The main purpose of the paper is to determine a general behavior of a multi-agent model capable of describing the process of deliberation of an investors group witch may repeatedly decide whether to buy or sell an asset. Each adaptive agent was modeled asProgramming Models, Genetic algorithms, Information efficiency

Boldea Bogdan Ion

Boldea Costin-Radu

Stanculescu Mircea

English

Research Papers in Economics

897 
AN ADAPTATIVE EVOLUTIONARY MODEL OF FINANCIAL INVESTORS 
 
Boldea Bogdan Ion 
West University of Timişoara,  
Faculty of Economy and Business Administration,  
Str. J.H.Pestalozzi, no. 16, 300115, Timisoara, 
E-mail: bbi77@yahoo.com  
Boldea Costin-Radu 
 University  of  Craiova,  Faculty  of  Mathematics  and  Computer  Sciences,  Str.  A.I.Cuza,  no.  13,  cod.  200585, 
Craiova, E-mail: cboldea@inf.central.ucv.ro  
Stanculescu Mircea 
“Spiru Haret” University, Faculty of Management and Accounting,  Str.Brazda lui Novac, no. 4, 200585, Craiova 
E-mail: mircea.stanculesscu@yahoo.com  
 
The main purpose of the paper is to determine a general behavior of a multi-agent model capable of describing the process of 
deliberation of an investors group witch may repeatedly decide whether to buy or sell an asset. Each adaptive agent was 
modeled  as  a  collection  of  strategies  which  is  optimized  by  an  evolutionary  algorithm  (EA).  The  paper  investigates  the 
implications  and  the  effect  of  the  learning  process  for  the  information  strategies  used  by  the  agents  in  the  process  of 
deliberation of buy and sells order.   
 
Key words: Programming Models, Genetic algorithms, Information efficiency 
JEL Classification: C61, G14 
 
1.INTRODUCTION 
Behavioral traits of market participants and their effects on the properties of markets have been under growing 
investigation in recent years. Much of this work (e.g. [3], [4], [5], [6] and [8]) which emphasizes deviations from 
efficient markets theory, tries to elicit an internal market dynamics via computational approaches, and is loosely 
linked to the subject known as behavioral finance. These interdisciplinary studies try to explain market phenomena 
by formalizing a diverse array of behavioral and psychological considerations. Shiller, [11], strongly argues against 
a priori excluding any behavioral information from the explanations of financial markets. 
An agent-based approach contains a model for individual behavior combined with a price formation rule. Despite 
great emphasis on the characteristics of real market participants, however, models of individual behavior have 
remained rather simple. In an agent-based model, an agent’s psychology as distinct from manifest behavior is often 
not separately represented. Further, an agent is often modeled as a unit capable of being in one of a small number 
of states. 
We used a multi-agent model capable of describing the process of deliberation and how it leads to action. In our 
model, each agent is described by an internal state with an intrinsic dynamics and influenced by external events 
within or outside the population of agents. Actions result when the agent’s state crosses a predetermined threshold 
in state space. Specifically, we consider a population consisting of investors repeatedly deciding whether to buy or 
sell an asset. 
Each adaptive agent was modeled as an informational strategy which is optimized by an evolutionary algorithm 
(EA) ([1], [10]). EAs transfer the principles of natural evolution, first discovered by Darwin, to a computational 
setting. These algorithms have been used in the past, with considerable success, to solve difficult optimization 
problems  (Bäck  [1]).  Adaptive  agents  learn  in  different  ways  in  an  evolutionary  setting:  by  selection  and 
reproduction of successful strategies, and by random experimentation (by “mutating” existing strategies) or by 
recombining or “crossing over” previously-tested strategies. 
The  main  purpose  of  the  paper  is  to  determine  a  general  behavior  of  the  model  system  described  before  by 
evolutionary simulation. In the spirit of genetic algorithms methods, we classify the result of a simulation in group 
of strategies efficiency. A secondary important goal it will be to find some stable states of the system, states who 
generate a description of the wining strategies of the agents. 
 
