Journal of Computational Science 1 (2010) 24–32
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Journal of Computational Science
journal homepage: www.elsevier.com/locate/jocs
A chaotic gas-like model for trading markets
C. Pellicer-Lostao ∗ , R. López-Ruiz
Department of Computer Science and BIFI, Universidad de Zaragoza, Zaragoza 50009, Spain
a r t i c l e
i n f o
Article history:
Received 22 March 2010
Accepted 26 March 2010
Modeling and simulation
Econophysics
Money dynamics
Adaptation and self-organizing systems
Complex systems
a b s t r a c t
This paper considers the ideal gas-like model for trading markets, where each individual interacts with
others trading in money-conservative collisions. Traditionally this model introduces different rules of
random selection and exchange between pairs of agents, what leads to different money distributions in
the community. Real economic transactions are complex but obviously non-random. Therefore, unlike
the traditional model, this work introduces chaotic elements in the evolution of the economic system.
As a result, it is found that the chaotic gas-like model can reproduce the referenced wealth distributions
observed in real economies, i.e. the Gamma, Exponential and Pareto distributions.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Econophysics is a relatively new interdisciplinary science
[19] that applies many-body techniques developed in Statistical
Mechanics to the understanding of self-organizing economic systems [25]. One of its main objectives is to provide economists
with new tools and insights to deal with the inherent complexity of economic systems. Computational science is the heart of the
techniques developed in this field, as they mainly use advanced
computer multi-agent simulations [9,5,2]. Data analysis, innovative computation and numerical methods are used to understand
the multi-scale behavior of economic markets. As a result of that,
the statistical distributions of money, wealth and income can be
obtained for a community of agents under some rules of trade and
after an asymptotically high number of economic transactions.
The conjecture of a kinetic theory with (ideal) gas-like behavior
for trading markets was first discussed in 1995 by econo-physicists
[3]. This model considers a closed economic community of individuals where each agent is identified as a gas molecule that interacts
randomly with others, trading in elastic or money-conservative collisions. Randomness is an essential ingredient in this model, where
agents interact in pairs chosen at random by exchanging a random quantity of money. The interesting point of this model is that,
similarly to Energy, the equilibrium probability distribution of
money follows the exponential Boltzmann–Gibbs law for a wide
variety of trading rules [25].
This result is coherent with real economic data in some capitalist countries up to some extent, for in high ranges of wealth,
evidences are shown of heavy-tail distributions [10,16]. Nowadays
it is well established that the income and wealth distributions follow a characteristic pattern with two different phases. The first one
presents an exponential (Boltzmann–Gibbs) distribution of richness and covers about 90–95% of the individuals, those with low
and medium incomes. The other phase, which is integrated by the
individuals with the highest incomes, shows a power law (Pareto)
distribution. This is so in different countries, irrespective of their
differences in cultural or social structures, [10,24], from older societies [1,20,6,7] to modern ones [8,4,11].
This paper introduces chaotic-driven dynamics in the economic
gas-like model. This is a novel approach, where the rules of selection of agents and money transfers are no longer random, but
driven by nonlinear maps in chaotic regime. Three different scenarios are considered. Their resulting money distributions are
compared with those observed in real economies. These results
are analyzed at micro and macro-scales, in an effort to uncover
the causes responsible of the unequal distribution of money in
society.
The paper is organized as follows: Section 2 introduces the basic
theory of gas-like economic models. Section 3 describes the three
simulation scenarios considered in this work, and the following
section shows the results obtained in the simulations. Conclusions
are discussed in the final section.
2. The gas-like model: Boltzmann–Gibbs distribution of
money
∗ Corresponding author.
E-mail addresses: carmen.pellicer@unizar.es (C. Pellicer-Lostao),
rilopez@unizar.es (R. López-Ruiz).
1877-7503/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jocs.2010.03.005
The conjecture of a kinetic theory of (ideal) gas-like trading markets was first discussed in 1995 [3]. Then it was in year 2000, when
C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
several noteworthy papers dealing with the distribution of money
and wealth [9,5,2], presented this theory in more detail.
The gas-like model assimilates the dynamics of a perfect gas,
where particles exchange energy at every collision, with the
dynamics of an economic community, where individuals exchange
money at every trade. When both systems are closed and the
magnitude of exchange is conserved, the expected equilibrium
distribution of these statistical systems may be the exponential
Boltzmann–Gibbs distribution.
P(x) = ae−x/b ,
25
linear maps can be easily tuned to break the pairing symmetry of
agents (i, j) ⇔ (j, i), characteristic of the random model.
