the market setting without taxation

Table 3 Average empiric market price variance per simulation run, af p,k

N o Tax 118003.07 104151.73 79389.98 101820.32 60403.43 67626.24 81111.87 59976.77 48700.65 94719.41 152669.52 46369.23 55847.27

N o Tax 132235.36 69911.94 72228.12 117911.64 77576.66 80197.59 50803.49 25645.55 52342.53 59428.20 58833.92 122269.71

4 Conclusion

We presented a model based on SF-ASM where the amount of each agent's wealth influences the future price In this model high fluclxiations occur due to extreme con-centrations of wealth and stock shares We used a parameter setting where agents can easily adapt their strategy to a change in the price function of the risky asset

^ Commonly used in order to prove that the probability distribution fijnctions of two data sets have significantly different variances is Fisher's "F-tesf The Test statistic is

with (Ji > a i

Hypothesis HQ, i.e af and a l have the same variance a^, can be rejected at confidence level a if F > f(,.^-u2,i-a), where /(^^

;i/2;i-a) dcnotcs the (1 — a) quantile with Ui =

1 — rii degrees of freedom at m observations

14 Thomas Stumpert et al

Prohibition of trading restrictions empowers agents to possess and trade all shares each period The interaction of the auctioneer with each single agent is reflexive and without any trading restrictions high fluctuations occur in our model The high fluc-tuations depend less on the use of technical trading bits but on an arising extreme wealth ratio By introducing a Tobin tax this effect can be avoided, the tax leads

to market stabilization reducing the occurrence of longer periods with high spreads between the market price and the fundamental price

References

1 LeBaron B, Arthur WB, Palmer R, Holland JH, Tayler P (1997) Asset pricing under endogenous expectations in an artificial stock market In: Arthur WB, Durlauf S, Lane D (eds) The economy as an evolving complex system II Addison-Wesley

2 LeBaron B, Arthur WB, Palmer R (1999) Time series properties of an artificial stock market Journal of Economic Dynamics and Control 23:1487-1516

3 Tesfatsion L (2002) Notes on the Santa Fe artificial stock market model Econ 308x: Agent-based Computational Economics

4 Seese D, Stumpert T (2003) Influence of heterogeneous agents on market structure in an artificial stock market Proceedings of Annual Workshop on Economics with Heteroge-neous Interacting Agents, WEHIA 2003, Kiel

5 LeBaron B (2002) Building the Santa Fe Artificial Stock Market Working Paper, deis University

Bran-6 Ehrentreich N (2003) A corrected version of the Santa Fe institute artificial stock ket model Complexity 2003: Second Workshop of the Society for Computational Eco-nomics

mar-7 Cont R (2001) Emperical properties of asset returns: Styhzed facts and statistical issues Quantitative Finance Volume 1:223-236

8 Walras L (1874) Elements d'economie pohtique pure [English Translation: Elements of Pure Economics of the Theory of Social Wealth Irwin RD, Homewood, 1926] Corbaz L (eds) Lausanne

9 Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning Addison-Wesley

Market Dynamics and Agents Behaviors: a

Computational Approach

Julien Derveeuw^

Laboratoire d'lnformatique Fondamentale de Lille - UMR USTL/CNRS 8022

Universite des Sciences et Technologies de Lille

Batiment M3, Bureau 14a

59655 Villeneuve d'Ascq Cedex

julien.derveeuw@lifl.frandhttp://www.lif1.fr/~derveeuw

Summary We explore market dynamics generated by the Santa-Fe Artificial Stock Market

model It allows to study how agents adapt themselves to a market dynamic without knowing its generation process It was shown by Arthur and LeBaron, with the help of computer ex-periments, that agents in bounded rationality can make a rational global behavior emerge in this context In the original model, agents do not ground their decision on an economic logic Hence, we modify indicators used by agents to watch the market to give them more economic rationality This leads us to divide agents in two groups: fundamentalists agents, who watch the market with classic economic indicators and speculator agents, who watch the market with technical indicators This split allows us to study the influence of individual agents behaviors

on global price dynamics In this article, we show with the help of computational simulations that these two types of agents can generate classical market dynamics as well as perturbed ones (bubbles and kraches)

