Stratified economic exchange on networks

We investigate a model of stratified economic interactions between agents when the notion of spatial location is introduced. The agents are placed on a network with near-neighbor connections. Interactions between neighbors can occur only if the difference in their wealth is less than a threshold value that defines the width of the economic classes. By employing concepts from spatiotemporal dynamical systems, three types of patterns can be identified in the system as parameters are varied: laminar, intermittent and turbulent states. The transition from the laminar state to the turbulent state is characterized by the activity of the system, a quantity that measures the average exchange of wealth over long times. The degree of inequality in the wealth distribution for different parameter values is characterized by the Gini coefficient. High levels of activity are associated to low values of the Gini coefficient. It is found that the topological properties of the network have little effect on the activity of the system, but the Gini coefficient increases when the clustering coefficient of the network is increased.     

Pareto and Boltzmann-Gibbs behaviors in a deterministic multi-agent system

A deterministic system of interacting agents is considered as a model for economic dynamics. The dynamics of the system is described by a coupled map lattice with nearest neighbor interactions. The evolution of each agent results from the competition between two factors: the agent’s own tendency to grow and the environmental influence that moderates this growth. Depending on the values of the parameters that control these factors, the system can display Pareto or Boltzmann-Gibbs statistical behaviors in its asymptotic dynamical regime. The regions where these behaviors appear are calculated on the space of parameters of the system. Other statistical properties, such as the mean wealth, the standard deviation, and the Gini coefficient characterizing the degree of equity in the wealth distribution are also calculated.     

Evolution of the Distribution of Wealth in an Economic Environment Driven by Local Nash Equilibria

We present and analyze a model for the evolution of the wealth distribution within a heterogeneous economic environment. The model considers a system of rational agents interacting in a game theoretical framework, through fairly general assumptions on the cost function. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse gamma distribution as an equilibrium in the particular case of quadratic cost functions which has been previously considered in the literature.     

Class formation in a social network with asset exchange

We study two kinds of economic exchanges, additive and multiplicative, in a system of N agents. The work is divided into two parts. In the first one, the agents are free to interact with each other. The system evolves to a Boltzmann-Gibbs distribution with additive exchange and condenses with a multiplicative one. If bankruptcy is introduced, both types of exchange lead to condensation. Condensation times have been studied. In the second part, the agents are placed in a social network. We analyze the behavior of wealth distributions in time, and the formation of economic classes is observed for certain values of network connectivity.     

Kinetic market models with single commodity having price fluctuations

We study here numerically the behavior of an ideal gas like model of markets having only one non-consumable commodity. We investigate the behavior of the steady-state distributions of money, commodity and total wealth, as the dynamics of trading or exchange of money and commodity proceeds, with local (in time) fluctuations in the price of the commodity. These distributions are studied in markets with agents having uniform and random saving factors. The self-organizing features in money distribution are similar to the cases without any commodity (or with consumable commodities), while the commodity distribution shows an exponential decay. The wealth distribution shows interesting behavior: gamma like distribution for uniform saving propensity and has the same power-law tail, as that of the money distribution, for a market with agents having random saving propensity.     

Economic exchanges in a stratified society: End of the middle class?

We study the effect of the social stratification on the wealth distribution on a system of interacting economic agents that are constrained to interact only within their own economic class. The economical mobility of the agents is related to its success in exchange transactions. Different wealth distributions are obtained as a function of the width of the economic class. We find a range of widths in which the society is divided in two classes separated by a deep gap that prevents further exchange between poor and rich agents. As a consequence, the middle wealth class is eliminated. The high values of the Gini indices obtained in these cases indicate a highly unequal society. On the other hand, lower and higher widths induce lower Gini indices and a fairer wealth distribution.     