The simulation scenarios presented here follow the traditional
gas-like model. A community of N agents is given an initial equal
quantity of money, m0 for each agent. The total amount of money,
M = Nm0 , is conserved in time. For each transaction, a pair of agents
(i, j) is selected, and an amount of money m is transferred from
one to the other. In this work two simple and well known rules
are used. Both consider a variable in the interval (0, 1), in the
following way:
(1)
where a is a normalizing constant and b is related to the mean
energy or money in the system (b = x). The derivation, and so
the significance, of this distribution is based on the statistical
behavior of the system and the conservation of the total magnitude of exchange. This can be obtained from a maximum entropy
condition [15] or from purely geometric considerations on the
equiprobability over all accessible states of the system [18].
Different agent-based computer models of money transfer presenting asymptotic exponential money distributions can be found
in literature [9,21,13]. In these simulations, a community of N
agents with an initial quantity of money, m0 per agent, trade among
them. The system is closed, hence the total amount of money M is a
constant (M = Nm0 ). Then, a pair of agents is selected (i, j) and a bit
of money m is transferred from one to the other. This process of
exchange is repeated many times, typically of order N 2 , until statistical equilibrium is reached and the final asymptotic distribution
of money is obtained.
In these models, the rule of agents selection in each transaction
is random, i.e. there is no local preference or intelligent agents. The
money exchanged at each time m is regarded basically under two
possibilities: a fixed or a random quantity. From an economic point
of view, this means respectively that agents trade products with a
fixed price or with prices can vary freely.
These models have in common that they generate a final
stationary distribution that fits well the exponential function. Perhaps one would be tempted to affirm that this final distribution is
universal despite the different rules for the money exchange, but
this is not the case as it can be seen in [9,21].
3. Simulation scenarios
This paper introduces chaos in the dynamics of the economic
gas-like model. This is done upon two facts that seem particularly relevant for this purpose. The first one is, that real economy
is not purely random. Economic transactions are driven by some
specific interest (of profit) between the interacting parts. Thus, on
one hand, there is some evidence of markets being not purely random. On the other hand, everyday life shows us the unpredictable
component of real economy with its recurrent crisis. Hence, it
can be sustained that the short-time dynamics of economic systems evolves under deterministic forces though, in the long term,
these kind of systems show an inherent instability. Therefore, the
prediction of the future situation of an economic system resembles somehow to the weather prediction. It can be concluded that
determinism and unpredictability, the two essential components of
chaotic systems, take part in the evolution of Economy and Financial Markets.
The second fact is, that the transition from a Boltzmann–Gibbs to
a Pareto distribution may require the introduction of some kind of
inhomogeneity that breaks the random indistinguishability among
the individuals of the market. Though this is not a necessary condition in order to have such kind of transition [12]. Nonlinear maps
can be an ideal candidate to obtain that, as they can be quite
flexible drivers of the evolution in the market. In particular, non-
• Rule 1: agents undergo an exchange of money, in a way that agent
i ends with a -dependent portion of the total money of the two
agents, ((mi + mj )), and agent j takes the rest ((1 − )(mi + mj ))
[21].
• Rule 2: a -dependent portion of the average amount of money
of the two agents, m = (mi + mj )/2, is taken from i and given
to j [9]. If i doesn’t have enough money, the transfer doesn’t take
place.
As it was seen, in the gas-like models there are two different
simulation parameters involved: the selection of the agents and
the money transferred in the economic transactions. Consequently,
apart from the full random model, three different scenarios can
be obtained depending on the random-like or chaotic selection of
these parameters. These scenarios are the ones considered in the
following sections and are described as:
• Scenario I: random selection of agents with chaotic money
exchange.
• Scenario II: chaotic selection of agents with random money
exchange.
• Scenario III: chaotic selection of agents with chaotic money
exchange.
To produce chaotically a pair of agents (i, j) for each interaction, a
2D nonlinear map under chaotic regime is considered. The pair (i, j)
is easily obtained from the coordinates of a chaotic point at instant t,
Xt = [xt , yt ], by a simple float to integer conversion (xt and yt to i and
j, respectively). Additionally, to obtain a chaotic money exchange, a
float number in the interval [0, 1] is obtained form the coordinates
of a chaotic point at instant t by taking one or a combination of
both. This number produces the chaotic quantity of money m
that is traded between agents i and j. The simulation scenario uses
two particular 2D nonlinear maps under chaotic regime. These are
the Hénon map (2) described in [14] and the Logistic bimap (3)
described in [17]. These maps are given by the following equations:
TH : × → × xt+1 = axt2 + yt + 1,
(2)
yt+1 = bxt .
TA : [0, 1] × [0, 1] → [0, 1] × [0, 1]
xt+1 = a (3yt + 1)xt (1 − xt ),
(3)
yt+1 = b (3xt + 1)yt (1 − yt ).
These maps show chaotic behavior for some values of their
parameters. An interacting demonstration of both maps can be seen
in [22]. In this work, the following values are considered: the Hénon
map in its canonical form with a = 1.4 and b = 0.3 and one arbitrary
initial condition, x0 = −0.75 and y0 = −0.02. For the Logistic bimap,
a and b are considered in the interval where chaotic regime can
be obtained, [1.032, 1.0843]. Here one arbitrary initial condition is
also selected,x0 = 0.4913 and y0 = 0.6913.