1 Introduction

Market simulations with the help of computer agents have become in the last few

years a growing field of interest under the impulsion of the Santa-Fe Institute for

ex-ample These simulations allow to predict market evolutions, to validate theoretical hypotheses or to test models in perfectly controlled virtual worlds [8] The most used

approach to study these complex systems is the use of agents with bounded rationality

who learn and make their behaviors evolve in time Following the founding works of [1] or [10], who showed that it is possible to make rational global behaviors emerge with simple, bounded individual behaviors, numerous models of markets have been developped These models aim to reproduce real economic phenomena (for example, bubbles and crashes: [13]) or to study the impact of these phenomena on the agents

population [5] The early version of the Santa-Fe Artificial Stock Market (also known

as SF-ASM) by [15] remains a major reference in this field: it shows that, a global, rational economic behavior can emerge from an agents population that build its be-

in time [4], [6]) or to add more realism to the underlying economic logic of the model [11], [12] Though, these modifications remain minor technical corrections

We develop our model considering two major modifications of the original SF-ASM: the first improves the agents behavior by putting a stronger economic rationality in

their decisions There, we define two canonical subpopulations: fundamentalists and

speculators Thus, the second modification consists in mixing those subpopulations

to observe and characterize some interesting global market dynamics

In this article, we want to address the question of markets dynamics, bubbles and

crashes using a bottom-up approach We show that critical events may be caused by bounded rationality individuals that ground their behaviors on market trends and liq-

uidity signals Our results are consistant with the general thesis of [9] and the french

neo-keynesian school of finance leaded by [14] According to those last approaches, critical events are caused by the interaction of rational investors that do not arbi-

trate prices considering a so-caUQd fundamental value but that try to obtain profits in

catching the market mood If they trust the market will raise despite it is yet evaluated, they will have a global buying attitude that will therefore push the prices

over-up The main issue they face is that the market cannot offer enough liquidity if all the agents perform the same decision: thus, if all of them want to sell at the same time, the market breaks down abruptly^

The article is organized as follows: a first part presents the architecture of the

original model of the SF-ASM as well as some minor modifications we bring to

it The second part details the learning process used by the agents to make their decisions and the differences between our two subpopulations of agents, the fiinda- mentalists and the speculators The last part presents our results and discusses some consequences of this research

2 Market Model Presentation

Our work is directly based on the articles of [15] and [12] Let us show common parts as well as differences between the original SF-ASM and our modifications

^ as instance, one can report to the tulipomania bubble which occured in Netherlands

Market Dynamics and Agents Behaviors: a Computational Approach 17

2.1 The Original Model of the SF-ASM

The model architecture is reduced to the essential: it is composed of one type of stock (5) and of n heterogeneous agents a i , , a^: they all have a different behavior Each step of time can be considered as a market day Agents do not know at which iteration

the simulation will stop At each step of time t, the stock has a current price pt and pays a dividend dt per asset to each stockholder Each agent a^ owns a certain amount

of money rui^t, and a number of shares hi^t- Their goal is to choose between keeping their shares to earn the dividend dt, to sell them to raise their funds or to buy new

ones Another possibility for agents is to invest their cash money in a risk free asset

which pays a moderated interest rate r at each step of time t

At the end of each time period, agents are asked their desires: they can either

bid to buy new shares (in this case, we have a bid: bi^t = 1 and no stock offered: Oi^t = 0), offer a share (bi^t = 0 and oi^t = 1) or do nothing (bi^t = 0 and Oi^t = 0)

We then obtain the cumulated offer (Ot) and supply (Bt) by summing the bi^t and Oi^t' The balance between cumulated offer and supply has a direct influence on the stock price and on the quantities exchanged by the agents If Bt = Ot, then all

offers and demands are satisfied: each agent who asked for a share receives it and each agent who offered a share sells it For remaining cases, we have to introduce

a process to distribute offered shares function of the number of asks: it is a market clearing process Each agent who asked for a share is given the maximum fraction

of share available (offered):

mm{Bt,Ot), , mm[Bt,Ot) iii,t-\-i = i^i,t H 5 ^i,t H 7^; Oi^t

J=>t Ut One can notice that bi^t = 1 do not mean that agent ai will receive a complete share at

t + 1, but that he will receive at most a share Hence, bi^t must be seen as a proposal

to buy the maximum fraction of share available At the end of each time step, the price is updated function to the offer and supply rule (the more the share is asked, the more its price raise) using the following formula:

fvt = ^ c Z t ( l + a ) " j,ty± -r uc) t=0

a is usually considered as equal to r This equation is hence simplified as:

fvt = —

r

18 Julien Derveeuw

2.2 Modifications on the Original Model

We have seen that at each step of time, agents receive a dividend dt per each share In

the original model of [15], the dividend generation process is relatively complex We have chosen to simplify it by choosing a well known generation process in economics for the generation of random process poorly evolving in time: the random walk [16]

