Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol25.Issue6-7.May2001:
Journal of Economic Dynamics & Control
25 (2001) 1019}1037
Genetic algorithm learning and evolutionary
games
Thomas Riechmann*
Universita( t Hannover, Institut fu( r Volkswirtschaftslehre, Ko( nigsworther Platz 1, 30 167 Hannover,
Germany
Accepted 12 June 2000
Abstract
This paper links the theory of genetic algorithm (GA) learning to evolutionary game
theory. It is shown that economic learning via genetic algorithms can be described as
a speci"c form of an evolutionary game. It will be pointed out that GA learning results in
a series of near Nash equilibria which during the learning process build up to "nally
approach a neighborhood of an evolutionarily stable state. In order to characterize this
kind of dynamics, a concept of evolutionary superiority and evolutionary stability of
genetic populations is developed, which allows for a comprehensive analysis of the
evolutionary dynamics of the standard GA learning processes. 2001 Elsevier Science
B.V. All rights reserved.
JEL classixcation: C63}D73}D83
Keywords: Learning; Genetic algorithms; Evolutionary games
1. Introduction
Genetic algorithms (GAs) have frequently been used in economics to characterize a well de"ned form of social learning. They have been applied
* I would like to thank Shu-Heng Chen and two anonymous referees for their invaluable help, and
the Evangelisches Studienwerk Haus Villigst for "nancial support.
E-mail address: [email protected] (T. Riechmann).
Some frequently cited papers are Andreoni and Miller (1995), Arifovic (1994,1995,1996), Axelrod
(1987), Birchenhall (1995), and Bullard and Du!y (1998).
0165-1889/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 6 6 - X
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
to mainstream economics problems and mathematically analyzed as to
their speci"c dynamic and stochastic properties (Dawid, 1994; Riechmann,
1999). But, although widely seen as to conduct a rather evolutionary economic
line of thought, up to now there is no piece of work explicitly focusing on what
it is that makes GA learning an evolutionary kind of behavior. At the same
time, although there is a number of papers employing GA learning in a game
theoretic context, there is still a lack of an explicit comparison of the similarities
and di!erences between GA-learning and game theory. This paper "lls both
gaps.
In this paper, it is shown that genetic algorithm learning is an evolutionary
process and even more than this: It is an evolutionary game. As will be
demonstrated, these conceptual clari"cations are of considerable help in explaining the economic dynamics of the GA learning process.
The paper starts with a brief explanation of the genetic algorithm in focus. In
the following three sections, three propositions will be given that will be needed
in order to carry out a precise analysis of the GA learning process. These
propositions are (a) every GA is a dynamic game; (b) every GA is an evolutionary game; and (c) in GA learning processes, populations tend to converge
towards a Nash equilibrium. In Section 6, a concept of evolutionary superiority
and evolutionary stability is developed, which serves to apply a weak ordering
on the space of genetic populations. After this, GA learning can be characterized
by means of evolutionary concepts, which is comprehensively done in the fourth
section.
2. The canonical genetic algorithm
As there is a growingly large number of variants of genetic algorithms
in economic research, this paper will mainly deal with the most basic GA,
the so-called canonical GA, which is described in detail by Goldberg (1989).
As genetic algorithms have been well introduced into economic research, this
paper will not explicitly review the speci"c structure and working principles of
GAs.
More precisely, most of the "ndings of this paper will only apply to canonical,
one-population, economic GAs. In addition to the canonical GA being the most
standard one, there are two more decisive characteristics of genetic algorithms
dealt with in this paper. The "rst one is the fact that this paper does not
focus two- or more-population-GAs, like those used by e.g. Arifovic (1995) or
Birchenhall (1995). The second one is the fact, that this paper will only face
&economic' GAs.
The reader will be provided to have a basic understanding of genetic algorithms, which can be
gained from e.g. Goldberg (1989) or Mitchell (1996).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1021
Dexnition 1 (Economic genetic algorithm). An economic genetic algorithm is
a genetic algorithm with a state-dependent "tness function.
This means that in economic GAs the "tness of an economic agent does not
only depend on her own strategy, but also on the strategies of all other agents
involved in the model.
The genetic algorithms in focus are algorithms modeling processes of social
learning via interaction within a single population of economic agents. The
above sentence contains an implicit de"nition of social learning: Social learning
means all kinds of processes, where agents learn from one another. Examples for
social learning are learning by imitation or learning by communication. Learning by experiment, on the contrary, is no form of social learning. It is a form of
isolated individual learning.
The canonical GA is a stochastic process which repeatedly turns one population of bit strings into another. These bit strings are called genetic individuals. In
economic research they normally stand for some economic strategy or routine
in the sense of Nelson and Winter (1982). These routines are said to be used by
economic agents.
Each repeated &turn' of the genetic algorithm essentially consists of two kinds
of stochastic processes, which are variety generating and variety restricting
processes. Variety generating processes are processes which describe the development of new strategies by economic agents. These processes are models of
learning. In the canonical GA these processes are reproduction, which is interpreted as learning by imitation, crossover, which is interpreted as learning by
communication, and mutation, which is interpreted as learning by experiment.
All of these processes, or genetic operators, make use of some of the old
economic strategies (&parents') in order to "nd new ones (&children'). Then, the
parent strategies are replaced by their children, thus enhancing the variety of
strategies within the current population. In the canonical GA, there is one
variety restricting process, which is the genetic operator of selection. Selection
decreases the number of di!erent economic strategies within the population. It
"rst evaluates the economic success of each strategy, thus often being interpreted
as playing the role of the market as an information revealing device. Then it
selects strategies to be part of the next population. The selection operator of the
This means that GA models of e.g. the travelling salesman problem, which surely have an
economic subject, are nevertheless not &economic' GAs in the above meaning. For more information
on state dependency see Dawid (1999).
Sometimes, these &strategies' are so myopic, that a game theorist would rather call them &actions'.
It is important to clarify the following point: a genetic individual is not interpreted as an
economic agent, but as an economic strategy used by an economic agent. This interpretation allows
for several agents employing the same strategy.
Note, that this function of the market has already been described by Hayek (1978).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
Fig. 1. Structure of the canonical genetic algorithm.
canonical GA (often called roulette-wheel selection) does so by applying a
biased stochastic process: Selection for the next generation is done by repeatedly
drawing with replacement strategies from the pool of the old population to be
reproduced into the next one. The chance of each strategy to be drawn is equal
to its relative "tness, which is the ratio of its market success to the sum of the
market success of all strategies in the population. Thus, the number of di!erent
strategies within a population is reduced again. Fig. 1 shows an outline of the
genetic algorithm described above.
3. Economic genetic algorithms as dynamic games
Genetic algorithms have been applied to analyze learning in games before.
There is a number of papers which use a genetic algorithm to explicitly
formulate agents' behavior in economic games. This section shows that in fact
every economic genetic algorithm, even if it is not explicitly using a game
theoretic setting, is a game.
Game theoretic situations are su$ciently characterized as situations in which
the payo! for one player does not only depend on her own action but also on the
The "rst and most famous paper of this type is the one by Axelrod (1987). Another one is the one
by Marks (1992).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1023
actions of all other players involved in the same situation. In economic genetic
algorithms, all agents that are members of the same population are involved
within the same situation. Moreover, in models of economic GA-learning (see
De"nition 1 above), the "tness (alias economic success, alias game theoretic
payo!) of an economic agent depends on two things: (a) her own strategy and (b)
the strategies of all other members of the same population. In other words: The
"tness of an economic agent depends on the state of her population. In GAlearning-lingo this reads: Agents' "tness is state dependent.
Proposition 2 (GA-games). Every simple, one-population, economic genetic algorithm is a dynamic game.
Note, that it is state dependency, which is the common characteristic for both,
game theoretic situations and economic GA-learning models. More than this,
the game is repeated in every round of the genetic algorithm, so that every
individual is given the chance to alter and hopefully improve her strategy. Thus,
it can be concluded that in fact every simple, one-population, economic genetic
algorithm is a dynamic game.
As an illustration, imagine a cobweb model of the supply side of a market. In
every period of time, each agent represents a "rm which faces the problem of
maximizing its pro"t by chosing the appropriate quantity of supply. Each "rm's
"tness, which is the same as one period's pro"t, is given by
price
!unit costs).
(1)
"tness"agent's quantity ) (
GHI
GFFHFFI
'
The quantity the agent supplies re#ects her own strategy, whereas the ex-post
market price re#ects the state of the whole population, which means, given total
demand, it re#ects aggregate supply, i.e. the sum of all individual supply
strategies. While total supply has an important in#uence on each agent's "tness,
the cobweb model turns out to be a model of state-dependent "tness which is the
same as an economic game.
In technical terms, even more can be said about the types of games, GA
learning can be interpreted as. GA learning models describe a repeated economic game. Imagine a genetic algorithm using a population of M genetic
individuals with the length of each individual's bit string of ¸. Due to the binary
coding of genetic individuals, this means that each genetic individual represents
one out of 2* di!erent values. This means that the GA is able to deal with every
economic strategy in the set of all available strategies S, where S has the size
N"S"2*. Thus, the GA can be interpreted as a repeated symmetric one
population M person game with up to N pure strategies. But, compared to
A model of this type can be found in Arifovic (1994).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
&normal' evolutionary games, within most economic GA learning models, the
&rules' of the game are di!erent. Whereas in evolutionary games most of the time
a strategy is repeatedly paired with single competing strategies, in genetic
algorithm learning, each strategy plays against the whole aggregate rest of the
population. There is no direct opponent to a single strategy. Instead, every
economic agent aims to "nd a strategy i3S performing as good as possible
relative to its environment, which is completely determined by the current
population n and the objective function R( ) ).