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C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
To conclude the description of the simulation scenarios, it is
interesting to remark, that the rules of trade considered here
exhibit different symmetry properties. Rule 2 is asymmetric in
the sense that it takes money from i agent and gives it to the
other agent. When considering the chaotic maps, the Logistic bimap
exhibits symmetric behavior along the diagonal axis (straight
line y = x) when a = b . On the other hand the Hénon map is
asymmetric. These properties of symmetry are going to be modified in the simulations, to show how asymmetry can affect final
money distributions in society.
4. Asymptotic money distributions
The (ideal) gas-like model, as described in the previous sections shows an equilibrium distribution of money that follows the
exponential law for a wide variety of trading rules [25,18]. In this
section, chaos is introduced in this model. Different scenarios are
presented depending on the way the chaotic dynamics is brought
into the model. These scenarios where first proposed by the authors
in [23]. The asymptotic money distributions obtained this way,
are presented. The dynamics of the system is analyzed at microlevel in order to provide illuminating ideas about the cause of each
particular result.
4.1. Scenario I: random selection of agents with chaotic money
exchange
In this scenario, the selection of agents is random as in the
traditional gas-like model. Individuals don’t have preferences in
choosing an interaction partner, commercial transactions are neither restricted, nor driven by particular interests. But unlike this
model, the exchange of money evolves according to nonlinear patterns. Economically, this means that the exchange of money has
a deterministic component, although it varies chaotically. Put it in
another way, prices of products and services are discrete and evolve
in a limited range but in a complex way.
A community of N = 5000 individuals is considered with an initial quantity of money of m0 = 1000$. This community takes a total
time of 50 Millions of transactions. For each transaction two random numbers from a standard random generator are used to select
a pair of agents. Additionally, a chaotic float number is produced
to obtain the float number in the interval [0, 1]. The value of is
calculated as |xt |/1.5 for the Hénon map and as xt for the Logistic
bimap. This value and the rule selected for the exchange determine
the amount of money m that is transferred from one agent to the
other.
Two different simulations are considered, using rule 1 or 2. For
each rule, the final distributions of money are obtained with the
Hénon chaotic map (with a = 1.4, b = 0.3, x0 = −0.75 and y0 =
−0.02) or the Logistic bimap (with a = b = 1.05392, x0 = 0.4913
and y0 = 0.6913). New features appear in this scenario. These can
be observed in Fig. 1(a) and (b).
The first feature is that the chaotic behavior of is producing different final distributions for each rule. The figures show the
probability distribution function of money. The results of rule 1
present a very low proportion of the population in the state of total
poorness, and a high percentage of it in the middle of the richness
scale, near to the value of the mean wealth. Rule 1 gives a more equitable distribution of wealth. It is a Gamma-like distribution. Fig. 1(a)
shows the detail of the fitting to this distribution for the results
obtained with Hénon map. In this case the final money distribution
fits to the Gamma function, expressed as P(x) = cxa e−bx . The fitting
parameters obtained are a = 0.5724, b = 0.0016 and c = 10.973 for
the Hénon map, and for the Logistic bimap a = 1.0067, b = 0.0021
and c = 1.0549.
Rule 2 produces exponential distributions. Fig. 1(b) shows the
fitting to the function P(x) = ae−x/b of the results obtained with the
Hénon map. In this case, the fitting parameters are a = 245.407
and b = 1000. For the Logistic bimap, these are a = 242.81 and b =
1000. Observe that the fitted value of b is precisely the average
money per agent, b = x = M/N.
Rule 1 seems to lead to a more equitable distribution of wealth.
Basically, this is due to the fact that Rule 2 is asymmetric. Each
transaction with rule 2 represents an agent i trying to buy a
product to j and consequently i always ends with less or equal
money. On the contrary, with rule 1 both agents (i, j) end up
with a fraction of their total wealth. Then, rule 1 is symmetric
in the sense that each interaction is like an economic joint venture.
Analyzing the dynamics of the system at micro-level, one
observes that a chaotic evolution of means restricting its value
to the chaotic attractor. In the traditional gas-like model with rule
1 [21], where is random, the final money distribution is exponential. In contrast, in another completely different scenario with
a fixed , let say = 0.5, rule 1 would produce a delta distribution with all agents having the same initial money, m0 = 1000$.
Consequently, a restricted variation of should produce a transition from both distributions, an expansion of the delta distribution
Fig. 1. Probability distribution of money obtained in scenario I. (a) Rule 1 and chaotic trade selection with the Hénon map or the Logistic bimap. Detail of the fitting to the
Gamma distribution. (b) Rule 2 and chaotic trade selection with the Hénon map or the Logistic bimap. Detail of the fitting to the exponential distribution.