A random walk is defined as follow:

dt+i = dt-{-et

et is a gaussian noise parametered by its mean (null here) and its variance (S'^)

3 Agents Reasoning Process

As the agents do not know the generation process of dividend d and of price p, they

are forced to elaborate their strategies only with their experiences (the past values of dividend and prices) Their challenge is to maximize their satisfaction (here, maxi-

mize the amount of money won at the end of the game) This is seen here as trying

to recognize a particular market state to take the best decision function of this state

We will describe in a first part which representation the agents use to observe the market, and in a second part how this representation is used to take a decision

3.1 Market Representation

Each agent has a stock of m rules which describe market states and tell him which

decision to take A rule is composed of three subparts: the first one describes a

spe-cific market state (called condition part of the rule) The second part describes the

decision to take withing this specific context: bid a share, offer a share or do nothing

(called action part The last part represents the current evaluation of the rule's

ade-quateness in market activity (called^brc^) One have to keep in mind that each agent possess his own stock of rule that is, it is hardly possible that two agents are exactly similar

The /c-th rule of an agent ai is composed of:

1 a condition part that can be viewed as a 7 bit chromosome or a string made of

7 {0,1,#} symbols Each gene or character contributes to the description of a

specific market state that is therefore, completely expressed, with 7 statements Those statements are said to be true (the value of the gene is 1), false (0) or unrelevant (#) The space of conditions is hence 3"^ size To give an idea of what

a statement is, one can consider the following: Stock price is over 200$

2 an action ai^k to take if the rule is selected We have: aik = 1 "^ bid one share,

a^fc = — 1 <^ offer one share and aik = 0 ^ do nothing

3 a strength Si^k which tells how good this rule was in making the agent earning

money in the past

At t = 0, the rules are generated following those steps:

Market Dynamics and Agents Behaviors: a Computational Approach 19

1 The condition part is randomly built

2 The corresponding action is determined following a rational process which will

be explained further

3 Initial strengths are 0

When t > 0, the rules are generated, evaluated and updated using a genetic gorithm The genetic algorithm maintains diversity in the rules population, improves

al-them and allows to easily destroy the worst ones Let us now focus on the agents' decision process Since the agents have to perform the best possible choice, they first identify in their stocks of rules which of them correctly describe the current mar-

ket state These rules are said to be activated In other words, the chromosomes are matched against the current market state and a rule is said to be selectable if all of its bits (genes) are non contradictory with this state (they are said to be activated) Hence, a rule is activated if all of its bits are activated too A bit hi is activated:

1 6i = 0 and the z-th condition of the market state is f a l s e

2 hi = 1 and the i-th condition of the market state is t r u e

3 hi = # and the i-th condition of the market state is either t r u e of f a l s e

Among those activated rules some of them present a positive strength Si^k > 0

He then elects one of these rules with a random process proportional to their strength

The action aik associated with the elected rule gives the agent decision If there are

no rules activated by the current market state, then the agent's decision is to stay

unchanged {hi{t) = 0 and Oi{t) = 0) At the end of the time step, the agent updates

the previously activated rules according to how much money they would have make him earn, giving:

Sik{t + 1) = (1 - c)sik{t) + caik{p{t + ! ) - ( ! + r)p{t) + d{t + 1))

The parameter c controls the speed at which the rules strength is updated

Each time the genetic algorithm is run, the worst rules (e.g with the smallest strength) are deleted They are replaced by new rules generated using a classical genetic process: the best rules are selected to be the parents' of the new rules A new rule can be generated either by mutation (only a bit of the parent's chromosome is changed) or by a crossover process (reproduction between two parents rules) This mechanism permits, on the one hand, to delete the rules that don't make our agent earn money and to build up new rules using good genetical material This process is aimed to increase the adaptation of the agents to the market activity