4. Economic genetic algorithms as evolutionary games
Close relationships between economic learning models and models of evolutionary theory have been recognized before. Marimon (1993) gives a clear notion
of the similarities of learning models on the one hand and evolutionary processes on the other. As genetic algorithms, too, have been broadly interpreted as
models of economic learning, in this section it is argued that they can be
regarded as evolutionary processes as well.
At a "rst glance, it is the structure of genetic algorithms and evolutionary
models that suggests a close relationship between GAs and evolutionary
economic theory: Both face the central structure of a population of economic
agents interacting within some well de"ned economic environment and aiming
to optimize individual behavior.
As the aim of this paper is to describe economic genetic algorithms as
evolutionary processes, the "rst question to be answered is the question of
whether GAs are evolutionary processes at all. In the following, it will be argued
that GAs are a speci"c form of evolutionary processes, i.e. evolutionary games.
Proposition 3 (GA-evolutionary games). Every simple, one-population, economic
genetic algorithm is an evolutionary game.
In order to give some evidence for this proposition, a de"nition is needed that
clearly states what an evolutionary game is. This paper makes use of the
de"nition by Friedman (1998, p. 16), who gives three characteristics for an
evolutionary game:
Dexnition 4 (Evolutionary game; Friedman). An evolutionary game is a
dynamic model of strategic interaction with the following characteristics:
While this notion is true for most of the economic GA learning models, it is not true for GA
models that explicitly describe economic games, including Axelrod (1987) and Andreoni and Miller
(1995).
For such an interpretation see e.g. Dawid (1999).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1025
(a) higher payo! strategies tend over time to displace lower payo! strategies;
(b) there is inertia; (c) players do not intentionally in#uence other players' future
actions.
Prior to checking these three points, it is important to note that economic
GAs are in fact models of strategic interaction, which has implicitly already been
stated in Proposition 2. In the interpretation as models of social learning, GAs
deal with a number of economic agents, each trying to "nd a behavioral strategy
which, due to her surrounding, gives her the best payo! possible. GAs are
models of social learning, which in fact is a way of learning by interaction. Thus,
it can be stated, that GAs are in fact models of &strategic interaction'.
Now, let us focus on Friedman's (De"nition 4) three characteristics:
(a) Higher payo! strategies tend over time to displace lower payo! strategies.
Displacement of strategies in genetic algorithms is a process of change in the
genetic population over time. This, in turn, is a question of selection and
reproduction. GAs are in fact dynamic processes which reproductively prefer
higher payo! strategies to lower payo! ones. It has been shown that in the
canonical GA, the probability of a strategy i to be reproduced from its current
population n into the population of the next period, P(in ), depends only on its
relative "tness R(in ), which is the strategy's payo! or market success relative to
the aggregate payo! of the population n . Higher relative "tness leads to
a higher reproduction probability
dP(in )
'0.
dR(in )
(2)
Thus, Friedman's condition (a) is satis"ed.
(b) There is inertia.
According to Friedman, inertia means that &2changes in behavior do not
take place too abruptly'. (Friedman, 1998, p. 16).
Looking at the genetic or game theoretic population, it is mutation, or
learning by experiment, which causes the most abrupt changes. Whereas the
strategy of a single economic agent might be changed more dramatically by
imitation or communication, this is not true for the population as a whole. At
the level of the population, only learning by experiment is able to introduce
strategies or at least parts of strategies which have not occurred in society
before. Thus, it should be proved that small changes by mutation are more likely
than big ones. For the standard GA, using binary representation of genetic
For a more precise description, cf. Riechmann (1999).
This is valid for all variants of GA-selection processes, not only for the standard roulette-wheel
selection (Goldberg, 1989). For more, di!erent types of GA-selection operators see e.g. Goldberg and
Deb (1991) and Michalewicz (1996).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
individuals, the mutation operator is quite simple. Mutation randomly alters
(&#ips') single bits of the bit string encoding an economic strategy. Each bit
of the string has a small probability to be changed. , which is called the
mutation probability, is uniform for every bit of any genetic individual of
every population.
To carry out an analysis concerning the connection of mutation and abruptness of change, a measure for the abruptness is needed. This paper will make use
of the Hamming distance between two genetic individuals. According to the
Hamming distance, a change from individual i to individual j is said to be more
abrupt than the change from individual i to k, if the Hamming distance between
i and j, H(i, j) is greater than the Hamming distance between i and k, H(i, k).
The question of whether small changes by mutation are more likely than big
ones can be answered as follows: The probability of an economic strategy i to be
turned into strategy j by mutation, P (i, j), depends on the length of the genetic
K
individuals' bit string, ¸, the mutation probability , and the number of bits that
have to be #ipped in order to turn i into j, i.e. the Hamming distance between
i and j, H(i, j)
P (i, j)"&G H(1!)*\&G H.
K
Di!erentiation yields
(3)
(0 for (,
P (i j)
K
"&G H(1!)*\&G H[ln !ln(1!)] "0 for ",
(4)
H(i, j)
'0 for '.
For the usual parameter values of , this means the obvious: Small changes in
strategy are more likely than big changes. Thus, it becomes evident that GA
learning processes are processes which contain some inertia.
(c) Players do not intentionally in#uence other players' future actions.
This point can be proved more verbally. The agents modeled by a simple
economic GA have very restricted knowledge. By the time an economic agent
forms her latest economic strategy she does not know anything about the plans
The Hamming distance between the genetic individuals i and j, in short, is the number of places,
in which these genetic individuals di!er, i.e.the number of bits to be #ipped in order to turn
individual i into j. See e.g. Mitchell (1996, p. 7).
Normally, the mutation probability ranges somewhere between and .
The result given in (3) deserves two further remarks. First, the fact that for ' big changes are
more likely than small ones explains the fact that for relatively large values of , GA results are close
to white noise. Second, the result yields an interesting interpretation for the "eld of economic
learning. If mutation is interpreted as learning by experiment, (3) shows that a little experimenting is
a good thing to do, while too many experiments will disturb the generation of valuable new
economic strategies. If mutation is interpreted as making mistakes in imitation or communication
(see e.g. Alchian, 1950), Eq. (3) simply means that one should not make too many of those mistakes.
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1027
of the other agents in her population. All an economic agent in a GA model can
do is to try her best to adopt to her neighbors' past actions, for the near past is all
such an economic agent can remember. Taking into account these very limited
individual abilities, it becomes obvious, that there is no room for intentional
in#uences on other agents' actions.
From the above it can be concluded, that models of economic GA learning
are in fact models which can be interpreted as evolutionary games.
5. Genetic populations as near Nash equilibria
The main structure (information scientists would call it &data structure') in
genetic algorithm learning models is the genetic population. A population
contains all the agents that are economically active in a certain period of time.
The agents can completely be characterized by their economic strategy. Thus,
a population can be described by counting how often each of the di!erent
possible strategies is used by the members of the population. Thus, one way of
representing a population is to give the (absolute or relative) frequency of each
possible strategy within the population. This means that a population is nothing
more than a distribution of di!erent economic or behavioral strategies. This is
true for genetic populations as well as for populations in their game theoretic
interpretation. It can be said that a genetic population is a game theoretic
population.
From Section 4 it is known, that every economic agent aims to "nd the best
performing strategy i3S with respect to the objective function R( ) ) and given
the strategies of the rest of her population n . This means that every economic
agent faces problem (5)
max R(in ).
(5)
GZ1
This immediately leads to the concept of Nash equilibria. A Nash strategy is
de"ned as the best strategy given the strategies of the competitors, and a Nash
strategy is exactly what every economic agent, alias genetic individual, is trying
to reach.
Proposition 5 (GA populations * near Nash equilibrium). In every simple, onepopulation, economic genetic algorithm, the population tends over time to move to
a Nash equilibrium without fully reaching it.
In fact, genetic algorithms can be shown to be Markov processes, the main characteristic of
which is the &no memory property'. This property says that there is no memory to the history of the
GA process.
For an in-depth discussion of this, refer to Davis and Principe (1993) or Riechmann (1999).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
As an illustration, in the cobweb model by Arifovic (1994), there is no single,
direct opponent to the "rm. Instead, it is the aggregate output of all the
competitors which each "rm has to take into account. This is to say: Although
there is no direct opponent to agents in a market, which is di!erent from most of
the &normal' evolutionary games, markets can still be seen as games, even if the
rules are slightly di!erent. The main point still remains the same: Agents are
trying to play Nash.
It has already been shown that selection and reproduction work in favor of
relatively well performing strategies. This means that via selection and reproduction strategies which are not Nash strategies are eliminated from the population, whereas the population share of Nash strategies is increased. At the same
time, due to the fact that agents do not stop experimenting, some agents carry
out experiments that result in non-Nash strategies. Thus, although selection and
reproduction drive the population towards a Nash equilibrium, learning by
experiment prevents the population from fully reaching such an equilibrium.
This means that genetic populations represent a state which is not a real Nash
equilibrium but which is not far from it. This state will be called &near-Nash
equilibrium' throughout this paper.