C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
27
Fig. 2. Detailed analysis of the values of used in the simulations. (a) Histogram of 20000 values of or the Logistic bimap. (b) Chaotic trade points from the Logistic Bimap
represented as pairs in the range [0, 1] × [0, 1].
around the initial value of m0 . Then it should produce a Gamma-like
distribution.
Fig. 2(a) shows an histogram of values of generated with the
Logistic map in the conditions of the simulation. The distribution
of values shows a high proportion in the interval [0.6, 0.7] where
the system has a hyperbolic fixed point. Fig. 2(b) shows the values
of depicted as pairs of values produced in consecutive instants of
time. It can be observed that the values of are limited in range
and correlated in time. The pairs [(t), (t + 1)] follow a specific
trajectory, consequence of its dynamics in the chaotic attractor.
4.2. Scenario II: chaotic selection of agents with random money
exchange
In this scenario, the selection of agents follows chaotic
dynamics. Individuals do have preferences in choosing an interaction partner, driven by their particular interests. On the other hand
the exchange of money is random, meaning that prices of products and services are not limited and distributed uniformly in the
market.
A community of N = 5000 agents with initial money of m0 =
1000$ is taken and the chaotic map variables xt and yt are used
as simulation parameters. This community takes a total time of 50
Millions of transactions. For each transaction two chaotic floats in
the interval [0, 1] are produced from the coordinates of a chaotic
point at instant t, Xt = [xt , yt ]. The value of these floats are |xt |/1.5
and |yt |/0.4 for the Hénon map and xt and yt for the Logistic bimap.
These values are used to obtain a pair of agents (i, j). The pair is easily obtained, by a simple float to integer conversion (xt and yt to i
and j, respectively). Additionally, a random number from a standard
random generator is used to obtain the float number in the interval [0, 1]. The value of and the selected rule determine the amount
of money m that is transferred from one agent to the other.
Two different simulations are considered, using rule 1 or 2.
For each rule, the final distributions of money are obtained with
the Hénon chaotic map (with a = 1.4, b = 0.3, x0 = −0.75 and y0 =
−0.02) or the Logistic bimap (with a = b = 1.05392, x0 = 0.4913
and y0 = 0.6913). An interesting point appears in this scenario
with both rules. This is, a high number of agents that keep their
initial money (183 agents for the Hénon map and 654 for the
Logistic bimap). The reason for this is that they don’t exchange
money at all. The chaotic numbers used to choose the interacting
agents are forcing trades between a deterministic group of them
and hence some commercial relations become restricted. These non
Fig. 3. Probability distribution of money obtained in scenario II. (a) Rule 1 and chaotic agents selection with the Hénon map or the Logistic bimap. Detail of the fitting to the
exponential distribution. (b) Rule 1 and chaotic agents selection with the Hénon map or the Logistic bimap. Detail of the fitting to the exponential distribution.
28
C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
Table 1
List of values of used in different simulation cases with the Logistic Bimap
Case
1
2
3
4
5
6
7
8
a
b
1.032
1.032
1.032
1.037
1.032
1.043
1.032
1.049
1.032
1.061
1.032
1.072
1.032
1.078
1.032
1.084
interacting agents are removed from the model and so, not considered in the final distributions of money. The results can be observed
in Fig. 3(a) and (b). The figures show the probability distribution
function of money obtained in the final state.
It can be observed in Fig. 3(a) and (b), that the asymptotic distributions in this scenario again resemble the exponential function.
This is so for the all the distributions except one. Rule 1 and rule
2 with the Logistic bimap produce exponential distributions. The
values of the fitting parameters obtained in each case are: a =
250.181 and b = 1000 for rule 1 and Hénon map, a = 218.798 and
b = 1000 for rule 1 and Logistic bimap and, a = 214.867 and b =
1000 for rule 2 and Logistic bimap. As in scenario I, b = x = M/N.
Amazingly, the Hénon Map with Rule 2 in Fig. 3(b) leads to a
distribution where a high proportion of the population (around
4282 agents) finishes in the state of poorness. There is a minority
of agents with great fortunes distributed up to the range of 60000$.
These numbers take a longer range, compared with the exponential distribution obtained for the Logistic bimap. The distribution
obtained for the Hénon case resembles a heavy tail, a Pareto-like
distribution.
This case produces an extremely unfair distribution of money
in the community of agents. This is due to the asymmetric conditions simulated in the market. On one hand, the trading rule 2 is
asymmetric, for agent i always decrements its money with every
transaction. On the other hand, the asymmetry of the Hénon map
coordinates used for the selection of agents is selecting the pair of
agents in an asymmetric way. This double asymmetry makes some
agents prone to loose in the majority of the transactions, while a
few others always win.
The Logistic Bimap is symmetric for coordinates x and y with
a = b , and this is why it produces the effect of no preference
on agents selection. The result is the same as in scenario I where
the selection of agents was random. This can be fully appreciated
when the Logistic bimap becomes gradually asymmetric and rule
2 is used. This leads to distributions with a gradual heavier tail,
Pareto-like distributions.