There are two types of agents in our simulations: some who try to be as close

as possible to the fundamental value of the stock (will be refered as fundamentalist agents in the following) and some who try to make the maximum benefit without taking care of the fundamental value (will be refered as speculators agents in the

following) This is a point that largely make our work different from those ously cited We think that in [3] and [7] one issue is that the decision rules of agents are excessively dominated by randomness: whatever the market statements are, the corresponding action is decided randomly It is true that along market activity, the evolving process selects best responses to those statements, but nothing grants that

previ-20 Julien Derveeuw

the corresponding actions are relevant with respect to an economic logic For ple, it is very probable that although a stock is mispriced (let's say underevaluated), the agents will never try to arbitrate this spread (here with buying it) The other issue

exam-is that technical statements as well as fundamental statements are melted and no ical behavior is clearly observable We try to improve the agent model by defining

typ-a minimum economic logic thtyp-at letyp-ads etyp-ach subpopultyp-ation typ-actions: fundtyp-amenttyp-alists try to arbitrate any price deviation whereas speculators ground their decisions on subjective, technical informations

As said before, the main characteristic of the fundamentalist agents is that they have appropriate decisions considering the spread between the observed prices and the fundamental value Let's consider the composition of the chrosome and what kind of statements are coded inside

Corresponding action

1 =^ to buy

- 1 =^ to sell

0 =^ stay unchanged

0 =^ stay unchanged (ibid.)

0 => stay unchanged (ibid.)

Fig 1 Fundamentalists' chromosome Fig 2 Rules for fundamentalist rationalization

Let's consider the seventh gene; the corresponding statement, depending on its

wahxQ {l, 0, ^} is: The price {is, i s n o t , i s o r i s not} at least forty percent above the fundamental value

We have added to the original SASM a rationalize procedure This procedure

aims to achieve a minimal economic rationality for the agents Fundamentalists are

assumed to arbitrate significant spreads between fv and p, that is to bid for priced shares and to ask for overpriced stock This procedure is based on some rules

under-presented in table 3.1 One has to keep in mind that this procedure is run each time

a new rule is generated (consequently, when the genetic algorithm is initialized and run)

Let's consider now the second subpopulation: the speculator agents As said

be-fore, those agents do not arbitrate prices but rather try to make profit using trends or subjective knowledges Therefore, their chromosome is constructed using this kind

of market representations as shown in table 1

The chromosome is thought to code general sentiment on the market trend which

is very different than the identification of a market state What we mean here is that

Market Dynamics and Agents Behaviors: a Computational Approach 21

ment is bull market a rational behavior for a speculator agent is to buy (symetrically,

if the market is bear, the rational behavior is to sell) We have coded this logic in

the speculator rationalization To have a global sentiment on the market trend, we simply appreciate the dominant trend given by the indicators or groups of indicators The decision making process for speculator agents is relatively complex and can be divided into two major steps

For bits 1, 2, 7, 8 and 9, we simply consider if the belief of the agent validates the condition or not Let's consider the example of bit 8: we explicitly test if the price is over or above the median of the interval bounded by the highest and the lowest quotation during the lasts 100 days If the price is above, it is thought that the price will decrease and alternatively, if it under this median, it is believed that the price will rise As instance, this last situation pushes the agent to bid new shares Bits 3 to 6 receive a special treatment: bits 3 and 4 are considered together as well

as bits 5 and 6 The first pair allows the estimation of short range trend while the

second pair allows the estimation of a long range trend, pairs, hiU is the first one

e.g bit number 3 and bit number 5 while 6^+1 is the second one e.g 4 and 6 To appreciate the trend, one has to consider the situation of the current price relatively

to those bits As example, let's consider the situation where the chromosome's bits

3 and 4 are respectively 0 and 1 In this case, it is false to assert that the current price is above the mobile average on the past five days whereas it is clearly above the mobile average on the past ten days We therefore consider that this information

is not sufficiently clear to influence the decision and bid and ask positions have to be weighted with the same absolute value scalar: 0.5 When those bits are respectively

1 and 1, the trend is clearly bull and the agents will be temptated to follow it, e.g to

buy The nine possibilities for each pair are summed up in table 2

A first step in the speculators' rationalization process is then achieved: our agent can form an initial belief on the possible tendency of the market summing the values

of each indicator One has to keep in mind that some of them are positive (giving bid signals) negative (ask signals) or null (do nothing) If the number of positive