Thus, at a "rst step, an economic genetic algorithm can be viewed as modeling
a system of economic agents, each of them trying to play a Nash strategy against
the rest of the population. In economic terms this means that every agent tries to
coordinate her strategy with the other agents' ones, for this is the best way of
maximizing her pro"t (or utility or payo! or whatever the model wants the
agent to maximize). The population is driven to the neighborhood of a Nash
equilibrium by the forces of the the market, which are represented by selection in
economic genetic algorithms. But, due to the e!ects of ongoing experimentation,
the population will never be able to fully reach the Nash equilibrium.
6. Evolutionary stability of genetic populations
While a genetic population represents a primarily static concept, learning is of
course a genuinely dynamic process. Thus, in order to analyze GA learning as
an evolutionary learning process, questions regarding the dynamics and the
concepts of stability have to be addressed.
This is, of course a slight simpli"cation: Ruling out non-Nash strategies may alter the
population in a way that make former Nash strategies non-Nash and vice versa.
Replicator dynamics (see e.g. Weibull, 1995 or Hofbauer and Sigmund, 1988, 1998), which have
often been used to characterize evolutionary dynamics, seem to be unsuited for some economic
problems. (Mailath, 1992, p. 286 even suggests that &there is nothing in economics to justify
replicator dynamics'.) Applied to the analysis of GA learning, replicator dynamics, not directly
accounting for stochastics, are simply not precise enough to cover the whole GA learning process.
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1029
This paper will make use of the concept of evolutionary stability, especially
the notion of evolutionarily stable strategies or evolutionarily stable states
(ESS). See i.e. Maynard Smith (1982), Hofbauer and Sigmund (1988, 1998),
Samuelson (1997, Weibull (1995), Marks (1992), or Mailath (1992). In short,
a strategy is evolutionarily stable if, relative to its population, it performs better
than any new, &invading' strategy. Though widely used in economic dynamics,
the concept of ESS has a serious weakness which makes it only limitedly suitable
for the analysis of genetic algorithms: There is no explicit formulation of the
selection process underlying the concept of evolutionary stability. ESS are based
on the notion that invading &mutant' strategies are somehow rejected or eliminated from the population. It is not clear how this rejection is carried out.
Genetic algorithms, in contrast, present a clear concept of rejection:
Every strategy will be exposed to a test, which has been described as a
one-against-the-rest game in the previous sections. Then the strategy will be
reproduced or rejected with a probability depending on its performance (i.e.
market performance) in this game. Thus, the rejection of strategies in GA
learning models is a question of reproduction. GA reproduction has two main
features, it selects due to performance and it selects due to probability, which
means that a bad strategy will be rejected almost surely although not with
probability one.
Thus, a re"ned concept of evolutionary stability is needed for genetic algorithms. A possible way of setting up a concept of evolutionary stability
for genetic algorithms which is keeping the spirit of the ESS is the following:
A genetic population is evolutionarily stable if the process of the genetic
algorithm rejects an invasion by one or more strategies from the genetic
population. Invasion itself can either take the form of a totally new strategy
entering the population or it can simply mean a change in the frequency of
the strategies already contained within the population. Thus, a more precise
de"nition of an evolutionarily stable population might be: A population
is evolutionarily stable if it is resistant against changes in its composition (see
De"nition 7).
6.1. Evolutionary superiority
More formally, a genetic population n will be called evolutionarily superior to
m ) if it exhibits two characteristics:
population m , (denoted as n '
(a) Every strategy i contained within population n gains at least the same "tness
in the basic population n as it gains in the invaded population m , while at
least one strategy gains even more "tness in n than in m .
(b) The invading strategies k3m n are the worst performing strategies contained in m , so that they will be most surely rejected.
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Note that the above characterizes a kind of weak dominance concept. This
de"nition of evolutionary superiority induces a partial ordering on the space of
genetic populations, which resembles the concept of Pareto superiority.
In more mathematical terms, the de"nition reads:
Dexnition 6 (Evolutionary superiority). A genetic population n is called evolutionarily superior to population m , if
R(in )5R(im )
∀i3n
(6)
j with R(jn )'R(jm )
(7)
R(km )(R(im )
(8)
∀i3n ; ∀k3m n .
To illustrate this concept, consider the following example of an oligopoly
game. Assume a population of "ve "rms, which covers the whole supply side of
the market. Demand is given exogenously and does not change over time. The
current population is characterized by the vector of output quantities, i.e.
n "(3, 2, 2, 3, 2).
The "rst element of n gives the output level of "rm number one, the second
element gives the output of "rm number two and so on. Let us further suppose
that the resulting aggregate output level of 12 leads to a market price that in turn
leads to positive pro"ts for every single "rm within the population. Now, at the
end of the current period, "rm number two starts experimenting and decides to
change its output level to 10. In other words: Infection is taking place, changing
population n to
m "(3, 10, 2, 3, 2).
Due to this infection, aggregate supply increases, inducing a decrease in the
market price. This of course leads to diminishing pro"ts of each xrm within the
population. For simplicity, assume that the price decreases so much, that it falls
below unit costs. In this case, instead of making pro"ts, "rms run at a loss. This
is especially true for "rm number two, the loss of which is the biggest throughout
the population. Accordingly, what will happen in the near future is the following: Firm number two recognizes that it is currently performing very poorly and
adopts a di!erent strategy from one of its competitors. In terms of genetic
algorithm theory: Sooner or later the selection operator will discard strategy
This point has, in earlier drafts of this paper, led to misconception: Neither the concept of
evolutionary superiority nor the concept of Pareto superiority make any statements about some
kind of welfare. Originally, Pareto superiority is just a means to order points within a highly
dimensioned space. Applying evolutionary superiority analogously to the original meaning of the
Pareto criterion has no welfare implication at all. It is just used in order to make genetic populations
weakly comparable with respect to the process of the genetic algorithm and GA's way of turning one
population into another.
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1031
number two and replace it by a strategy from the rest of the population. By that,
the infection that turned population n into population m is rejected. Though this
does not inevitably mean that population n is directly regained by that process, it
shows at least, that in the light of GA dynamics, n is &better' than m . Population
n is evolutionarily superior to population m . This is veri"ed by application of the
two criteria from above: Strategies one and three to "ve (i.e. the output strategy of
"rm number one and "rms number two to "ve) lead to greater payo! in
population n than in population m . Hence, criterion (a) is met. Moreover, strategy
number two in population m is the worst performing strategy in this population.
By this, criterion (b) is met as well. Population n is evolutionarily superior to m . At
last, let us focus evolutionary superiority of n over m by using De"nition 6. The
"tness of each of the "ve "rms' strategies is higher in population n than in m (Eqs.
(6) and (7)). Note that this is especially true for "rm number two, which su!ers
from a particularly high loss due to its change in strategy. Firm number two is the
only one that has changed its strategy and at the same time it is the worst
performing "rm, thus ful"lling condition (8).
For full validity, a further remark is necessary: Within genetic algorithms,
invading strategies can only result from reproduction (&imitation'), crossover
(&communication') or mutation (&experiment') within the population itself. This
means that the "nal outcome of GAs without mutation (i.e. processes with
learning by imitation and communication only), which are uniform populations,
may have other populations being superior to them, but } without mutation
} better populations simply cannot arise.
6.2. Evolutionarily stable populations
A population is evolutionarily stable in the concept of De"nition 6 if there is
no other population within S, the set of all populations, which is evolutionarily
superior to it.
Dexnition 7 (Evolutionary stability of genetic populations). A genetic population n is called an evolutionarily stable population, if
CQ n .
x m 3S with m '
(9)
De"nition 7 is a generalization of the concept of evolutionary stability.
Note, that due to De"nition 7 more than just one evolutionarily stable genetic
population can exist. Thus, once an evolutionarily stable genetic population has
been reached, two things can result from an infection of the population: Either
the infecting strategy is rejected from the population and the original population
See Riechmann (1999) for the restrictions di!erent learning techniques impose on the set of
available strategies.
Note e.g. the similarity to (Weibull's 1995, pp. 36) de"nition.
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
is regained, or infection causes the transition to another evolutionarily stable
population, provided there is one.
Due to the fact that genetic algorithm selection is a probabilistic rather than
a deterministic process, invading strategies, even in an evolutionarily stable
population, may not be rejected within a single round of the algorithm. It can
only be stated that the invader will be driven out of the population within "nite
time. That is to say: If a genetic population is evolutionarily stable, it will recover
from an invasion within a "nite number of steps of the GA, which means that in
the long run the population will not lastingly be changed. Nevertheless, once an
evolutionarily stable population is invaded, there may appear a number of
evolutionarily inferior populations within the next few rounds of the GA. These
populations represent transitory states of the process of rejecting the invader.
Riechmann (1999) shows that there is in fact more than one population that will
occur in the long run. These may be transitory populations as well as di!erent
populations which are evolutionarily stable, too.
7. Evolutionary dynamics
According to the evolutionary superiority (De"nition 6), the GA always selects
in favor of the superior population. This notion can be used to characterize GA
learning dynamics: The genetic algorithm as a stochastic process continuously
discards populations in favor of better ones in the sense of evolutionary superiority. But this only describes the direction of the process, not the exact transition
path over time. In fact, due to the stochastic properties of genetic algorithms, the
exact path of the process highly depends on the initial conditions, i.e. the
composition of the very "rst genetic population. And although the path up to an
evolutionarily stable equilibrium may di!er, Markov chain theory shows that in
the long run a "xed and stable distribution of population will be reached. This will
even happen irrespective of the starting conditions. There may be path dependence, lock-ins, or related phenomena, but in the case of genetic algorithm learning
these will only be of temporary nature. In the long run, genetic algorithm theory
promises, the &best' state will be reached.