To illustrate it, one may vary the value of the parameter as
described in Table 1. This leads to eight simulation cases with
gradual asymmetry in the Logistic bimap.
Then, eight different simulations are run for the Logistic bimap
and rule 2 as previously but with the value of b varying from 1.032
to 1.084. The resulting money distributions are then obtained. It
is observed that as b increases, the number of non-participants
decreases. This is because the chaotic map expands and its resulting
projections on axis x and y grow in range, taking a greater group of
i and j values (see [22]). Eliminating these non-participants of the
final money distributions, and so their money too, one can obtain
the final cumulative distribution functions (CDF) for the different
values of b . Fig. 4 shows the CDF’s obtained. Here the probability of
having a quantity of money bigger or equal to the variable MONEY.
An interesting progression is shown as the asymmetry increases
progressively. Fig. 4(a) shows the representation of simulation
cases 1 to 5 in a natural log plot up to a range of $2000. Here it
is observed that as b increases, these distributions diverge from
the exponential shape. As b increases, the straight shape obtained
for the symmetric case bends progressively, increasing the probability of finding an agent in the state of poorness. It also can be seen
that for cases 3 and 4 where no agent can be found in a middle range
of wealth (from 1000$ to 2000$). This means that the distribution
of money is becoming progressively more unequal.
In Fig. 4(b) the CDF’s for simulation cases 6 to 8 are depicted
from a range of 2000$ and in double decimal logarithm plot. Here,
a minority of agents reach very high fortunes. This explains, how
other majority of agents becomes in the state of poorness. The data
seem to follow a straight line arrangement for case 6, which resembles a Pareto distribution. Cases 7 and 8 show two straight line
arrangements which can also be adjusted to two Pareto distributions of different slopes.
Fig. 4(a) and (b) show how the community is becoming more
unequal as b increases. The rich gets richer as the asymmetry of
the chaotic selection of agents increases. In the other side of the
society, the proportion of the population that ends in the state of
poverty increases with b .
Analyzing the dynamics of the system one may see what is happening at micro-level. This can be appreciated in Fig. 5 where the
number of transactions won or lost are depicted per agent.
Fig. 5 shows, in number of interactions, the times an agent has
been a looser (bottom graph) and the difference of winning over
losing times (top graph). The x axis shows the ranking of agents
ordered by its final money, in a way so that, agent number 0 is the
richest of the community and agent number 5000 is in the poorest
range.
Fig. 5(a) depicts the symmetric case, where a = b = 1.032.
Here, the number of wins and looses is uniformly distributed
among the community. There also is a range of agents that don’t
Fig. 4. CDF’s obtained for simulation cases of Table 1. (a) Representations up to 2000$ dollars of the final distributions for cases 1,2,3,4 and 5 in natural log plot. (b)
Representations from 2000$ dollars of the final cases 6,7, and 8 in double decimal logarithm plot.
C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
29
Fig. 5. Representation of the role of all agents after all the interactions. Agents are arranged in descending order according to their final wealth. The upper graphic shows the
total number of wins over looses of an agent. The bottom graphic shows the number of times an agent has been selected as i agent or looser. (a) Simulation case 1, b = 1.032.
(b) Simulation case 8, b = 1.084.
interact (1133 agents), this can be seen clearly in the figures now.
In this case, the chaotic selection of agents show no particular
preference and the final distribution becomes the exponential.
Similar to traditional simulations with random agents [25].
Fig. 5(b) shows the same magnitudes for case 8, where b =
1.084 and the asymmetry is maximum. Here it can be seen the
first group of agents in the range of maximum richness that never
loose. The chaotic selection is giving them maximum luck and this
makes them richer and richer at every transaction. These are 184
rich agents. Then it comes a lower range of agents than in Fig. 5(a),
that are passive and never interact (470 agents). There is no middle
class here, and the rest of the community (4346 agents) become in
state of poorness with a final wealth inferior to 500$ and of them,
1874 agents finish with no money at all.
It is also interesting to see in Fig. 5(b) that in the poor class there
are agents that have a positive difference of wins over looses, but
amazingly they are poor anyway. Consequently, one can deduce
that they are also bad luck guys. They are j agents in most part
of their transactions but unfortunately their corresponding trading
partners (i agents) are poor too, and they can effectively earn low
or no money in these interactions.
4.3. Scenario III: chaotic selection of agents with chaotic money
exchange
In this section, the selection of agents and the exchange of
money are chaotic. Economically, this means that commercial relations are complex and some transactions are restricted. The prices
of products and services are not random, they vary disorderly but
in a deterministic way.