22 Julien Derveeuw

Table 2 Speculators' rationalization when i G {3,5}

biti biti+i

#

0 -1

lead to ask and null signals lead to stay unchanged

One can easily imagine that such a logic may lead to constantly growing or falling

markets: bear signals are followed by bid positions that push the price up Why this

tendency should break down ? According to [14], one major indicator observed by

the traders is market liquidity The idea is that operators are very concerned with the

possibility of clearing their positions (to sell when they hold stocks or to buy if they are short) This implies that minimum volumes are realized at each time step When the market becomes illiquid, agents may be stucked with their shares Therefore, they follow the market only and only if they are confident on the liquidity level of the market This point has been included in the agents' logic with the following rules:

• each agent has her own treshold above which she considers that the market is unsufficiently liquid to clear her positions

• When this threshold is reached, she adopts a position opposite to the one she would have adopted without considering this treshold By the way, she decides

to reverse her investment strategy to go out of the market

4 Experimental Schedule and Results

As the model contains many numerical parameters, we have chosen to only vary the ones which directly impacts the global price dynamics, that is to say the speculator agents proportion and their liquidity fear parameters The other ones are considered

as constant as they can be seen as more technical model parameters All of the lowing experiments are realized on a time range of 10000 iterations Though, as the genetic algorithm used by the agents to adapt themselves to the market needs a learning period, only iterations between time step 2000 and 10000 are shown All statistics are conducted on this range unless the opposite is mentionned

fol-As our primarly goal is to study the influence of speculator agents' proportion

on price dynamics, we first run an experiment without speculators (i.e only with fimdamentalists) This first experiment allows us to validate our fundamentalist agent model by matching the experimental results with the ones obtained by [15] with the original SF-ASM model This experiment will also be used as a comparison base with other ones as it represents the baseline price dynamics of our model (i.e with the less variant price series) Other experiments are realized by gradually increasing the speculator agents proportion in the agents population and by adjusting their liquidity

Market Dynamics and Agents Behaviors: a Computational Approach 23 fear Many experiments have been run, but we only detail here the ones with the more significant results

4.1 A Fundamentalist Market

The figure 3 represents the price and fundamental value motions when the market is only made of fiindamentalist agents The two series perfectly overlap

\ y v M

Fig 3 Market dynamics with fundamentalist agents

The first step to test if those motions are somewhat consistent with what happens

in the real stock markets consist in testing whether they are driven by non-stationary processes or not The appropriate test to seek for a random-walk process in market

returns is an Augmented Dickey-Fuller unit root test (e.g ADF) Both fundamental values and prices have to be random walks if we want to qualify the simulations re- alistic since the immense part of academic researchs attest such motions for modem,

real stock market dynamics

In the following tests, the null hypothesis is time seriepresents one unit root (HQ) while the alternative is time serie has no unit root (Hi) Table 3 reports the results of

those tests Interpretation is the following: if t-Statistics is less than the critical value,

one can reject the HQ against the one-sided alternative Hi which is not possible in

our case

Table 3 ADF Unit Root Tests

time series t-Statistics Augmented Dickey-Fuller test fund val -2.3594

Augmented Dickey-Fuller test price -2.4154

Critical values: 1% level: -3.9591, 5% level:

Prob.*

0.4010 0.3713 -3.4103, 10%: -3.1269

*MacKinnon one-sided p-values

24 Julien Derveeuw

A Johansen integration test shows that prices and ftindamental values evolve We also observe that the spread between prices and fundamental values re- mains very weak (between -3.22% and +2.72% with a 0.02% mean and a 1.11%) standard deviation) This base line experiment exhibits therefore some interesting results if one considers its proximity with real market dynamics It also shows that bounded rationality agents can make emerge a random walk motion that is charac- teristic of efficient prices on stock markets This result is already documented by [15],[2] Nevertheless, our contribution is to obtain such results with agents follow- ing rules that make sense, which was less evident in the original studies

co-4.2 A Mixed Market

Figure 4 represents price and fundamental value motions when the market is made

of 25% of fundamentalist agents and 75% of speculators

Fig 4 Market dynamics with 25% fundamentalist and 75% speculators

It appears that the market is more volatile when it is flooded with fundamentalists which is an expected result If one considers the statistical properties of the price mo- tion globally (on the complete sample), it appears that a Null hypotheses of random

walk can be rejected with a very low risk (with p < 3%) This result is

understand-able as the agents population is composed of a majority of speculators Though, on smaller samples (for example on time range from 2000 to 3000), the result of the test

is inverted: the market is in a period where it behaves as if it follows a random walk

In such periods, the price and the fundamental value motion are co-integrated, which shows that market follows the fundamental value dynamics