Again, see Riechmann (1999) for a more detailed explanation.
This may be regarded as a weakness of the concept of genetic algorithm learning, as it neglects the
possibility of modelling path dependence or lock-ins. So it may be worthwhile to mention two further
points, which are mainly beyond the scope of this paper. First, depending on the underlying (economic)
problem, some GAs spend long times supporting populations which are not evolutionarily stable.
Some keywords pointing to this topic are &deceptiveness' of genetic algorithms and the problem of
&premature convergence'. Secondly, the lack of ability to model lasting lock-ins or path dependence
applies to the basic genetic algorithm. There are variations of genetic algorithms capable of modelling
these phenomena. One keyword pointing into this direction of research may be &niching mechanisms'.
Again, a good starting point for more descriptions of all of the various special cases and variants of
GAs is Goldberg (1989).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1033
Knowing the special form of the dynamic process of the GA and the direction
in which this process will lead, a few more words can be said about the role of
heterogeneity for the dynamics. It seems important to notice the way economic
change takes place. Starting with an arbitrary population, genetic operators (i.e.
learning mechanisms) cause changes in the population, while new types of
behavior are tested. The test is performed by exposing the strategies to the
market. The market reveals the quality of each tested strategy relative to all
other strategies within the population. Then selection leads economic agents to
abandon poorly performing strategies and adopt better ones (imitation) or even
create new ones by communication (crossover) or experimentation (mutation).
After that, again, strategies are tested and evaluated by the market, by that way
coordinating the agents' strategies, and so on.
There are two crucial aspects of this repeated process: First, it is the diversity
of strategies that drives economic change, i.e. the succession of populations
constantly altering their composition. Under the regime of genetic algorithm
learning, this change in individual as well as in social behavior heavily (while not
entirely) relies on learning by imitation and learning by communication.
Evidently, these kinds of learning can only take place within heterogeneous
populations. The more diverse a population is, the greater is the number of
di!erent strategies or even parts of strategies that a member of this population
can learn by imitating the other members or by communicating to them. Thus,
in a way, it can be said that it is heterogeneity that is the main driving force
behind economic change.
The second crucial aspect of the process of genetic algorithm learning is the
role of selection, which can be interpreted as the role of the market. While the act
of learning will be enough to achieve economic change, economic development,
i.e. the movement of genetic population towards an evolutionarily stable state,
can only be reached by the cooperation of learning and selection. In order to
turn the succession of di!erent populations into the succession of constantly
improving populations (in the sense of evolutionary superiority), a device is
needed that makes it possible to distinguish successful strategies from less
successful ones. Having at hand such a device, it is possible to decide
which strategies shall live and grow and which ones shall die. This device is the
market in economics as it is the selection operator within genetic algorithms. It
is the market and only the market that turns economic change into economic
development.
Birchenhall et al. (1997) clearly show that there is a connection between the extent of diversity in
a population and the learning speed that is achieved by members of that population.
To this interpretation, &development' is a term of stability rather than optimality.
This re#ects a rather classical economic thought, given, e.g., in Hayek (1969) (usually quoted as
Hayek, 1978).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
Summarizing, under the regime of the market, the evolutionary dynamic of
genetic algorithm learning is mainly driven by two forces: Heterogeneity, which
constantly induces behavioral (and by that, economic) change, and the market
as a coordination device, revealing information about the quality of each type of
behavior and ruling out poorly performing strategies, thus turning economic
change into economic development.
Finally, looking at genetic algorithm learning from an evolutionary point of
view, one more point has to be added. The evolutionary dynamic of the GA
learning process is essentially a two-stage process. It has been shown that, as
long as possible, genetic algorithm learning and market selection &improve'
individual and as a result social behavior. But, this is just the "rst stage of the
underlying dynamics. Yet, once an evolutionarily stable state of behavior has
been reached, there certainly is no room for further &improvement'. But, due to
the special structure of genetic algorithms, this does not mean that in this state
economic agents stop changing their behavior. Instead, at this point the second
stage of the dynamic process starts. Learning, or what has above been called
change, still continues and will not cease to continue. Still, there will appear new
ways of individual behavior within a population. Now it is the role of the market
(i.e. selection) to drive these strategies out of the population again. Due to the
probabilistic nature of the GA, this process may take more than one period, thus
producing one or even more transitory populations until an evolutionarily stable
population is regained. To put it in di!erent words: Even after an evolutionarily
stable state is reached, evolutionary stability is continuously challenged by new
strategies. While in the "rst phase of the GA learning process some of the new
strategies are integrated into the population, in the second phase all of the
invaders will be replaced again. So there is an ongoing near-equilibrium movement resulting from the continuous rejection of invading strategies.
In fact, genetic algorithm learning leads to an
interplay of coordinating tendencies arising from competitive adaptions in
the markets and de-coordinating tendencies caused by the introduction of
novelty
(Witt, 1993, p. xix), which has often been regarded as a key feature of evolutionarily economic analysis of the market. See Witt (1985).
By means of Markov chain theory, it has been shown before, that the
canonical GA does neither lead to a lasting state of total uniformity of individuals in their populations nor to a state where populations ultimately stop
changing over time. These long-run dynamics of GA learning processes have (in
mathematical terms) been characterized as a state of Lyapunov stability
(Riechmann, 1999). Although a plain technical explanation for this behavior can
be found, up to now it has not been made clear, what this explanation means in
terms of learning and market behavior. With the help of evolutionary game
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1035
theory, a clear-cut economic reasoning can be found, why this state of Lyapunov
stability shows up: It is a process of near-equilibrium dynamics, caused by the
continuously ongoing challenge of the ESS by newly learned strategies and the
rejection of these strategies by the market that prevents social behavior from
total convergence but still keeps it close enough to a stable state.
8. Conclusions
Economic genetic algorithm learning can be shown to be a speci"c form of an
evolutionary game. In this paper, this has been discussed for the most basic form
of genetic algorithm, the canonical GA. For populations of GAs, concepts of
evolutionary superiority and evolutionary stability are developed, which help to
explain the way processes of GA learning behave. This behavior turns out to be
an evolutionary, two step process: First, there is a movement of populations
towards an evolutionarily stable state. Second, once such a state has been
reached, the learning process turns into a near equilibrium dynamic of getting
out of evolutionarily stable states and returning there again.
The notion of GA learning as an evolutionary process can be transfered to the
analysis of modi"ed genetic algorithms. As an example, consider certain changes
to the selection operator. Elitist selection schemes, including the selection
within evolution strategies cf. BaK ck et al. (1991) and Arifovic's (1994) election
operator, ensure that at least the "rst best genetic individual of a population
will become a member of the next generation's population. In contrast to
roulette-wheel selection, elitist selection ensures that invading strategies which
turn out to be the worst strategies throughout the population will be replaced at
once. This means that there is no room for transitory populations. Bad strategies, i.e. strategies obeying condition (8), are ruled out even before they can
enter a population. This certainly leads, in most cases to asymptotic behavioral stability.
Summarizing, the results of this paper demonstrate that research in the
dynamics of economic GA learning models can be equipped with the whole tool
box of evolutionary game theory. Using these instruments, in this "eld of
science, much more work can and has to be done.
A survey on various selection schemes can be found in Goldberg and Deb (1991).
For an interpretation and an extension of the election operator, see Franke (1997).
It has been largely neglected throughout this paper, that the stability properties of at least some
variations of economic GAs do certainly depend on the underlying economic problem, too. This
means that there might be some problems that can cause lasting changes of the population even in
GAs with elitist selection. Examples for this might be found at genuinely cyclic problems like cases of
Lotka}Volterra dynamics, see, e.g. Hofbauer and Sigmund, 1998, pp. 11, which, in economics, have
been applied to models of business cycles, Goodwin (1967).
1036
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
References
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211}221.
Andreoni, J., Miller, J.H., 1995. Auctions with arti"cial adaptive agents. Games and Economic
Behavior 10, 39}64.
Arifovic, J., 1994. Genetic algorithm learning and the cobweb-model. Journal of Economic Dynamics and Control 18, 3}28.
Arifovic, J., 1995. Genetic algorithms and in#ationary economies. Journal of Monetary Economics
36, 219}243.
Arifovic, J., 1996. The behavior of the exchange rate in the genetic algorithm and experimental
economies. Journal of Political Economy 104, 510}541.
Axelrod, R., 1987. The evolution of strategies in the iterated prisoner's dilemma. In: Davis, L. (Ed.),
Genetic Algorithms and Simulated Annealing. Pitman, London, pp. 32}41.
BaK ck, T., Ho!meister, F., Schwefel, H.-P., 1991. A survey of evolution strategies. In: Belew, R.K.,
Booker, L.B. (Eds.), Proceedings of the 4th International Conference on Genetic Algorithms.
Morgan Kaufmann, San Mateo, CA, pp. 2}9.
Birchenhall, C., 1995. Modular technical change and genetic algorithms. Computational Economics
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Davis, T.E., Principe, J.C., 1993. A markov chain framework for the simple genetic algorithm.
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Dawid, H., 1994. A Markov chain analysis of genetic algorithms with a state dependent "tness
function. Complex Systems 8, 407}417.