2.THE MODEL OF INVESTORS BEHAVIOR 
2.1. Single investor mathematical model 
Here we try to quantitatively capture a simple, perhaps superficial, part of a financial agent’s psychology as well as 
its connection to actual behavior. This is kept to a bare minimum in order not to obscure our general approach. The 
model is based partly on results of so-called neural decision field theory (Ormutag, Knight and Sirovich [9]), 
adapted for financial investors.  
We assume that the state of an agent at time t is fully determined by the value of a variable x(t). This is the agent’s 
momentary internal condition. It may be viewed, in general, as the agent’s disposition or tendency to behave in a 
number of specific ways. ‘x’ determines the coordinates of state (or phase) space where each agent is represented 
as a point. The agent acts only when its representative point crosses a predetermined boundary or threshold in state 898 
space. More specifically we consider an investor repeatedly faced with the problem of deciding whether to buy or 
sell a particular asset. The agent’s ongoing process of deliberation is described by the dynamical system  
 
(1) 
where γ is the personal decision reaction speed, supplemented by the condition that at |x| = 1 the state is reset to x = 
0. The reset condition constrains the state to remain within a finite interval, −1≤x≤1. The function I(t) describes the 
effect of external influence on the agent. This may originate from all kinds of sources information such as the news 
media or personal contacts. Note that, despite appearances, (1) is non-linear, due to the reset condition. If we regard 
x as the fraction of certainty in buying/selling, then the choice of ±1 as thresholds appears less arbitrary. 
The effect of external information arrival was reprezented by a series of instantaneous jumps of size e(k) in the 
state received at arrival times tk,, k = 1, 2, .... This formulation is valid so long as the time scale of the impact of 
information is much shorter than that of other changes in x. The agent’s state is therefore driven by a term 
 
(2) 
where the sum is from the last moment of reset condition (the last buy or sell action) until the curent time t. It will 
be assumed the existence of a large number of uncorrelated sources of information and accordingly take arrivals as 
being Poisson distributed in time. For simplicity each jump will be taken as positive, e+ , or negative, e- . Following 
this formulation, we will refer below to positive and negative pieces of information. Generalizations to jumps with 
stochastic size or jump size which depends on the current state are straightforward. We take the magnitudes e± as 
measures of informational impact on a typical agent. In the special case of e+=−e−, one can speak of informational 
impact being symmetric. The informational impact magnitudes e±  describes the informational  strategy of the 
agent and characterize it. 
In this model the dynamics of the state leads to action in the following way: the agent places an order to buy (sell) 
when its state crosses the boundary at x=1 (respectively −1). Accordingly, when x(t) is in the positive (negative) 
half of the  domain, the  agent  has  a  greater  tendency  to  buy  (sell) and  may  be  viewed  as being  “optimistic” 
(“pessimistic”).  
 
2.2. Interactions 
We next consider a large collection of N agents trading the same asset. Each agent is indexed by i = 1,...,N, and its 
state described by the equation (1). Market participants clearly do not collectively listen to the same news and are 
not exposed to the same channels of information, then inter-agent differences in exogenous information are to be 
expected. In our model we allow every agent i to be driven by its private stream of information Ii(t). Informational 
heterogeneity along with nonlinear dynamical evolution and possibly different initial conditions, injects a great 
variety of individual behaviors into the population.   
The average rate of arrival of exogenous information is the same for all agents. Every agent, in addition, is affected 
by the behavior of other agents in the population. We denote by n the average number of agents affecting any one 
agent. Hence mutual interactions in the population occur through the influence of the actions of agents on the states 
of other agents. Specifically, if an agent places i an order to buy (resp. sell) at the moment tk , the state of another 
agent j is instantaneously incremented by eji+(−)(k) if that agent is affected by the action at that moment. The 
stream of information characterising the agent j  at the time t will be 
 
(3) 
Note that the eji term mesure the influence of the actions of agent i on the agent j and was taken sub-unitary. This 
formalizes the intuition that the more an investor sees others buying (selling) the more bullish (bearish) he or she 
will become.  
 