Network evolution based on minority game with herding behavior

The minority game (MG) is used as a source of information to design complex networks where the nodes represent the playing agents. Differently from classical MG consisting of independent agents, the current model rules that connections between nodes are dynamically inserted or removed from the network according to the most recent game outputs. This way, preferential attachment based on the concept of social distance is controlled by the agents wealth. The time evolution of the network topology, quantitatively measured by usual parameters, is characterized by a transient phase followed by a steady state, where the network properties remain constant. Changes in the local landscapes around individual nodes depend on the parameters used to control network links. If agents are allowed to access the strategies of their network neighbors, a feedback effect on the network structure and game outputs is observed. Such effect, known as herding behavior, considerably changes the dependence of volatility σ on memory size: it is shown that the absolute value of σ as well as the corresponding value of memory size depend both on the network topology and on the way along which the agents make their playing decisions in each game round.     

Dynamics of wealth inequality

We study an agent-based model of evolution of wealth distribution in a macroeconomic system. The evolution is driven by multiplicative stochastic fluctuations governed by the law of proportionate growth and interactions between agents. We are mainly interested in interactions increasing wealth inequality, that is, in a local implementation of the accumulated advantage principle. Such interactions destabilise the system. They are confronted in the model with a global regulatory mechanism that reduces wealth inequality. There are different scenarios emerging as a net effect of these two competing mechanisms. When the effect of the global regulation (economic interventionism) is too weak, the system is unstable and it never reaches equilibrium. When the effect is sufficiently strong, the system evolves towards a limiting stationary distribution with a Pareto tail. In between there is a critical phase. In this phase, the system may evolve towards a steady state with a multimodal wealth distribution. The corresponding cumulative density function has a characteristic stairway pattern that reflects the effect of economic stratification. The stairs represent wealth levels of economic classes separated by wealth gaps. As we show, the pattern is typical for macroeconomic systems with a limited economic freedom. One can find such a multimodal pattern in empirical data, for instance, in the highest percentile of wealth distribution for the population in urban areas of China.     

Effects of randomness on chaos and order of coupled logistic maps

Natural systems are essentially nonlinear being neither completely ordered nor completely random. These nonlinearities are responsible for a great variety of possibilities that includes chaos. On this basis, the effect of randomness on chaos and order of nonlinear dynamical systems is an important feature to be understood. This Letter considers randomness as fluctuations and uncertainties due to noise and investigates its influence in the nonlinear dynamical behavior of coupled logistic maps. The noise effect is included by adding random variations either to parameters or to state variables. Besides, the coupling uncertainty is investigated by assuming tinny values for the connection parameters, representing the idea that all Nature is, in some sense, weakly connected. Results from numerical simulations show situations where noise alters the system nonlinear dynamics.     

The distribution of wealth in the presence of altruism in simple economic models

We study the effect of altruism in two simple asset exchange models: the yard sale model (winner gets a random fraction of the poorer player's wealth) and the theft and fraud model (winner gets a random fraction of the loser's wealth). We also introduce in these models the concept of bargaining efficiency, which makes the poorer trader more aggressive in getting a favorable deal thus augmenting his winning probabilities. The altruistic behavior is controlled by varying the number of traders who behave altruistically and by the degree of altruism that they show. The resulting wealth distribution is characterized using the Gini index. We compare the resulting values of the Gini index at different levels of altruism in both models. It is found that altruistic behavior does lead to a more equitable wealth distribution but only for unreasonable high values of altruism that are difficult to expect in a real economic system.     

Effects of introduction of new resources and fragmentation of existing resources on limiting wealth distribution in asset exchange models

Pareto law, which states that wealth distribution in societies has a power-law tail, has been the subject of intensive investigations in the statistical physics community. Several models have been employed to explain this behavior. However, most of the agent based models assume the conservation of number of agents and wealth. Both these assumptions are unrealistic. In this paper, we study the limiting wealth distribution when one or both of these assumptions are not valid. Given the universality of the law, we have tried to study the wealth distribution from the asset exchange models point of view. We consider models in which (a) new agents enter the market at a constant rate (b) richer agents fragment with higher probability introducing newer agents in the system (c) both fragmentation and entry of new agents is taking place. While models (a) and (c) do not conserve total wealth or number of agents, model (b) conserves total wealth. All these models lead to a power-law tail in the wealth distribution pointing to the possibility that more generalized asset exchange models could help us to explain the emergence of a power-law tail in wealth distribution.     