Dawid, H., 1999. Adaptive Learning by Genetic Algorithms, 2nd Edition. Springer, Berlin.
Franke, R., 1997. Behavioral heterogeneity and genetic algorithm learning in the cobweb
model. Discussion Paper 9, IKSF } Fachbereich 7 } Wirtschaftswissenschaft, UniversitaK t
Bremen.
Friedman, D., 1998. On economic applications of evolutionary game theory. Journal of Evolutionary Economics 8, 15}43.
Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning.
Addison-Wesley, Reading, MA.
Goldberg, D.E., Deb, K., 1991. A comparative analysis of selection schemes used in genetic
algorithms. In: Rawlins, Gregory J.E. (Ed.), Foundations of Genetic Algorithms. Morgan
Kaufmann, San Mateo, CA, pp. 69}93.
Goodwin, R.M., 1967. A growth cycle. In: Feinstein, C.H. (Ed.), Socialism, Capitalism and Economic
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Hayek, F.A.v., 1969. Freiburger Studien. Der Wettbewerb als Entdeckungsverfahren. J.C.B. Mohr
(Paul Siebeck), TuK bingen, pp. 249}265 (Chapter 15).
Hayek, F.A.v., 1978. New studies in philosophy, politics, economics and the history of ideas.
Competition as a Discovery Process. Routledge & Kegan Paul, London, pp. 179}190 (Chapter
12).
Hofbauer, J., Sigmund, K., 1988. The Theory of Evolution an
25 (2001) 1019}1037
Genetic algorithm learning and evolutionary
games
Thomas Riechmann*
Universita( t Hannover, Institut fu( r Volkswirtschaftslehre, Ko( nigsworther Platz 1, 30 167 Hannover,
Germany
Accepted 12 June 2000
Abstract
This paper links the theory of genetic algorithm (GA) learning to evolutionary game
theory. It is shown that economic learning via genetic algorithms can be described as
a speci"c form of an evolutionary game. It will be pointed out that GA learning results in
a series of near Nash equilibria which during the learning process build up to "nally
approach a neighborhood of an evolutionarily stable state. In order to characterize this
kind of dynamics, a concept of evolutionary superiority and evolutionary stability of
genetic populations is developed, which allows for a comprehensive analysis of the
evolutionary dynamics of the standard GA learning processes. 2001 Elsevier Science
B.V. All rights reserved.
JEL classixcation: C63}D73}D83
Keywords: Learning; Genetic algorithms; Evolutionary games
1. Introduction
Genetic algorithms (GAs) have frequently been used in economics to characterize a well de"ned form of social learning. They have been applied
* I would like to thank Shu-Heng Chen and two anonymous referees for their invaluable help, and
the Evangelisches Studienwerk Haus Villigst for "nancial support.
E-mail address: [email protected] (T. Riechmann).
Some frequently cited papers are Andreoni and Miller (1995), Arifovic (1994,1995,1996), Axelrod
(1987), Birchenhall (1995), and Bullard and Du!y (1998).
0165-1889/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 6 6 - X
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
to mainstream economics problems and mathematically analyzed as to
their speci"c dynamic and stochastic properties (Dawid, 1994; Riechmann,
1999). But, although widely seen as to conduct a rather evolutionary economic
line of thought, up to now there is no piece of work explicitly focusing on what
it is that makes GA learning an evolutionary kind of behavior. At the same
time, although there is a number of papers employing GA learning in a game
theoretic context, there is still a lack of an explicit comparison of the similarities
and di!erences between GA-learning and game theory. This paper "lls both
gaps.
In this paper, it is shown that genetic algorithm learning is an evolutionary
process and even more than this: It is an evolutionary game. As will be
demonstrated, these conceptual clari"cations are of considerable help in explaining the economic dynamics of the GA learning process.
The paper starts with a brief explanation of the genetic algorithm in focus. In
the following three sections, three propositions will be given that will be needed
in order to carry out a precise analysis of the GA learning process. These
propositions are (a) every GA is a dynamic game; (b) every GA is an evolutionary game; and (c) in GA learning processes, populations tend to converge
towards a Nash equilibrium. In Section 6, a concept of evolutionary superiority
and evolutionary stability is developed, which serves to apply a weak ordering
on the space of genetic populations. After this, GA learning can be characterized
by means of evolutionary concepts, which is comprehensively done in the fourth
section.
2. The canonical genetic algorithm
As there is a growingly large number of variants of genetic algorithms
in economic research, this paper will mainly deal with the most basic GA,
the so-called canonical GA, which is described in detail by Goldberg (1989).
As genetic algorithms have been well introduced into economic research, this
paper will not explicitly review the speci"c structure and working principles of
GAs.
More precisely, most of the "ndings of this paper will only apply to canonical,
one-population, economic GAs. In addition to the canonical GA being the most
standard one, there are two more decisive characteristics of genetic algorithms
dealt with in this paper. The "rst one is the fact that this paper does not
focus two- or more-population-GAs, like those used by e.g. Arifovic (1995) or
Birchenhall (1995). The second one is the fact, that this paper will only face
&economic' GAs.
The reader will be provided to have a basic understanding of genetic algorithms, which can be
gained from e.g. Goldberg (1989) or Mitchell (1996).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1021
Dexnition 1 (Economic genetic algorithm). An economic genetic algorithm is
a genetic algorithm with a state-dependent "tness function.
This means that in economic GAs the "tness of an economic agent does not
only depend on her own strategy, but also on the strategies of all other agents
involved in the model.
The genetic algorithms in focus are algorithms modeling processes of social
learning via interaction within a single population of economic agents. The
above sentence contains an implicit de"nition of social learning: Social learning
means all kinds of processes, where agents learn from one another. Examples for
social learning are learning by imitation or learning by communication. Learning by experiment, on the contrary, is no form of social learning. It is a form of
isolated individual learning.
The canonical GA is a stochastic process which repeatedly turns one population of bit strings into another. These bit strings are called genetic individuals. In
economic research they normally stand for some economic strategy or routine
in the sense of Nelson and Winter (1982). These routines are said to be used by
economic agents.
Each repeated &turn' of the genetic algorithm essentially consists of two kinds
of stochastic processes, which are variety generating and variety restricting
processes. Variety generating processes are processes which describe the development of new strategies by economic agents. These processes are models of
learning. In the canonical GA these processes are reproduction, which is interpreted as learning by imitation, crossover, which is interpreted as learning by
communication, and mutation, which is interpreted as learning by experiment.
All of these processes, or genetic operators, make use of some of the old
economic strategies (&parents') in order to "nd new ones (&children'). Then, the
parent strategies are replaced by their children, thus enhancing the variety of
strategies within the current population. In the canonical GA, there is one
variety restricting process, which is the genetic operator of selection. Selection
decreases the number of di!erent economic strategies within the population. It
"rst evaluates the economic success of each strategy, thus often being interpreted
as playing the role of the market as an information revealing device. Then it
selects strategies to be part of the next population. The selection operator of the
This means that GA models of e.g. the travelling salesman problem, which surely have an
economic subject, are nevertheless not &economic' GAs in the above meaning. For more information
on state dependency see Dawid (1999).
Sometimes, these &strategies' are so myopic, that a game theorist would rather call them &actions'.
It is important to clarify the following point: a genetic individual is not interpreted as an
economic agent, but as an economic strategy used by an economic agent. This interpretation allows
for several agents employing the same strategy.
Note, that this function of the market has already been described by Hayek (1978).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
Fig. 1. Structure of the canonical genetic algorithm.
canonical GA (often called roulette-wheel selection) does so by applying a
biased stochastic process: Selection for the next generation is done by repeatedly
drawing with replacement strategies from the pool of the old population to be
reproduced into the next one. The chance of each strategy to be drawn is equal
to its relative "tness, which is the ratio of its market success to the sum of the
market success of all strategies in the population. Thus, the number of di!erent
strategies within a population is reduced again. Fig. 1 shows an outline of the
genetic algorithm described above.
3. Economic genetic algorithms as dynamic games
Genetic algorithms have been applied to analyze learning in games before.
There is a number of papers which use a genetic algorithm to explicitly
formulate agents' behavior in economic games. This section shows that in fact
every economic genetic algorithm, even if it is not explicitly using a game
theoretic setting, is a game.
Game theoretic situations are su$ciently characterized as situations in which
the payo! for one player does not only depend on her own action but also on the
The "rst and most famous paper of this type is the one by Axelrod (1987). Another one is the one
by Marks (1992).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1023
actions of all other players involved in the same situation. In economic genetic
algorithms, all agents that are members of the same population are involved
within the same situation. Moreover, in models of economic GA-learning (see
De"nition 1 above), the "tness (alias economic success, alias game theoretic
payo!) of an economic agent depends on two things: (a) her own strategy and (b)
the strategies of all other members of the same population. In other words: The
"tness of an economic agent depends on the state of her population. In GAlearning-lingo this reads: Agents' "tness is state dependent.
Proposition 2 (GA-games). Every simple, one-population, economic genetic algorithm is a dynamic game.
Note, that it is state dependency, which is the common characteristic for both,
game theoretic situations and economic GA-learning models. More than this,
the game is repeated in every round of the genetic algorithm, so that every
individual is given the chance to alter and hopefully improve her strategy. Thus,
it can be concluded that in fact every simple, one-population, economic genetic
algorithm is a dynamic game.