In Table 4, we have reported some basic statistics related the spreads between observed prices and fundamental values It clearly appears that prices are much more volatile in the second regime (with speculators) than in the first one (standard, maximum and minimum deviations) The over-returns mean is also strictly positive Moreover, returns distribution does not follow a Normal distribution

Market Dynamics and Agents Behaviors: a Computational Approach 25 Table 4 Prices deviations relatively to the fundamental value

Critical sub-sample

0% 75%

0.152548 3.180839 0.150780 1.450169 2.114522 5.535200 -0.230643 1.228508 2.236049 3.489872

On a critical period where we can visually identify a bubble, for example during

time period 5000-5400, prices are still a random walk Table 4 reports prices ations during this critical event Here the standard deviation is greater than the one observed on the complete sample A bubble is hence characterized by a great de- viation between the stock price and its fundamental value during a long time range This typical dynamic, obtained with 75% of speculators and 25% of fundamentalists, can be found with other sets of parameters as long as speculator agents proportion is great (> 70%)

devi-In the speculative regime (when speculators compose the main part of tion), we obtain a highly volatile price dynamic with bubbles and crashes These phenomena would rather be undetectable if we could not watch the fundamental value Moreover, as the prices follow most of the time a random walk, nothing can distinguish such a dynamic from the one observed with a fundamentalist population except the comparison between the prices and the fundamental value Hence, there could be speculative bubbles in real market while the technical efiiency properties would be respected

popula-5 Conclusion

In our simulations, we obtain price dynamics specific to our two agents populations These behaviors were designed to illustrate two main economic logic: the first fol- lows the classical economic theory which is grounded on agents arbitrating differ- ences between the fundamental values and the current stock prices, whereas the sec- ond is mainly based on ideas from the keynesian theory of speculation

The first market dynamics is obtained when the agents population is only posed of fundamentalists We show that in this case, the price dynamics follows a random walk which co-evolve with the fundamental values This first result can be related to the ones of [15]: inductive agents in bounded rationality can make effi- cient prices emerge The difference here is that fundamentalists only ground their decisions on classic market indicators and that these decisions are made following constitent behavioral rules, which is not the case in many simulated stock markets When speculator agents compose the main part of the agents population, we ob- tain another type of dynamics: prices still follow a random walk process, but during

com-26 Julien Derveeuw

some periods, the system reaches a critical state This critical state is characterized

by the emergence of a new phenomenom: the stock starts to be more and more priced (bubble) before falling back violently to its fundamental value (crash) More-over, these market dynamics are very volatile

over-Next steps in our research could be to introduce a third agent behavior which will act as a market regulator to arbitrate the market and prevent bubbles from happening This could for example be realized by introducing a behavior who would ponctually decrease the market liquidity to force the speculators to reverse their decisions One can also imagine to study the impact of social interaction between agents on market dynamics to see if it would arbitrate the price deviations or amplify them

Durlauf, editors The Economy as an Evolving Complex System II, pages 15-44, 1997

4 N Ehrentreich A corrected version of the santa fe institute artificial stock market

model Working Paper, Martin Luther Universitat, Dept of Banking and Finance, Wittenberg (Germany), September 2003

Halle-5 S Focardi, S Cincotti, and M Marchesi Self-organization and market crashes Journal

of Economic Behavior and Organization, 49(2):241-267, 2002

6 L Gulyas, B Adamcsek, and A Kiss An early agent-based stock market : Replicaton

and participation Proceedings of the NEU 2003, 2003

7 L Gulyas, B Adamcsek, and A Kiss An early agent-based stock market : Replicaton

and participation Proceedings of the NEU 2003, 2003

8 N.F Johnson, D Lamper, P Jeffries, M.L Hart, and S Howison Application of

multi-agent games to the prediction of financial time-series Oxford Financial Research Centre Working Papers Series N° 2001mf04., 2001