As in the previous sections, the chaotic maps of 2 and 3 are used
and the map variables, xt and yt , are used as simulation parameters. The computer simulations are performed in the following
manner. A community of N = 5000 agents with an initial quantity
of money of m0 = 1000$ is considered. The simulations take a total
of 50 Millions of transactions. For each transaction, three chaotic
floats in the interval [0, 1] are produced. Two of them are used to
select chaotically a pair of agents (i, j) for each interaction. The pair
(i, j) is easily obtained from the coordinates of a chaotic point at
instant t, Xt = [xt , yt ], by a simple float to integer conversion (xt and
yt to i and j, respectively). Additionally, to obtain a chaotic money
exchange, a float number in the interval [0, 1] is obtained form
the coordinates of a chaotic point by taking one coordinate or a
combination of them. This number produces the chaotic quantity
of money m that is traded between agents i and j. The floats used
for the selection of agents are |xt |/1.5 and |xt+1 |/1.5 for the Hénon
map or xt and xt+1 for the Logistic Bimap. Additionally, the float
number used to produce is calculated as (|yt | + |yt+1 |)/0.8 for the
Hénon map or as (yt + yt+1 )/2 for the Logistic Bimap. This values
and the selected rule of exchange determine the money m that is
transferred between agents.
Two different simulations are considered, using rule 1 or 2.
For each rule, the final distributions of money are obtained with
the Hénon chaotic map (with a = 1.4, b = 0.3, x0 = −0.75 and y0 =
−0.02) or the Logistic bimap (with a = b = 1.05392, x0 = 0.4913
and y0 = 0.6913). Fig. 6(a) and (b) show the probability distributions of money obtained in this scenario.
The results obtained are quite similar to those obtained in scenario I (sub-section 4.1), where agents were selected randomly. The
only difference is that some agents do not interact. Precisely 718
agents in the Hénon case and 654 in the Logistic bimap case. These
are removed of the model as in previous scenarios. Fig. 6(a) shows
how rule 1 gives a Gamma-like distribution. The final money distribution is fitted to the Gamma function expressed as P(x) = cxa e−bx .
The detail of the fitting to this distribution can also be seen in the
figure. The Hénon map fits with parameters a = 1.9401, b = 0.0032
and c = 0.0051 and the Logistic bimap with a = 3.0638, b = 0.0041
and c = 0.000007. Fig. 6(b) shows how rule 2 produces exponential
distributions. In also shows the detail of the fitting to this distribution expressed as P(x) = ae−x/b . In this case, the Hénon map money
distribution fits with parameters a = 206.412 and b = 1000 and the
Logistic bimap with a = 208.949 and b = 1000. As in scenarios I and
II, b = x = M/N.
Showing these results one may be tempted to conclude that the
chaotic selection of the trading parameter is dominant over the
chaotic selection of agents. This may be so, but in scenario III the
selection of agents is slightly different to the one used in scenario
II. Consequently, to validate this hypothesis, a new simulation case
is required with the selection the agents of scenario II. This is done
by modifying the simulation parameters as follows: selection of
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C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
Fig. 6. Probability distribution of money obtained in scenario III. (a) Rule 1 with chaotic agents and trade selection using the Hénon map or the Logistic bimap. Detail of
the fitting to the Gamma distribution. (b) Rule 2 with chaotic agents and trade selection using the Hénon map or the Logistic bimap. Detail of the fitting to the Exponential
distribution.
agents i and j, is done with |xt |/1.5 and |yt |/0.4 for the Hénon map
or xt and yt for the Logistic Bimap. Additionally, |xt+1 |/1.5 is taken
to produce for the Hénon map or as xt+1 for the Logistic Bimap.
Fig. 7(a) and (b) show the distributions of money obtained for this
new case.
These results show how the previous hypothesis is invalid. In
this case, the parameters used for the selection of agents seem to
be determinant. This is so, in the sense that taking both coordinates
at different instants of time produces a random-like selection (see
Fig. 6) as in scenario I. When the selection of agents is done with
the coordinates of the chaotic map at the same instant, the intrinsic
correlation of the chaotic coordinates introduces a correlation factor in the economic transactions that determines a result different
from the random scenario (see Fig. 7).
Now, one may appreciate that both aspects, the trading parameter and the selection of agents are affecting the final result. Here
Rule 1 shows a mixed behavior. The Hénon maps resembles a
Gamma-like distribution as in the previous simulation, but the
Logistic bimap gives rise to an Pareto shape. When rule 2 is used,
the results resemble those of scenario II. The selection of agents
becomes a predominant factor.
Analyzing at micro-level, one may look at the number of times
that an agent wins or looses. Fig. 8 shows, in number of interactions,
the times an agent has been a looser (bottom graph) and the difference of winning over losing times (top graph). The x axis shows
the ranking of agents ordered by its final money, in a way so that,
agent number 0 is the richest of the community and agent number
5000 is in the poorest range.