A coupled map lattice model for rheological chaos in sheared nematic liquid crystals

A variety of complex fluids under shear exhibit complex spatiotemporal behavior, including what is now termed rheological chaos, at moderate values of the shear rate. Such chaos associated with rheological response occurs in regimes where the Reynolds number is very small. It must thus arise as a consequence of the coupling of the flow to internal structural variables describing the local state of the fluid. We propose a coupled map lattice model for such complex spatiotemporal behavior in a passively sheared nematic liquid crystal using local maps constructed so as to accurately describe the spatially homogeneous case. Such local maps are coupled diffusively to nearest and next-nearest neighbors to mimic the effects of spatial gradients in the underlying equations of motion. We investigate the dynamical steady states obtained as parameters in the map and the strength of the spatial coupling are varied, studying local temporal properties at a single site as well as spatiotemporal features of the extended system. Our methods reproduce the full range of spatiotemporal behavior seen in earlier one-dimensional studies based on partial differential equations. We report results for both the one- and two-dimensional cases, showing that spatial coupling favors uniform or periodically time-varying states, as intuitively expected. We demonstrate and characterize regimes of spatiotemporal intermittency out of which chaos develops. Our work indicates that similar simplified lattice models of the dynamics of complex fluids under shear should provide useful ways to access and quantify spatiotemporal complexity in such problems, in addition to representing a fast and numerically tractable alternative to continuum representations.     

Chaos and unpredictability in evolutionary dynamics in discrete time

A discrete-time version of the replicator equation for two-strategy games is studied. The stationary properties differ from those of continuous time for sufficiently large values of the parameters, where periodic and chaotic behavior replace the usual fixed-point population solutions. We observe the familiar period-doubling and chaotic-band-splitting attractor cascades of unimodal maps but in some cases more elaborate variations appear due to bimodality. Also unphysical stationary solutions can have unusual physical implications, such as the uncertainty of the final population caused by sensitivity to initial conditions and fractality of attractor preimage manifolds.     

Emergence of power-law in a market with mixed models

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A Yardsale (YS) model in which amount put on the bet is a fraction of minimum of the two players leads to condensation of wealth in hands of some agent while theft and fraud (TF) model in which the amount to be exchanged is a fraction of loser's wealth leads to an exponential distribution of wealth. We show that if we allow few agents to follow a different model than others, i.e., there are some agents following TF model while rest follow YS model, it leads to distribution with power-law tails. Similar effect is observed when one carries out transactions for a fraction of one's wealth using TF model and for the rest YS model is used. We also observe a power-law tail in wealth distribution if we allow the agents to follow either of the models with some probability.     

Entropy and equilibrium state of free market models

Many recent models of trade dynamics use the simple idea of wealth exchanges among economic agents in order to obtain a stable or equilibrium distribution of wealth among the agents. In particular, a plain analogy compares the wealth in a society with the energy in a physical system, and the trade between agents to the energy exchange between molecules during collisions. In physical systems, the energy exchange among molecules leads to a state of equipartition of the energy and to an equilibrium situation where the entropy is a maximum. On the other hand, in a large class of exchange models, the system converges to a very unequal condensed state, where one or a few agents concentrate all the wealth of the society while the wide majority of agents shares zero or almost zero fraction of the wealth. So, in those economic systems a minimum entropy state is attained. We propose here an analytical model where we investigate the effects of a particular class of economic exchanges that minimize the entropy. By solving the model we discuss the conditions that can drive the system to a state of minimum entropy, as well as the mechanisms to recover a kind of equipartition of wealth.     