9 J M Keynes The General Theory of Interest, Employment and Money MacMillan,

Uni-13 H Levy, M Levy, and S Solomon A microscopic model of the stock market: Cycles,

booms, and crashes Economic Letters, A5{\)'A()2>-\\\, May 1994

14 A OrlQan Le pouvoir de la finance 1999

15 R.G Palmer, W.B Arthur, J.H Holland, B LeBaron, and P Tayler Artificial economic

life : A simple model of a stockmarket Physica D, 15:264-21 A, 1994

16 P.A Samuelson Proof that properly anticipated prices fluctuate randomly Industrial Management Review, (6):41-49, 1965

Traders Imprint Themselves by Adaptively

Updating their Own Avatar

Gilles Daniel^, Lev Muchnik^, and Sorin Solomon^

^ School of Computer Science, University of Manchester, UK

gilles@cs.man.ac.uk

^ Department of Physics, Bar Ilan University, Ramat Gan, Israel

LevMuchnik@gmail.com

^ Racah Institute of Physics, Hebrew University of Jerusalem and

Lagrange Laboratory for Excellence in Complexity, ISI Foundation, Torino

We propose here the following Avatar-Based Method (ABM) The subjects

im-plement and maintain their Avatars (programs encoding their personal decision ing procedures) on NatLab, a market simulation platform Once these procedures are fed in a computer edible format, they can be operationally used as such without the need for belabouring, interpreting or conceptualising them Thus ABM short- circuits the usual behavioural economics experiments that search for the psychologi- cal mechanisms underlying the subjects behaviour Finally, ABM maintains a level of objectivity close to the classical behaviourism while extending its scope to subjects' decision making mechanisms

mak-We report on experiments where Avatars designed and maintained by humans from different backgrounds (including real traders) compete in a continuous double- auction market Instead of viewing this as a collectively authored computer simula- tion, we consider it rather as a new type of computer aided experiment Indeed we consider the Avatars as a medium on which the subjects can imprint and refine inter- actively representations of their internal decision making processes Avatars can be objectively validated (as carriers of a faithful replica of the subject decision making process) by comparing their actions with the ones that the subjects would take in sim- ilar situations We hope this unbiased way of capturing the adaptive evolution of real subjects behaviour may lead to a new kind of behavioural economics experiments with a high degree of reliability, analysability and reproducibility

28 Gilles Daniel et al

1 Introduction

In the last decade, generic stylized facts were reproduced with very simple agents

by a wide range of models [3, 12, 14, 6, 16, 8] By the very nature of their generic properties, those models teach us little on real particular effects taking place as result

of real particular conditions within the market In order to understand such specific market phenomena, one may need to go beyond "simple-stupid" traders behaviour [1] Thus the task of the present generation of models is to describe and explain the observed collective market phenomena in terms of the actual behaviour of the individuals

For a long while, classical economics assumed individuals were homogeneous and behaved rationally Thus it was not necessary to study real people behaviour since (presumably) there is only one way to be rational Even after the conditions of rationality and homogeneity were relaxed, many models did it by postulating arbi- trary departures not necessarily based on actual experiments When the connection

to the real subjects behaviour was considered [11], an entire host of puzzles and paradoxes appeared even in the simplest artificial (laboratory) conditions Thus the inclusion of real trader behaviour in the next generation of models and simulations is hampered by the inexistence of comprehensive, systematic, reliable data Given the present state of the art in psychological experiments, where even the behaviour of single subjects is difficult to assess, we are lead to look for alternative ways to elicit the necessary input for agent-based market modelling

In this paper we propose a way out of this impasse Rather than considering the computer as a passive receiver of the behavioural information elicited by psycholog- ical experiments, we use the computer itself as an instrument to extract some of the missing information More precisely, we ask the subjects to write and update adap- tively, between simulation runs (or virtual trading sessions) their own avatars By gradual corrections, those avatars converge to satisfactory representations of the sub- jects' behaviour, in situations created by their own collective co-evolution The fact that the co-evolution takes place through the intermediary of the avatars interaction provides an objective detailed documentation of the process

More important, the dialogue with the avatars, their actions and their collective consequences assist the subjects in expressing in a more and more precise way their take on the evolving situation and validate the avatar as an expression of the subject internal decision mechanisms Ultimately, the avatar becomes the objective repos- itory of the subject's decision making process Thus we extend, with the help of computers, the behaviorist realm of objectivity to a new area of decision making d3mamics The classical behaviourism limits legitimate research access to external overt behaviour, restraining its scope to the external effects produced by a putative mental dynamics The method above enables us to study the subjects decision mak- ing dynamics without relying on ambiguous records of overt subjects behaviour nor

on subjective introspective records of their mental state and motivations

Far from invalidating the psychological experimental framework, the present method offers psychological experiments a wide new source of information in prob- ing humans mind The competitive ego-engaging character of the realistic NatLab