As an illustration, imagine a cobweb model of the supply side of a market. In
every period of time, each agent represents a "rm which faces the problem of
maximizing its pro"t by chosing the appropriate quantity of supply. Each "rm's
"tness, which is the same as one period's pro"t, is given by
price
!unit costs).
(1)
"tness"agent's quantity ) (
GHI
GFFHFFI
'
The quantity the agent supplies re#ects her own strategy, whereas the ex-post
market price re#ects the state of the whole population, which means, given total
demand, it re#ects aggregate supply, i.e. the sum of all individual supply
strategies. While total supply has an important in#uence on each agent's "tness,
the cobweb model turns out to be a model of state-dependent "tness which is the
same as an economic game.
In technical terms, even more can be said about the types of games, GA
learning can be interpreted as. GA learning models describe a repeated economic game. Imagine a genetic algorithm using a population of M genetic
individuals with the length of each individual's bit string of ¸. Due to the binary
coding of genetic individuals, this means that each genetic individual represents
one out of 2* di!erent values. This means that the GA is able to deal with every
economic strategy in the set of all available strategies S, where S has the size
N"S"2*. Thus, the GA can be interpreted as a repeated symmetric one
population M person game with up to N pure strategies. But, compared to
A model of this type can be found in Arifovic (1994).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
&normal' evolutionary games, within most economic GA learning models, the
&rules' of the game are di!erent. Whereas in evolutionary games most of the time
a strategy is repeatedly paired with single competing strategies, in genetic
algorithm learning, each strategy plays against the whole aggregate rest of the
population. There is no direct opponent to a single strategy. Instead, every
economic agent aims to "nd a strategy i3S performing as good as possible
relative to its environment, which is completely determined by the current
population n and the objective function R( ) ).
4. Economic genetic algorithms as evolutionary games
Close relationships between economic learning models and models of evolutionary theory have been recognized before. Marimon (1993) gives a clear notion
of the similarities of learning models on the one hand and evolutionary processes on the other. As genetic algorithms, too, have been broadly interpreted as
models of economic learning, in this section it is argued that they can be
regarded as evolutionary processes as well.
At a "rst glance, it is the structure of genetic algorithms and evolutionary
models that suggests a close relationship between GAs and evolutionary
economic theory: Both face the central structure of a population of economic
agents interacting within some well de"ned economic environment and aiming
to optimize individual behavior.
As the aim of this paper is to describe economic genetic algorithms as
evolutionary processes, the "rst question to be answered is the question of
whether GAs are evolutionary processes at all. In the following, it will be argued
that GAs are a speci"c form of evolutionary processes, i.e. evolutionary games.
Proposition 3 (GA-evolutionary games). Every simple, one-population, economic
genetic algorithm is an evolutionary game.
In order to give some evidence for this proposition, a de"nition is needed that
clearly states what an evolutionary game is. This paper makes use of the
de"nition by Friedman (1998, p. 16), who gives three characteristics for an
evolutionary game:
Dexnition 4 (Evolutionary game; Friedman). An evolutionary game is a
dynamic model of strategic interaction with the following characteristics:
While this notion is true for most of the economic GA learning models, it is not true for GA
models that explicitly describe economic games, including Axelrod (1987) and Andreoni and Miller
(1995).
For such an interpretation see e.g. Dawid (1999).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1025
(a) higher payo! strategies tend over time to displace lower payo! strategies;
(b) there is inertia; (c) players do not intentionally in#uence other players' future
actions.
Prior to checking these three points, it is important to note that economic
GAs are in fact models of strategic interaction, which has implicitly already been
stated in Proposition 2. In the interpretation as models of social learning, GAs
deal with a number of economic agents, each trying to "nd a behavioral strategy
which, due to her surrounding, gives her the best payo! possible. GAs are
models of social learning, which in fact is a way of learning by interaction. Thus,
it can be stated, that GAs are in fact models of &strategic interaction'.
Now, let us focus on Friedman's (De"nition 4) three characteristics:
(a) Higher payo! strategies tend over time to displace lower payo! strategies.
Displacement of strategies in genetic algorithms is a process of change in the
genetic population over time. This, in turn, is a question of selection and
reproduction. GAs are in fact dynamic processes which reproductively prefer
higher payo! strategies to lower payo! ones. It has been shown that in the
canonical GA, the probability of a strategy i to be reproduced from its current
population n into the population of the next period, P(in ), depends only on its
relative "tness R(in ), which is the strategy's payo! or market success relative to
the aggregate payo! of the population n . Higher relative "tness leads to
a higher reproduction probability
dP(in )
'0.
dR(in )
(2)
Thus, Friedman's condition (a) is satis"ed.
(b) There is inertia.
According to Friedman, inertia means that &2changes in behavior do not
take place too abruptly'. (Friedman, 1998, p. 16).
Looking at the genetic or game theoretic population, it is mutation, or
learning by experiment, which causes the most abrupt changes. Whereas the
strategy of a single economic agent might be changed more dramatically by
imitation or communication, this is not true for the population as a whole. At
the level of the population, only learning by experiment is able to introduce
strategies or at least parts of strategies which have not occurred in society
before. Thus, it should be proved that small changes by mutation are more likely
than big ones. For the standard GA, using binary representation of genetic
For a more precise description, cf. Riechmann (1999).
This is valid for all variants of GA-selection processes, not only for the standard roulette-wheel
selection (Goldberg, 1989). For more, di!erent types of GA-selection operators see e.g. Goldberg and
Deb (1991) and Michalewicz (1996).
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
individuals, the mutation operator is quite simple. Mutation randomly alters
(&#ips') single bits of the bit string encoding an economic strategy. Each bit
of the string has a small probability to be changed. , which is called the
mutation probability, is uniform for every bit of any genetic individual of
every population.
To carry out an analysis concerning the connection of mutation and abruptness of change, a measure for the abruptness is needed. This paper will make use
of the Hamming distance between two genetic individuals. According to the
Hamming distance, a change from individual i to individual j is said to be more
abrupt than the change from individual i to k, if the Hamming distance between
i and j, H(i, j) is greater than the Hamming distance between i and k, H(i, k).
The question of whether small changes by mutation are more likely than big
ones can be answered as follows: The probability of an economic strategy i to be
turned into strategy j by mutation, P (i, j), depends on the length of the genetic
K
individuals' bit string, ¸, the mutation probability , and the number of bits that
have to be #ipped in order to turn i into j, i.e. the Hamming distance between
i and j, H(i, j)
P (i, j)"&G H(1!)*\&G H.
K
Di!erentiation yields
(3)
(0 for (,
P (i j)
K
"&G H(1!)*\&G H[ln !ln(1!)] "0 for ",
(4)
H(i, j)
'0 for '.
For the usual parameter values of , this means the obvious: Small changes in
strategy are more likely than big changes. Thus, it becomes evident that GA
learning processes are processes which contain some inertia.
(c) Players do not intentionally in#uence other players' future actions.
This point can be proved more verbally. The agents modeled by a simple
economic GA have very restricted knowledge. By the time an economic agent
forms her latest economic strategy she does not know anything about the plans
The Hamming distance between the genetic individuals i and j, in short, is the number of places,
in which these genetic individuals di!er, i.e.the number of bits to be #ipped in order to turn
individual i into j. See e.g. Mitchell (1996, p. 7).
Normally, the mutation probability ranges somewhere between and .
The result given in (3) deserves two further remarks. First, the fact that for ' big changes are
more likely than small ones explains the fact that for relatively large values of , GA results are close
to white noise. Second, the result yields an interesting interpretation for the "eld of economic
learning. If mutation is interpreted as learning by experiment, (3) shows that a little experimenting is
a good thing to do, while too many experiments will disturb the generation of valuable new
economic strategies. If mutation is interpreted as making mistakes in imitation or communication
(see e.g. Alchian, 1950), Eq. (3) simply means that one should not make too many of those mistakes.
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1027
of the other agents in her population. All an economic agent in a GA model can
do is to try her best to adopt to her neighbors' past actions, for the near past is all
such an economic agent can remember. Taking into account these very limited
individual abilities, it becomes obvious, that there is no room for intentional
in#uences on other agents' actions.
From the above it can be concluded, that models of economic GA learning
are in fact models which can be interpreted as evolutionary games.
5. Genetic populations as near Nash equilibria
The main structure (information scientists would call it &data structure') in
genetic algorithm learning models is the genetic population. A population
contains all the agents that are economically active in a certain period of time.
The agents can completely be characterized by their economic strategy. Thus,
a population can be described by counting how often each of the di!erent
possible strategies is used by the members of the population. Thus, one way of
representing a population is to give the (absolute or relative) frequency of each
possible strategy within the population. This means that a population is nothing
more than a distribution of di!erent economic or behavioral strategies. This is
true for genetic populations as well as for populations in their game theoretic
interpretation. It can be said that a genetic population is a game theoretic
population.
From Section 4 it is known, that every economic agent aims to "nd the best
performing strategy i3S with respect to the objective function R( ) ) and given
the strategies of the rest of her population n . This means that every economic
agent faces problem (5)
max R(in ).
(5)
GZ1
This immediately leads to the concept of Nash equilibria. A Nash strategy is
de"ned as the best strategy given the strategies of the competitors, and a Nash
strategy is exactly what every economic agent, alias genetic individual, is trying
to reach.
Proposition 5 (GA populations * near Nash equilibrium). In every simple, onepopulation, economic genetic algorithm, the population tends over time to move to
a Nash equilibrium without fully reaching it.
