Agent-based models have demonstrated their power and flexibility in Econophysics. However their major challenge is still to devise more realistic simulation scenarios. The complexity of Economy makes appealing the idea of introducing chaotic number generators as simulation engines in these models. Chaos based number generators are easy to use and highly configurable. This makes them just perfect for this application.
Economy is difficult to model. Intermittent crisis have demonstrated throughout decades, the inherent complexity of economic systems and the fatal flaws of contemporary models. Nevertheless, significant methodologies have constantly been developed to move forward. In today's Economy there are two dominant paradigms; one is based on Econometry and the other on dynamic stochastic general equilibrium [1]. But Economy seems unpredictable and quantitative econometric predictions fail not far in time. Also equilibrium based models have shown their limitations to predict disruptive crisis of real economies.
To this particular matter, Econophysics offer agent based models. These provide computerized simulation scenarios of great number of decision makers (agents) and institutions, which interact through prescribed rules. Computational power gives to these models a richness of scenarios not necessarily in equilibrium, able of reproducing more realistic and complex situations and handling a far wider range of non linear behaviour [2]. However, agent based models are not a panacea. Their major challenge lies on specifying how agents behave and in choosing the rules they use to make decisions. In fact, simplifying these specifications may lead to impractical simulation scenarios.
One essential constituent of many agent based models is their stochastic nature, for one way or another they may use stochastic simulation tools to run [3,4]. Typically a number of agents are randomly selected or awakened from sleep to take economic decisions. Not only agents, but also the economic variables they possess may be changed randomly. The random approach can be very handy as a simulation engine and it also takes into account the variety and unpredictability ingredients of real situations.
However, random number generators provide a statistical uniformity that may unwillingly disguise real situations. Going a little bit further, let’s say that Economy may be complex but it hardly seems random.
To contribute with a new approach, this work illustrates how chaos based number generators can be used in agent based models. As it will be seen, these tools are highly powerful and flexible. They offer the possibility of producing numbers with different statistical random quality or even chaotic. Moreover, they possess a series of control parameters to manipulate the dynamics of the simulation. This paper explores the main characteristics of chaos based number generators and gives an illustrative example of their application in Econophysics dedicated to the ideal gas-like models for wealth distributions.
In general sense, the term chaos refers to physical phenomena that are fully deterministic and even so, unpredictable and erratic [5]. Determinism and unpredictability, the two essential components of chaotic systems seem to be present in real economic systems. On one hand, in the short term or at micro level, economic transactions are mainly deterministic, based on ‘rational expectations’ to maximize the long-run personal advantage. On the other, in the long term or at the macro level, economic variables seem to be capricious and erratic. This makes chaos an appealing feature to be considered in economic models.
Since 1990 many pseudo-random number generators based in chaotic dynamical systems have been proposed [6]. They are based on N-dimensional deterministic discrete-time dynamical systems, and this makes them able to offer a rich variety of possibilities as simulation engines in Monte Carlo processes. An N-dimensional deterministic discrete-time dynamical system is an iterative map F: R N R N of the form: (1) where k = 0, 1,…,n is the discrete time, X 0 ,X 1 ,…,X n , are the states of the system at different instants of time and Λ is a vector of control parameters. This kind of systems present different asymptotic behaviours for different values of X 0 and Λ, where nearby orbits converge to given compact sets of the space state called attractors. If these attractors display complex behaviour, they are said to be strange attractors. Moreover when the system exhibits sensibility to initial conditions (X 0 ) is said to be in chaotic regime.
To build a chaotic PRNG is necessary to construct a numerical algorithm that transforms the states of the system in chaotic regime into integer numbers, or typically bits. The existing designs use different techniques to pass from the continuum to the binary world:
Extracting bits from each state along the chaotic orbits.
Dividing the phase space into m sub-spaces, and output a number i = 0, 1,…,m if the chaotic orbit visits the i-th subspace.
Combining the outputs of two or more chaotic systems to generate the pseudo-random numbers.
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The construction of a chaos based PRNG involves a series of design parameters (such as for example, the number of bits extracted with technique 1). These design parameters are able to control the statistical quality of randomness of the numbers pro
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