Introducing Chaos in Economic Gas-like Models

The conjecture of a kinetic theory of (ideal) gas-like model for trading markets was first discussed in 1995 [6] by econophysicists. 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. The interest of this model is that, by analog with energy, the equilibrium probability distribution of money follows the exponential Boltzmann-Gibbs law for a wide variety of trading rules [2].

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 heavytail distributions [7], [8]. Different reasons can be argued for this failure of the gas-like model. In this work, the authors suppose that real economy is not purely random.

On one hand, there is some evidence of markets being not purely random. Real economic transactions are driven by some specific interest (or profit) between the different interacting parts. On the other hand, history shows 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 and, in the long term, these systems display inherent unpredictability and 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.

Consequently, one may consider of interest to introduce chaotic patterns in the theory of (ideal) gas-like model for trading markets. One can observe this way, which money distributions are obtained, how they differ from the referenced exponential distribution and how they resemble real economic distributions.

The paper presented here, follows precisely this approach. It focuses on the statistical distribution of money in a closed community of individuals, where agents exchange their money under a certain conservative rule. But unlike these traditional models, this work is going to introduce chaotic trade interactions. More specifically it introduces a chaotic procedure for the selection of agents that interact at each transaction. This chaotic selections of trading partners is going to determine the success of some individuals over others. In the end it will be seen that, as in real life, a minority of chaos-predilected people can follow heavy tail distributions.

The contents of this paper are organized as follows: section 2 describes the simulation scenario. Section 3 shows the results obtained in this scenario. Final conclusions are discussed in Section 4.

The simulation scenario considered here follows a traditional gas-like model, but the rules of trade intend to be less random and more chaotic. The study of these scenarios was first proposed by the authors in [9]. There, it is shown that the use of chaotic numbers produces the exponential as well as other wealth distributions depending on how they are injected to the system. This paper considers the scenario where the selection of agents is chaotic, while the money exchanged at each interaction is a random quantity.

In the computer simulations presented here, a community of N agents is given with an initial equal quantity of money, m 0 , for each agent. The total amount of money, M = N * m 0 , is conserved. For each transaction, at a given instant t, a pair of agents (i, j) is selected chaotically and a random amount of money ∆m is traded between them.

To produce chaotically a pair of agents (i, j) for each interaction, a 2D chaotic system is considered. The pair (i, j) is easily obtained from the coordinates of a chaotic point at instant t, X t = [x t , y t ], by a simple float to integer conversion (x t and y t to i and j, respectively). Additionally, a random number from a standard random generator is used to obtain a float number υ in the interval [0, 1]. This number produces the random the quantity of money ∆m traded between agents x t and y t .

The particular rule of trade is the following: let us consider two agents i and j with their respective wealth, m i and m j at instant t. At each interaction, the quantity ∆m = υ * (m i + m j )/2, is taken from i and given to j . Here, the transaction of money is quite asymmetric as agent j is the absolute winner, whi

…(Full text truncated)…