Fig. 8 (a) shows the selection of agents for the Hénon map
with one coordinate in consecutive instants of time |xt |/1.5 and
|xt+1 |/1.5. This resembles a random situation where there are no
preferences for any agent as winner or looser. This case resembles
the situation of random selection of agents as in Scenario I. Fig. 8 (b)
shows the selection of agents for the Hénon map with both coordinates at the same instant of time |xt |/1.5 and |yt |/0.4. One may
appreciate how Fig. 8(b) shows that there are agents that win more
than others. There demonstrates an intrinsic correlation between
the values of the chaotic coordinates at a given instant of time and
Fig. 7. Probability distribution of money obtained in scenario III where the selection of agents is performed exactly as in scenario II. (a) Rule 1 with chaotic agents and trade
selection using the Hénon map or the Logistic bimap. (b) Rule 2 with chaotic agents and trade selection using the Hénon map or the Logistic bimap.
C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
31
Fig. 8. Representation of the role of all agents after the simulation. Agents are arranged in descending order according to their final money. Upper graphics show the total
number of wins over looses for an agent. Bottom graphics show the number of times an agent has been selected as looser (i agent). (a) First simulation case with agents
selected with Hénon map’s coordinates in consecutive instants of time. (b) Second simulation case with agents selected with Hénon map at the same instant of time.
this causes agents to have preferences in their transactions, which
leads to an asymmetric situation.
5. Conclusions
The work presented here focuses on the statistical distribution of money in a closed community of individuals, where agents
exchange their money under certain economic laws. The various
models existing in this field are based on computational science
([25,9,5,2,21]). They implement computer multi-agent simulations
to understand the multi-scale behavior of economic markets. These
models simulate scenarios where market evolution parameters,
such as the selection of interacting agents or the money they
exchange, are traditionally random. As reality tends to be complex
rather than purely random, it may seem interesting to consider
chaotic parameters in the evolution of an economic system. The
paper introduces this new perspective and presents a series of
novel agent-based computational simulations, where the market
is driven by chaotic dynamics.
It is shown that chaos offers a simple model to reproduce the
various money distributions observed in real economies (Gamma,
Exponential or Pareto distributions). New and interesting results
are obtained in the simulation scenarios. Restriction of commercial
relations is observed, as well as a dependence on the rule of trade in
the final distribution of money in a chaotic market. This dependence
can be summarized in Table 2.
On one hand, in scenario I (chaotic prices of products) it is
seen that the rule of exchange used in the market is determinant
to produce a more or less equitable distribution of the money. A
symmetric rule of trading, leads to more equitable distributions
of wealth (Gamma-like distributions). Consequently, the effect of
chaos in prices is, that the policies of trade become responsible of
obtaining a fairer distribution of money in the society.
Table 2
Summary of the influence of chaos in the final money distribution
Rule 1
Rule 2
Chaotic trade
Chaotic selection of agents
Gamma
Exponential
Exponential
Pareto
On the other hand, in scenario II (chaotic transaction partners) demonstrates that asymmetric conditions in the formation
of interacting pairs lead to unequal distributions. It seems that the
asymmetry of the trading rule and also of the chaotic selection of
agents leads to less equitable distributions of money. Consequently,
this type of the markets operate under “unfair” or asymmetric
conditions. Moreover, under these assumptions, this scenario illustrates how a small group of people can be chaotically destined to
be very rich, while the bulk of the population ends up in state of
poverty. This may resemble some realistic conditions, showing how
some individuals can accumulate big fortunes in trading markets,
as a natural consequence of the intrinsic asymmetric conditions of
real economy.
Scenario III (chaotic prices of products and transaction partners) demonstrates how the chaotic selection of agents is able to
introduce a correlation between pairs of traders. In particular, this
mechanism is able to break the natural pairing symmetry (i, j) ⇔
(j, i) that appears in random scenarios. This scenario also shows
how both factors, the combination of chaotic prices or chaotic selection of partners, affect equally in the final result.
The authors hope that this work can trigger complementary
studies, that can help to uncover new clues in the nature of economic self-organizing systems.
Acknowledgements
The authors acknowledge some financial support by Spanish
grant DGICYT-FIS2009-13364-C02-01.
References
[1] A.Y. Abul-Magd, Wealth distribution in an ancient Egyptian society, Phys. Rev.
E. 66 (2002) 057104, doi:10.1103/PhysRevE.66.057104.
[2] J.P. Bouchaud, M. Mézard, Wealth condensation in a simple model economy,
Physica A 282 (2000) 536–545.
[3] B.K. Chakrabarti, A. Chakraborti, A. Chatterjee (Eds.), Econophysics and Sociophysics: Trends and Perpectives, Willey-VCH, Berlin, 2006.
[4] B.K. Chakrabarti, S. Marjit, Self-organization in game of life and economics,
Indian J. Phys. B 69 (1995) 681–698.
[5] A. Chakraborti, B.K. Chakrabarti, Statistical mechanics of money: how saving
propensity affects its distribution, Eur. Phys. J. B 17 (2000) 167–170.