An impact excitation system for repeatable,high-bandwidth modal testing of miniature structures

Miniature components and devices are increasingly seen in a myriad of applications. In general, the dynamic behavior of miniature devices is critical to their functionality and performance. However, modal testing of miniature structures poses many challenges. This paper presents a design and evaluation of an impact excitation system (IES) for repeatable, high-bandwidth, controlled-force modal testing of miniature structures. Furthermore, a dynamic model of the system is derived and experimentally validated to enable the identification of the system parameters that yield single-hit impacts with desired bandwidth and force magnitude. The system includes a small instrumented impact tip attached to a custom designed flexure-based body, an automated electromagnetic release mechanism, and various precision positioners. The excitation bandwidth and the impact force magnitude can be controlled by selecting the system parameters. The dynamic model of the system includes the structural dynamics of the flexure-based body, the electromagnetic force and the associated eddy-current damping, and the impact event. A validation study showed an excellent match between the model simulations and experiments in terms of impact force and bandwidth. The model is then used to create process maps that relate the system parameters to the number of hits (single vs. multiple), the impact force magnitudes and the excitation bandwidths. These process maps can be used to select system parameters or predict system response for a given set of parameters. A set of experiments is conducted to compare the performances of the IES and a (manual) miniature impact hammer. It is concluded that the IES significantly improves repeatability in terms of the impact bandwidth, location, and force magnitude, while providing a high excitation-bandwidth and excellent coherence values. The application of the IES is demonstrated through modal testing of a miniature contact-probe system.     

Statistical equilibrium in simple exchange games II. The redistribution game

We propose a simple stochastic exchange game mimicking taxation and redistribution. There are g agents and n coins; taxation is modeled by randomly extracting some coins; then, these coins are redistributed to agents following Polya's scheme. The individual wealth equilibrium distribution for the resulting Markov chain is the multivariate symmetric Polya distribution. In the continuum limit, the wealth distribution converges to a Gamma distribution, whose form factor is just the initial redistribution weight. The relationship between this taxation-and-redistribution scheme and other simple conservative stochastic exchange games (such as the BDY game) is discussed.     

Parameter renormalization of maps based on potential function

A systematic way for deriving the parameter renormalization group equation for one-dimensional maps is presented and the critical behavior of periodic doubling is investigated. Introducing a formal potential function in one-parameter cases, it is shown that accumulation points correspond to local potential maxima and universal constants are easily determined. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function shows scaling in the parameter space with the universal convergent rate at the accumulation point similar to the Feigenbaum universal function. For two-parameter cases, a parameter reduction transformation is found to be useful to determine some important fixed points. A locally defined potential function is introduced and its scaling property is discussed. (c) 1997 American Institute of Physics.     

Electronically-implemented coupled logistic maps

The logistic map is a paradigmatic dynamical system originally conceived to model thediscrete-time demographic growth of a population, which shockingly, shows that discretechaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we designand characterize a simple, low-cost, easy-to-handle, electronic implementation of thelogistic map. In particular, our implementation allows for straightforwardcircuit-modifications to behave as different one-dimensional discrete-time systems. Also,we design a coupling block in order to address the behavior of two coupled maps, although,our design is unrestricted to the discrete-time system implementation and it can begeneralized to handle coupling between many dynamical systems, as in a complex system. Ourfindings show that the isolated and coupled maps’ behavior has a remarkable agreementbetween the experiments and the simulations, even when fine-tuning the parameters with aresolution of ~10^-3. We support these conclusions by comparing the Lyapunovexponents, periodicity of the orbits, and phase portraits of the numerical andexperimental data for a wide range of coupling strengths and map’s parameters.     

Buying on margin,selling short in an agent-based market model

Credit trading, or leverage trading, which includes buying on margin and selling short, plays an important role in financial markets, where agents tend to increase their leverages for increased profits. This paper presents an agent-based asset market model to study the effect of the permissive leverage level on traders’ wealth and overall market indicators. In this model, heterogeneous agents can assume fundamental value-converging expectations or trend-persistence expectations, and their effective demands of assets depend both on demand willingness and wealth constraints, where leverage can relieve the wealth constraints to some extent. The asset market price is determined by a market maker, who watches the market excess demand, and is influenced by noise factors. By simulations, we examine market results for different leverage ratios. At the individual level, we focus on how the leverage ratio influences agents’ wealth accumulation. At the market level, we focus on how the leverage ratio influences changes in the asset price, volatility, and trading volume. Qualitatively, our model provides some meaningful results supported by empirical facts. More importantly, we find a continuous phase transition as we increase the leverage threshold, which may provide a further prospective of credit trading.