In fact, genetic algorithms can be shown to be Markov processes, the main characteristic of
which is the &no memory property'. This property says that there is no memory to the history of the
GA process.
For an in-depth discussion of this, refer to Davis and Principe (1993) or Riechmann (1999).
1028
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
As an illustration, in the cobweb model by Arifovic (1994), there is no single,
direct opponent to the "rm. Instead, it is the aggregate output of all the
competitors which each "rm has to take into account. This is to say: Although
there is no direct opponent to agents in a market, which is di!erent from most of
the &normal' evolutionary games, markets can still be seen as games, even if the
rules are slightly di!erent. The main point still remains the same: Agents are
trying to play Nash.
It has already been shown that selection and reproduction work in favor of
relatively well performing strategies. This means that via selection and reproduction strategies which are not Nash strategies are eliminated from the population, whereas the population share of Nash strategies is increased. At the same
time, due to the fact that agents do not stop experimenting, some agents carry
out experiments that result in non-Nash strategies. Thus, although selection and
reproduction drive the population towards a Nash equilibrium, learning by
experiment prevents the population from fully reaching such an equilibrium.
This means that genetic populations represent a state which is not a real Nash
equilibrium but which is not far from it. This state will be called &near-Nash
equilibrium' throughout this paper.
Thus, at a "rst step, an economic genetic algorithm can be viewed as modeling
a system of economic agents, each of them trying to play a Nash strategy against
the rest of the population. In economic terms this means that every agent tries to
coordinate her strategy with the other agents' ones, for this is the best way of
maximizing her pro"t (or utility or payo! or whatever the model wants the
agent to maximize). The population is driven to the neighborhood of a Nash
equilibrium by the forces of the the market, which are represented by selection in
economic genetic algorithms. But, due to the e!ects of ongoing experimentation,
the population will never be able to fully reach the Nash equilibrium.
6. Evolutionary stability of genetic populations
While a genetic population represents a primarily static concept, learning is of
course a genuinely dynamic process. Thus, in order to analyze GA learning as
an evolutionary learning process, questions regarding the dynamics and the
concepts of stability have to be addressed.
This is, of course a slight simpli"cation: Ruling out non-Nash strategies may alter the
population in a way that make former Nash strategies non-Nash and vice versa.
Replicator dynamics (see e.g. Weibull, 1995 or Hofbauer and Sigmund, 1988, 1998), which have
often been used to characterize evolutionary dynamics, seem to be unsuited for some economic
problems. (Mailath, 1992, p. 286 even suggests that &there is nothing in economics to justify
replicator dynamics'.) Applied to the analysis of GA learning, replicator dynamics, not directly
accounting for stochastics, are simply not precise enough to cover the whole GA learning process.
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1029
This paper will make use of the concept of evolutionary stability, especially
the notion of evolutionarily stable strategies or evolutionarily stable states
(ESS). See i.e. Maynard Smith (1982), Hofbauer and Sigmund (1988, 1998),
Samuelson (1997, Weibull (1995), Marks (1992), or Mailath (1992). In short,
a strategy is evolutionarily stable if, relative to its population, it performs better
than any new, &invading' strategy. Though widely used in economic dynamics,
the concept of ESS has a serious weakness which makes it only limitedly suitable
for the analysis of genetic algorithms: There is no explicit formulation of the
selection process underlying the concept of evolutionary stability. ESS are based
on the notion that invading &mutant' strategies are somehow rejected or eliminated from the population. It is not clear how this rejection is carried out.
Genetic algorithms, in contrast, present a clear concept of rejection:
Every strategy will be exposed to a test, which has been described as a
one-against-the-rest game in the previous sections. Then the strategy will be
reproduced or rejected with a probability depending on its performance (i.e.
market performance) in this game. Thus, the rejection of strategies in GA
learning models is a question of reproduction. GA reproduction has two main
features, it selects due to performance and it selects due to probability, which
means that a bad strategy will be rejected almost surely although not with
probability one.
Thus, a re"ned concept of evolutionary stability is needed for genetic algorithms. A possible way of setting up a concept of evolutionary stability
for genetic algorithms which is keeping the spirit of the ESS is the following:
A genetic population is evolutionarily stable if the process of the genetic
algorithm rejects an invasion by one or more strategies from the genetic
population. Invasion itself can either take the form of a totally new strategy
entering the population or it can simply mean a change in the frequency of
the strategies already contained within the population. Thus, a more precise
de"nition of an evolutionarily stable population might be: A population
is evolutionarily stable if it is resistant against changes in its composition (see
De"nition 7).
6.1. Evolutionary superiority
More formally, a genetic population n will be called evolutionarily superior to
m ) if it exhibits two characteristics:
population m , (denoted as n '
(a) Every strategy i contained within population n gains at least the same "tness
in the basic population n as it gains in the invaded population m , while at
least one strategy gains even more "tness in n than in m .
(b) The invading strategies k3m n are the worst performing strategies contained in m , so that they will be most surely rejected.
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
Note that the above characterizes a kind of weak dominance concept. This
de"nition of evolutionary superiority induces a partial ordering on the space of
genetic populations, which resembles the concept of Pareto superiority.
In more mathematical terms, the de"nition reads:
Dexnition 6 (Evolutionary superiority). A genetic population n is called evolutionarily superior to population m , if
R(in )5R(im )
∀i3n
(6)
j with R(jn )'R(jm )
(7)
R(km )(R(im )
(8)
∀i3n ; ∀k3m n .
To illustrate this concept, consider the following example of an oligopoly
game. Assume a population of "ve "rms, which covers the whole supply side of
the market. Demand is given exogenously and does not change over time. The
current population is characterized by the vector of output quantities, i.e.
n "(3, 2, 2, 3, 2).
The "rst element of n gives the output level of "rm number one, the second
element gives the output of "rm number two and so on. Let us further suppose
that the resulting aggregate output level of 12 leads to a market price that in turn
leads to positive pro"ts for every single "rm within the population. Now, at the
end of the current period, "rm number two starts experimenting and decides to
change its output level to 10. In other words: Infection is taking place, changing
population n to
m "(3, 10, 2, 3, 2).
Due to this infection, aggregate supply increases, inducing a decrease in the
market price. This of course leads to diminishing pro"ts of each xrm within the
population. For simplicity, assume that the price decreases so much, that it falls
below unit costs. In this case, instead of making pro"ts, "rms run at a loss. This
is especially true for "rm number two, the loss of which is the biggest throughout
the population. Accordingly, what will happen in the near future is the following: Firm number two recognizes that it is currently performing very poorly and
adopts a di!erent strategy from one of its competitors. In terms of genetic
algorithm theory: Sooner or later the selection operator will discard strategy
This point has, in earlier drafts of this paper, led to misconception: Neither the concept of
evolutionary superiority nor the concept of Pareto superiority make any statements about some
kind of welfare. Originally, Pareto superiority is just a means to order points within a highly
dimensioned space. Applying evolutionary superiority analogously to the original meaning of the
Pareto criterion has no welfare implication at all. It is just used in order to make genetic populations
weakly comparable with respect to the process of the genetic algorithm and GA's way of turning one
population into another.
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1031
number two and replace it by a strategy from the rest of the population. By that,
the infection that turned population n into population m is rejected. Though this
does not inevitably mean that population n is directly regained by that process, it
shows at least, that in the light of GA dynamics, n is &better' than m . Population
n is evolutionarily superior to population m . This is veri"ed by application of the
two criteria from above: Strategies one and three to "ve (i.e. the output strategy of
"rm number one and "rms number two to "ve) lead to greater payo! in
population n than in population m . Hence, criterion (a) is met. Moreover, strategy
number two in population m is the worst performing strategy in this population.
By this, criterion (b) is met as well. Population n is evolutionarily superior to m . At
last, let us focus evolutionary superiority of n over m by using De"nition 6. The
"tness of each of the "ve "rms' strategies is higher in population n than in m (Eqs.
(6) and (7)). Note that this is especially true for "rm number two, which su!ers
from a particularly high loss due to its change in strategy. Firm number two is the
only one that has changed its strategy and at the same time it is the worst
performing "rm, thus ful"lling condition (8).
For full validity, a further remark is necessary: Within genetic algorithms,
invading strategies can only result from reproduction (&imitation'), crossover
(&communication') or mutation (&experiment') within the population itself. This
means that the "nal outcome of GAs without mutation (i.e. processes with
learning by imitation and communication only), which are uniform populations,
may have other populations being superior to them, but } without mutation
} better populations simply cannot arise.
6.2. Evolutionarily stable populations
A population is evolutionarily stable in the concept of De"nition 6 if there is
no other population within S, the set of all populations, which is evolutionarily
superior to it.
Dexnition 7 (Evolutionary stability of genetic populations). A genetic population n is called an evolutionarily stable population, if
CQ n .
x m 3S with m '
(9)
De"nition 7 is a generalization of the concept of evolutionary stability.
Note, that due to De"nition 7 more than just one evolutionarily stable genetic
population can exist. Thus, once an evolutionarily stable genetic population has
been reached, two things can result from an infection of the population: Either
the infecting strategy is rejected from the population and the original population
See Riechmann (1999) for the restrictions di!erent learning techniques impose on the set of
available strategies.
Note e.g. the similarity to (Weibull's 1995, pp. 36) de"nition.
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T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
is regained, or infection causes the transition to another evolutionarily stable
population, provided there is one.