3. GENETIC ALGORITHM: LEARNING AND SIMULATION 
3.1. Evolutionary algorithms method 
Genetic algorithms constitute a class of search, adaptation, and optimization techniques based on the principles of 
natural evolution. Genetic algorithms were developed by Holland ([7]). For an introduction to genetic algorithms, 
see Bäck, T. ([1]). 
An evolutionary algorithm maintains a population of solution candidates and evaluates the quality of each solution 
candidate according to a problem-specific fitness function, which defines the environment for the evolution. New 
solution candidates are created by selecting relatively fit members of the population and recombining them through 
various  operators.  Specific  evolutionary  algorithms  differ  in  the  representation  of  solutions,  the  selection 
mechanism, and the details of the recombination operators. 
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Evolutionary algorithms offer a number of advantages over more traditional optimization methods. They can be 
applied to problems with a non-differentiable or discontinuous objective function, to which gradient-based methods 
such as Gauss–Newton would not be applicable. They are also useful when the objective function has several local 
optima.  
More formally, the GA scheme can be summarized as follows. P denotes the current population (i.e., a set of 
individuals), and H is the set of children. Note g the current generation number. The Genetic Algorithm can be 
described by the pseudocode scheme: 
   g := 1; 
   generate initial population Pop; 
   compute fitness for individuals i Î Pop; 
  WHILE  g< G  DO 
  BEGIN 
     g := g+1; 
    produce children H  from Pop by crossover; 
    apply mutation to children IÎ  H ; 
    Pop := PopÈ H  ; 
    compute fitness for entire population IÎ Pop; 
    reduce population Pop by means of selection; 
  END. 
 
3.2 Genetic Representation: coding the agents population and evaluating individuals by a ‘fitness’ function 
For  the  multi-agent  model  described  in  the  Section  2,  a  valid  genetic  representation  is  given  by  a  vector  of 
information’s about the personal decision reaction speed γi and the coefficients describing the influence of the 
others agents eij (j=1..N): 
        (γi ; ei1 , …, eiN)                  (4) 
At the start of the first generation of agents, each one of this is random generated between 0 and 1, with the 
supplementary condition that eij=0,  with the probability p (fixed from the beginning). 
For  evaluating  the  efficiency  of  individual  agent  information  strategy,  a  simulation  of  the  trade  activity  is 
necessary. We consider a trade game consisting in m rounds of negotiation (t=tk, k=1..m). Each agent i start the 
simulation with a fixed capital Ci(0)=10000 units. A buy/sell action of an agent modify his capital with a value 
equal with the price -/+ Pk of a market “action” at the time  tk; alternatively, the price Pk is variable in time, 
according to the formula: 
        Pk+1= Pk(1+n1(k)/N- n2(k)/N)               (5) 
where n1(k) is the number of agents buying actions at the time tk, and n2(k) is the number of agents selling actions 
at the time tk. This condition simulates the variation of value of the price in condition of a free market. The fitness 
function associated to each agent is defined by the value of individual capital Ci(m) at the time tm. At the tome t0=0, 
the price was P0=250 units. 
 
3.3 Crossover, Mutation and Selection 
Let us assume that two individuals of the current population have been selected for crossover. We have a mother 
individual M =(LM ) and a father individual F =(LF ). Now two child individuals have to be constructed, a daughter 
D = (LD ) and a son S = (LS ), each one by the 2-point crossover procedure, choused to increment the speed of the 
algorithm.  
For the crossover procedure we draw two random integers q1 and q2 with 1 £ q1 < q2 £ N. Now the daughter’s 
activity list LD is determined by taking the coefficient eij list  of the positions i = 1,...,q1 from the mother, the 
positions i=q1+1,...,q2 are derived from the father and the remaining positions i = q2+1, . . . ,J are again taken from 
the mother. The son individual is computed analogously. For the son’s activity list, the first and the third part are 
taken from the father and the second one is taken from the mother.  
The following mutation operator is applied to each newly produced individual. The mutation operator modifies the 
all the genes (including the coefficient of decision speed γ) of the genotype with a probability of pmutation. The 
mutation operator modifies the value of a coefficient by a fixed fraction +(-)f (considered in the author simulation 
=1/10) of his initial value. The choice of sign is randomized. 
The selection operator act on the entire populations of agents, the parents and the new agents, selecting at each 
generation the first N best evaluated individuals, grouped by pairs (first with the second etc.) in order to produce a 
new generation. The evaluation use the fitness function, computed after a trade game simulation.  
 
   900 
4. COMPUTATIONAL RESULTS  
In this section we present the results of the computational studies. The self-adapting GA for the RCPSP has been 
coded in Dev C++, and tested under Windows XP. We considered an initial population formed by N=50 agents, 
generated with the conditions 0<γi<1, 0<eij<1 and p=3/4 (the probability that eij=0 at the beginning). The genetic 
algorithm computed G=250 generations. 
First  observation  is  that  the  simulation  of  the  trade,  at  each  generation,  gives  the  possibility  to  evaluate  the 
evolution of the market price Pt of one action in time. The Figure 1 give the evolution in time of the Pt for the 
resulting populations of agents after 250 generations. The similarity with a real chaotic market variation has a test 
value for the proposed model.  
 
