Marco Patriarca

Econophysics review: II. Agent-based models

This article is the second part of a review of recent empirical and theoretical developments usually grouped under the heading Econophysics. In the first part, we reviewed the statistical properties of financial time series, the statistics exhibited in order books and discussed some studies of correlations of asset prices and returns. This second part deals with models in Econophysics from the point of view of agent-based modeling. Of the large number of multi-agent-based models, we have identified three representative areas. First, using previous work originally presented in the fields of behavioral finance and market microstructure theory, econophysicists have developed agent-based models of order-driven markets that we discuss extensively here. Second, kinetic theory models designed to explain certain empirical facts concerning wealth distribution are reviewed. Third, we briefly summarize game theory models by reviewing the now classic minority game and related problems.

Statistical model with a standard Gamma distribution

We study a statistical model consisting of N basic units which interact with each other by exchanging a physical entity, according to a given microscopic random law, depending on a parameter lambda. We focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard Gamma distribution. The model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity lambda. Alternatively, from the nature of the equilibrium distribution, we show that the model can also be interpreted as a perfect gas at an effective temperature T(lambda), where particles exchange energy in a space with an effective dimension D(lambda).

Influence of geography on language competition

Competition between languages or cultural traits diffusing in the same geographical area is studied combining the model of Abrams and Strogatz with a model of human dispersal on an inhomogeneous substrate. Also, the effect of population growth is discussed. It is shown through numerical simulations that the final configuration of the languages can be strongly affected by geographical and historical factors. These factors are not related to the dynamics of culture transmission, but rather to initial population distributions as well as geographical boundaries and inhomogeneities, which modulate the diffusion process.

Fractional Fokker-Planck dynamics : Numerical algorithm and simulations

Anomalous transport in a tilted periodic potential is investigated numerically within the framework of the fractional Fokker-Planck dynamics via the underlying continuous-time random walk. An efficient numerical algorithm is developed which is applicable for an arbitrary potential. This algorithm is then applied to investigate the fractional current and the corresponding nonlinear mobility in different washboard potentials. Normal and fractional diffusion are compared through their time evolution of the probability density in state space. Moreover, we discuss the stationary probability density of the fractional current values.

Use and Abuse of a Fractional Fokker-Planck Dynamics for Time-Dependent Driving

We investigate a subdiffusive, fractional Fokker-Planck dynamics occurring in time-varying potential landscapes and thereby disclose the failure of the fractional Fokker-Planck equation (FFPE) in its commonly used form when generalized in an ad hoc manner to time-dependent forces. A modified FFPE (MFFPE) is rigorously derived, being valid for a family of dichotomously alternating force fields. This MFFPE is numerically validated for a rectangular time-dependent force with zero average bias. For this case, subdiffusion is shown to become enhanced as compared to the force free case. We question, however, the existence of any physically valid FFPE for arbitrary varying time-dependent fields that differ from this dichotomous varying family.

Influence of saving propensity on the power-law tail of the wealth distribution

Some general features of statistical multi-agent economic models are reviewed, with particular attention to the dependence of the equilibrium wealth distribution on the agents’ saving propensities. It is shown that in a finite system of agents with a continuous saving propensity distribution a power-law tail with Pareto exponent α=1 can appear also when agents do not have saving propensities distributed over the whole interval between zero and one. Rather, a power-law can be observed in a finite interval of wealth, whose lower and upper ends are shown to be determined by the lower and upper cutoffs, respectively, of the saving propensity distribution. It is pointed out that a cutoff of the power-law tail can arise also through a different mechanism, when the number of agents is small enough. Numerical simulations have been carried out by implementing a procedure for assigning saving propensities homogeneously, which results in a smoother wealth distributions and correspondingly wider power-law intervals than other procedures based on random algorithms.

Gibbs versus non-Gibbs distributions in money dynamics

We review a simple model of closed economy, where the economic agents make money transactions and a saving criterion is present. We observe the Gibbs distribution for zero saving propensity, and non-Gibbs distributions otherwise. While the exact solution in the case of zero saving propensity is already known to be given by the Gibbs distribution, here we provide the explicit analytical form of the equilibrium distribution for the general case of nonzero saving propensity. We verify it through comparison with numerical data and show that it can be cast in the form of a gamma-distribution.

Kinetic theory models for the distribution of wealth: power law from overlap of exponentials

Various multi-agent models of wealth distributions defined by microscopic laws regulating the trades, with or without a saving criterion, are reviewed. We discuss and clarify the equilibrium properties of the model with constant global saving propensity, resulting in Gamma distributions, and their equivalence to the Maxwell-Boltzmann kinetic energy distribution for a system of molecules in an effective number of dimensions

Relaxation in statistical many-agent economy models

D_\lambda

Computational study of a screw dislocation interacting with a stacking-fault tetrahedron

, related to the saving propensity