Due to the fact that genetic algorithm selection is a probabilistic rather than
a deterministic process, invading strategies, even in an evolutionarily stable
population, may not be rejected within a single round of the algorithm. It can
only be stated that the invader will be driven out of the population within "nite
time. That is to say: If a genetic population is evolutionarily stable, it will recover
from an invasion within a "nite number of steps of the GA, which means that in
the long run the population will not lastingly be changed. Nevertheless, once an
evolutionarily stable population is invaded, there may appear a number of
evolutionarily inferior populations within the next few rounds of the GA. These
populations represent transitory states of the process of rejecting the invader.
Riechmann (1999) shows that there is in fact more than one population that will
occur in the long run. These may be transitory populations as well as di!erent
populations which are evolutionarily stable, too.
7. Evolutionary dynamics
According to the evolutionary superiority (De"nition 6), the GA always selects
in favor of the superior population. This notion can be used to characterize GA
learning dynamics: The genetic algorithm as a stochastic process continuously
discards populations in favor of better ones in the sense of evolutionary superiority. But this only describes the direction of the process, not the exact transition
path over time. In fact, due to the stochastic properties of genetic algorithms, the
exact path of the process highly depends on the initial conditions, i.e. the
composition of the very "rst genetic population. And although the path up to an
evolutionarily stable equilibrium may di!er, Markov chain theory shows that in
the long run a "xed and stable distribution of population will be reached. This will
even happen irrespective of the starting conditions. There may be path dependence, lock-ins, or related phenomena, but in the case of genetic algorithm learning
these will only be of temporary nature. In the long run, genetic algorithm theory
promises, the &best' state will be reached.
Again, see Riechmann (1999) for a more detailed explanation.
This may be regarded as a weakness of the concept of genetic algorithm learning, as it neglects the
possibility of modelling path dependence or lock-ins. So it may be worthwhile to mention two further
points, which are mainly beyond the scope of this paper. First, depending on the underlying (economic)
problem, some GAs spend long times supporting populations which are not evolutionarily stable.
Some keywords pointing to this topic are &deceptiveness' of genetic algorithms and the problem of
&premature convergence'. Secondly, the lack of ability to model lasting lock-ins or path dependence
applies to the basic genetic algorithm. There are variations of genetic algorithms capable of modelling
these phenomena. One keyword pointing into this direction of research may be &niching mechanisms'.
Again, a good starting point for more descriptions of all of the various special cases and variants of
GAs is Goldberg (1989).
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1033
Knowing the special form of the dynamic process of the GA and the direction
in which this process will lead, a few more words can be said about the role of
heterogeneity for the dynamics. It seems important to notice the way economic
change takes place. Starting with an arbitrary population, genetic operators (i.e.
learning mechanisms) cause changes in the population, while new types of
behavior are tested. The test is performed by exposing the strategies to the
market. The market reveals the quality of each tested strategy relative to all
other strategies within the population. Then selection leads economic agents to
abandon poorly performing strategies and adopt better ones (imitation) or even
create new ones by communication (crossover) or experimentation (mutation).
After that, again, strategies are tested and evaluated by the market, by that way
coordinating the agents' strategies, and so on.
There are two crucial aspects of this repeated process: First, it is the diversity
of strategies that drives economic change, i.e. the succession of populations
constantly altering their composition. Under the regime of genetic algorithm
learning, this change in individual as well as in social behavior heavily (while not
entirely) relies on learning by imitation and learning by communication.
Evidently, these kinds of learning can only take place within heterogeneous
populations. The more diverse a population is, the greater is the number of
di!erent strategies or even parts of strategies that a member of this population
can learn by imitating the other members or by communicating to them. Thus,
in a way, it can be said that it is heterogeneity that is the main driving force
behind economic change.
The second crucial aspect of the process of genetic algorithm learning is the
role of selection, which can be interpreted as the role of the market. While the act
of learning will be enough to achieve economic change, economic development,
i.e. the movement of genetic population towards an evolutionarily stable state,
can only be reached by the cooperation of learning and selection. In order to
turn the succession of di!erent populations into the succession of constantly
improving populations (in the sense of evolutionary superiority), a device is
needed that makes it possible to distinguish successful strategies from less
successful ones. Having at hand such a device, it is possible to decide
which strategies shall live and grow and which ones shall die. This device is the
market in economics as it is the selection operator within genetic algorithms. It
is the market and only the market that turns economic change into economic
development.
Birchenhall et al. (1997) clearly show that there is a connection between the extent of diversity in
a population and the learning speed that is achieved by members of that population.
To this interpretation, &development' is a term of stability rather than optimality.
This re#ects a rather classical economic thought, given, e.g., in Hayek (1969) (usually quoted as
Hayek, 1978).
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Summarizing, under the regime of the market, the evolutionary dynamic of
genetic algorithm learning is mainly driven by two forces: Heterogeneity, which
constantly induces behavioral (and by that, economic) change, and the market
as a coordination device, revealing information about the quality of each type of
behavior and ruling out poorly performing strategies, thus turning economic
change into economic development.
Finally, looking at genetic algorithm learning from an evolutionary point of
view, one more point has to be added. The evolutionary dynamic of the GA
learning process is essentially a two-stage process. It has been shown that, as
long as possible, genetic algorithm learning and market selection &improve'
individual and as a result social behavior. But, this is just the "rst stage of the
underlying dynamics. Yet, once an evolutionarily stable state of behavior has
been reached, there certainly is no room for further &improvement'. But, due to
the special structure of genetic algorithms, this does not mean that in this state
economic agents stop changing their behavior. Instead, at this point the second
stage of the dynamic process starts. Learning, or what has above been called
change, still continues and will not cease to continue. Still, there will appear new
ways of individual behavior within a population. Now it is the role of the market
(i.e. selection) to drive these strategies out of the population again. Due to the
probabilistic nature of the GA, this process may take more than one period, thus
producing one or even more transitory populations until an evolutionarily stable
population is regained. To put it in di!erent words: Even after an evolutionarily
stable state is reached, evolutionary stability is continuously challenged by new
strategies. While in the "rst phase of the GA learning process some of the new
strategies are integrated into the population, in the second phase all of the
invaders will be replaced again. So there is an ongoing near-equilibrium movement resulting from the continuous rejection of invading strategies.
In fact, genetic algorithm learning leads to an
interplay of coordinating tendencies arising from competitive adaptions in
the markets and de-coordinating tendencies caused by the introduction of
novelty
(Witt, 1993, p. xix), which has often been regarded as a key feature of evolutionarily economic analysis of the market. See Witt (1985).
By means of Markov chain theory, it has been shown before, that the
canonical GA does neither lead to a lasting state of total uniformity of individuals in their populations nor to a state where populations ultimately stop
changing over time. These long-run dynamics of GA learning processes have (in
mathematical terms) been characterized as a state of Lyapunov stability
(Riechmann, 1999). Although a plain technical explanation for this behavior can
be found, up to now it has not been made clear, what this explanation means in
terms of learning and market behavior. With the help of evolutionary game
T. Riechmann / Journal of Economic Dynamics & Control 25 (2001) 1019}1037
1035
theory, a clear-cut economic reasoning can be found, why this state of Lyapunov
stability shows up: It is a process of near-equilibrium dynamics, caused by the
continuously ongoing challenge of the ESS by newly learned strategies and the
rejection of these strategies by the market that prevents social behavior from
total convergence but still keeps it close enough to a stable state.
8. Conclusions
Economic genetic algorithm learning can be shown to be a speci"c form of an
evolutionary game. In this paper, this has been discussed for the most basic form
of genetic algorithm, the canonical GA. For populations of GAs, concepts of
evolutionary superiority and evolutionary stability are developed, which help to
explain the way processes of GA learning behave. This behavior turns out to be
an evolutionary, two step process: First, there is a movement of populations
towards an evolutionarily stable state. Second, once such a state has been
reached, the learning process turns into a near equilibrium dynamic of getting
out of evolutionarily stable states and returning there again.
The notion of GA learning as an evolutionary process can be transfered to the
analysis of modi"ed genetic algorithms. As an example, consider certain changes
to the selection operator. Elitist selection schemes, including the selection
within evolution strategies cf. BaK ck et al. (1991) and Arifovic's (1994) election
operator, ensure that at least the "rst best genetic individual of a population
will become a member of the next generation's population. In contrast to
roulette-wheel selection, elitist selection ensures that invading strategies which
turn out to be the worst strategies throughout the population will be replaced at
once. This means that there is no room for transitory populations. Bad strategies, i.e. strategies obeying condition (8), are ruled out even before they can
enter a population. This certainly leads, in most cases to asymptotic behavioral stability.
Summarizing, the results of this paper demonstrate that research in the
dynamics of economic GA learning models can be equipped with the whole tool
box of evolutionary game theory. Using these instruments, in this "eld of
science, much more work can and has to be done.
A survey on various selection schemes can be found in Goldberg and Deb (1991).
For an interpretation and an extension of the election operator, see Franke (1997).
It has been largely neglected throughout this paper, that the stability properties of at least some
variations of economic GAs do certainly depend on the underlying economic problem, too. This
means that there might be some problems that can cause lasting changes of the population even in
GAs with elitist selection. Examples for this might be found at genuinely cyclic problems like cases of
Lotka}Volterra dynamics, see, e.g. Hofbauer and Sigmund, 1998, pp. 11, which, in economics, have
been applied to models of business cycles, Goodwin (1967).
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