Figure 1. The market simulated evolution of the price of actions  
The results reported in the Figure 2 are obtained after a process of learning, when the strategies have converged. It 
is important to note that, during learning, the number of significatives influences for each agent of new generation 
(i.e. number of agents j that influence an agent i, expressed by the condition eij≠0) is increasing.  
 
Figure 2. Statistical dependence of the information strategies efficiency of the relative number of sources 
Five strategies are statistically identifiable:  
- first (1) is the strategy of stability of the sources (the values of the influence coefficients eij are quite stable in 
time); 
- the second (2) is a periodic strategy (generated by a quasi-periodical rotation of the trusted sources); 
- the third (3) can be called the “mimetic” strategy (the temporary tendence of the market is reproduced bu the 
agent); 
- (4) is a combination of (1) and (3); 
- (5) is a combination of (2) and (3). 
The Figure 2 show the statistical repartition of the ‘fitness’ for the final generation  agent’s strategies in dependence 
with the percent of the significative influence agents (reported at their total number). The ‘fitness’ value, measuring 
the efficiency of a information strategy, was reported to the best value obtained for all generations of agent’s 
strategies.  
 
5. CONCLUSION 
In conclusion, the model described in this paper explicitly includes competition into a real options framework by 
using an agent-based approach of competing firms. The firms derive their investment triggers from a genetic 
algorithm  which  exploits  the  results  of  repeated  stochastic  simulations  of  the  market.  The  best  information 
strategies used by the learning trade agents under competition that we find are classifiable in five groups, and the 
mixed mimetic strategies are in top of efficiency. In other words, the combined strategy is very versatile.  
 
   901 
REFERENCES 
1. Bäck, T. (1996) -  Evolutionary Algorithms in Theory and Practice. Oxford University Press, Oxford. 
2. Bouchaud, J.-P. (2002) - „An introduction to statistical finance”, Physica A, pp. 238-51. 
3. Challet, D. and Zhang, Y-C. (1997) “Emergence of cooperation and organization in an evolutionary game”, 
Physica A, VOL. 246, pp. 407. 
4. Chan, N., LeBaron, B., Lo, A.W., and Poggio, T (1998) - "Information dissemination and aggregation in asset 
markets with simple intelligent traders", MIT AI Lab Technical Memorandum 1646, M.I.T., Boston. 
5. Farmer, J.D. (2001) - Agent-based models for Investment, Association for Investment Management Resesarch. 
6. Farmer, J.D. (2002) – „Market Force”, Ecology and Evolution, Industrial and Corporate Change, VOL. 11, issue 
5, pp. 895-953. 
7. Holland, J.H. (1975) - Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor. 
8. Lux, T. and Marchesi, M. (1999) – „Scaling and criticality in a stochastic multi-agent model of a financial 
market”, Nature, VOL. 397, pp.498–500,  
9. Omurtag, A., Knight, B.W. and Sirovich, L. (2000) – „On the Simulation of Large Populations of Neurons”, 
Journal of Computational Neuroscience, VOL. 8, Issue 1, pp. 51 - 63 
10. Riechmann, T. (1998) - ''Genetic Algorithms and Economic Evolutions", Preprint Diskussion-papier Nr219, 
University Hannover. 
11.  Shiller,  R.J.  (1999)  –  „Human  Behavior  and  the  Effciency  of  the  Financial  System”,  Handbook  of 
Macroeconomics, VOL. 1, edited by J.B. Taylor and M. Woodford.  
   

Adaptation in Natural and Artificial Systems.

Agent-based models for Investment, Association for Investment Management Resesarch.

An introduction to statistical finance”,

Emergence of cooperation and organization in an evolutionary game”,

Evolutionary Algorithms in Theory and Practice.

Genetic Algorithms and Economic Evolutions&quot;, Preprint Diskussion-papier Nr219,

Human Behavior and the Effciency of the Financial System”,

Information dissemination and aggregation in asset markets with simple intelligent traders&quot;,

On the Simulation of Large Populations of Neurons”,

Scaling and criticality in a stochastic multi-agent model of a financial market”,

AN ADAPTATIVE EVOLUTIONARY MODEL OF FINANCIAL INVESTORS

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AN ADAPTATIVE EVOLUTIONARY MODEL OF FINANCIAL INVESTORS

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