[6] D.G. Champernowne, A model of income distribution, Econ. J. 63 (1953)
318–351.
32
C. Pellicer-Lostao, R. López-Ruiz / Journal of Computational Science 1 (2010) 24–32
[7] D.G. Champernowne, F.A. Cowell, Economic inequality and income distribution,
Cambrige University Press, Cambridge, 1999.
[8] A. Chatterjee, S. Yarlagadda, B.K. Chakrabarti (Eds.), Econophysics of Wealth
Distributions, Springer Verlag, Milan, 2005.
[9] A.A. Dragulescu, V.M. Yakovenko, Statistical mechanics of money, Eur. Phys. J.
B 17 (2000) 723–729 (arXiv:cond-mat/0001432).
[10] A.A. Dragulescu, V.M. Yakovenko, Exponential and power-law probability distributions of wealth as income in the United Kingdom and the United States,
Physica A 299 (2001) 213–221 (arXiv:cond-mat/0103544v2).
[11] A. Dragulescu, V.M. Yakovenko, Evidence for the exponential distribution of
income in the USA, Eur. Phys. J. B 20 (2001) 585–589.
[12] J. González-Estévez, M. Cosenza, O. Alvárez-Llamoza, R. López-Ruiz, Transition
from Pareto to Boltzmann–Gibbs behavior in a deterministic economic model,
Physica A 388 (2009) 3521–3526.
[13] B. Hayes, Follow the money, Am. Sci. 90 (2002) 400–406.
[14] M. Hénon, A two-dimensional mapping with a strange attractor, Commun.
Math. Phys. 50 (1976) 69–77.
[15] J.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. E 106 (1957)
620–630.
[16] O.S. Klass, O. Biham, M. Levy, O. Malcai, S. Solomon, The Forbes 400 and the
Pareto wealth distribution, Econ. Lett. 90 (2006) 290–295.
[17] R. López-Ruiz, Pérez-Garcia, Dynamics of maps with a global multiplicative
coupling, Chaos Solitons Fractals 1 (1991) 511–528.
[18] R. López-Ruiz, J. Sañudo, X. Calbet, Geometrical derivation of the Boltzmann
factor, Am. J. Phys. 76 (2008) 780–781.
[19] R. Mantegna, H.E. Stanley, An Introduction to econophysics: correlations in
finance, Cambridge University Press, 2000, ISBN 0521620082.
[20] V. Pareto, Cours d’economie politique, F. Rouge, Lausanne and Paris, 1897.
[21] M. Patriarca, A. Chakraborti, K. Kaski, Gibbs versus non-Gibbs distributions in
money dynamics, Physica A 340 (2004) 334–339.
[22] C. Pellicer-Lostao, R. López-Ruiz, Orbit diagram of the Hénon map and Orbit diagram of two coupled logistic maps, from The Wolfram Demonstrations Project.
URL: http://demonstrations.wolfram.com/.
[23] C. Pellicer-Lostao, R. López-Ruiz, Economic models with chaotic money
exchange, Lect. Notes Comput. Sci. 5544 (2009) 43–52 (arXiv:0901.1038v1).
[24] S. Sinha, Evidence of the Power-law tail of the wealth distribution in India,
Physica A 359 (2006) 555–562.
[25] V.M. Yakovenko, Econophysics, statistical mechanics approach to, in: Encyclopedia of Complexity and System Science, Springer, 2009, ISBN 9780387758886,
(arXiv:0709.3662v4).
C. Pellicer-Lostao was born in Zaragoza (Spain) on
April the 5th in 1966. Graduate in Physics (area applied
physiscs, electronics) at the University Complutense of
Madrid in 1990. She was Assistant Professor in the Public
University of Navarra (Pamplona, Navarra) in the Department of Automatics and Computation (Area Telematics)
during 2003–2005. From 2006 she is Assistant Professor
in Department of Computer Science (DIIS) and member of
the Institute for Biocomputation and Physics of Complex
Systems (BIFI) in the University of Zaragoza (Zaragoza,
Spain). Her main research interests are: computation in
complex systems, economic models and chaotic cryptography.
R. López-Ruiz was born in Tudela (Navarra, Spain) on
October the 8th in 1967. Graduate in Physics at the University of Zaragoza in 1990 and Ph.D. in Physics at the
University of Navarra in 1994. His Postdoctoral Period was
spent in the Laboratoire de Physique Statistique at the
Ecole Normale Superieure of Paris during 1995–1996 and
in the Department of Physics at the Universidad of Buenos
Aires during 1997. From 1998 to 2000, he was Assistant Professor in the Department of Theoretical Physics
at the University of Zaragoza. Since 2001, he is Associate
Professor in the Department of Computer Science at the
University of Zaragoza, Spain. His main research interests are: computation in complex systems and biological,
economic and social models.