Self organization of social hierarchy and clusters in a challenging society with free random walks

Emergence of social hierarchy and clusters in a challenging equal-right society is studied on the basis of the agent-based model, where warlike individuals who have own power or wealth perform random walks in a random order on a lattice and when meeting others the individuals challenge the strongest among the neighbors. We assume that the winning probability depends on the difference in their wealth and after the fight the winner gets and the loser loses a unit of the wealth. We show that hierarchy is self organized when the population exceeds a critical value and the transition from egalitarian state to hierarchical state occurs in two steps. The first transition is continuous to the society with widespread winning-probability. At the second transition the variance of the winning fraction decrease discontinuously, which was not observed in previous studies. The second hierarchical society consists of a small number of extreme winners and many individuals in the middle class and losers. We also show that when the relaxation parameter for the wealth is large, the first transition disappears. In the second hierarchical society, a giant cluster of individuals is formed with a layered structure in the power order and some people stray around it. The structure of the cluster and the distribution of wealth are quite different from the results of the previous challenging model [M. Tsujiguchi and T. Odagaki, Physica A 375 (2007) 317] which adopts the preassigned order for random walk.     

Living in an irrational society: Wealth distribution with correlations between risk and expected profits

Different models to study the wealth distribution in an artificial society have considered a transactional dynamics as the driving force. Those models include a risk aversion factor, but also a finite probability of favoring the poorer agent in a transaction. Here, we study the case where the partners in the transaction have a previous knowledge of the winning probability and adjust their risk aversion taking this information into consideration. The results indicate that a relatively equalitarian society is obtained when the agents risk in direct proportion to their winning probabilities. However, it is the opposite case that delivers wealth distribution curves and Gini indices closer to empirical data. This indicates that, at least for this very simple model, either agents have no knowledge of their winning probabilities, either they exhibit an “irrational” behavior risking more than reasonable.     

Properties of wealth distribution in multi-agent systems of a complex network

We present a simple model for examining the wealth distribution with agents playing evolutionary games (the Prisoners’ Dilemma and the Snowdrift Game) on complex networks. Pareto’s power law distribution of wealth (from 1897) is reproduced on a scale-free network, and the Gibbs or log-normal distribution for a low income population is reproduced on a random graph. The Pareto exponents of a scale-free network are in agreement with empirical observations. The Gini coefficient of an ER random graph shows a sudden increment with game parameters. We suggest that the social network of a high income group is scale-free, whereas it is more like a random graph for a low income group.     

A unified framework for the pareto law and Matthew effect using scale-free networks

We investigate the accumulated wealth distribution by adopting evolutionary games taking place on scale-free networks. The system self-organizes to a critical Pareto distribution (1897) of wealth P(m)∼m^-(v+1) with 1.6 < v <2.0 (which is in agreement with that of U.S. or Japan). Particularly, the agent's personal wealth is proportional to its number of contacts (connectivity), and this leads to the phenomenon that the rich gets richer and the poor gets relatively poorer, which is consistent with the Matthew Effect present in society, economy, science and so on. Though our model is simple, it provides a good representation of cooperation and profit accumulation behavior in economy, and it combines the network theory with econophysics.     

<Emphasis Type="Bold">κ</Emphasis>-generalized models of income and wealth distributions: A survey

The paper provides a survey of results related to the “κ-generalized distribution”, a statistical model for the size distribution of income and wealth. Topics include, among others, discussion of basic analytical properties, interrelations with other statistical distributions as well as aspects that are of special interest in the income distribution field, such as the Gini index and the Lorenz curve. An extension of the basic model that is most able to accommodate the special features of wealth data is also reviewed. The survey of empirical applications given in this paper shows the κ-generalized models of income and wealth to be in excellent agreement with the observed data in many cases.     

A multi-agent dynamic model based on different kinds of bequests

We investigate how wealth transfer that happens at the end of an agent’s life affects its final distribution based on a multi-agent dynamic model. We discuss two kinds of wealth transfers: to a single agent and to charities. The first kind of bequest is common in our realistic world and is always regarded by the public as unequal inheritance. The bequests to charities will be gathered and then equally redistributed among the survivors in our model. We find that when all the decedents choose the second kind of bequest, the final distribution is the Gibbs exponential function. When all the decedents choose the first kind of bequest, the result is condensation that a single individual accumulates all the available wealth. When an increasing portion of decedents choose the one-heir bequests, the exponential distribution evolves towards a power law shape (accompanied by deteriorating inequality). This shape firstly appears from the intermediate range of wealth and extends towards the top end of the simulated distribution, while the distribution remains exponential for high values of the wealth. At the same time, the Gini coefficient increases and the wealth accumulation becomes serious. At last, we analyze the source of the inequality. We find that not only unequal inheritances, but also equal division of the charity’s wealth can relatively contribute to an inequality of wealth distribution.     

Study of a market model with conservative exchanges on complex networks

Many models of market dynamics make use of the idea of conservative wealth exchanges among economic agents. A few years ago an exchange model using extremal dynamics was developed and a very interesting result was obtained: a self-generated minimum wealth or poverty line. On the other hand, the wealth distribution exhibited an exponential shape as a function of the square of the wealth. These results have been obtained both considering exchanges between nearest neighbors or in a mean field scheme. In the present paper we study the effect of distributing the agents on a complex network. We have considered archetypical complex networks: Erdös–Rényi random networks and scale-free networks. The presence of a poverty line with finite wealth is preserved but spatial correlations are important, particularly between the degree of the node and the wealth. We present a detailed study of the correlations, as well as the changes in the Gini coefficient, that measures the inequality, as a function of the type and average degree of the considered networks.     

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.     

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.     

The unfair consequences of equal opportunities: Comparing exchange models of wealth distribution

Simple agent based exchange models are a commonplace in the study of wealth distribution of artificial societies. Generally, each agent is characterized by its wealth and by a risk-aversion factor, and random exchanges between agents allow for a redistribution of the wealth. However, the detailed influence of the amount of capital exchanged has not been fully analyzed yet. Here we present a comparison of two exchange rules and also a systematic study of the time evolution of the wealth distribution, its functional dependence, the Gini coefficient and time correlation functions. In many cases a stable state is attained, but, interesting, some particular cases are found in which a very slow dynamics develops. Finally, we observe that the time evolution and the final wealth distribution are strongly dependent on the exchange rules in a nontrivial way.     

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.     

Wealth condensation in a Barabasi-Albert network

We study the flow of money among agents in a Barabasi-Albert (BA) scale free network, where each network node represents an agent and money exchange interactions are established through links. The system allows money trade between two agents at a time, betting a fraction f of the poorer’s agent wealth. We also allow for the bet to be biased, giving the poorer agent a winning probability p. In the no network case there is a phase transition involving a relationship between p and f. In the networked case, we also found a condensation interface, however, this is not a complete condensation due to the presence of clusters in the network and its topology. As can be expected, the winner is always a well-connected agent, but we also found that the mean wealth decreases with the agents’ connectivity.     

Wealth distributions in asset exchange models

A model for the evolution of the wealth distribution in an economically interacting population is introduced, in which a specified amount of assets are exchanged between two individuals when they interact. The resulting wealth distributions are determined for a variety of exchange rules. For “random” exchange, either individual is equally likely to gain in a trade, while “greedy” exchange, the richer individual gains. When the amount of asset traded is fixed, random exchange leads to a Gaussian wealth distribution, while greedy exchange gives a Fermi-like scaled wealth distribution in the long-time limit. Multiplicative processes are also investigated, where the amount of asset exchanged is a finite fraction of the wealth of one of the traders. For random multiplicative exchange, a steady state occurs, while in greedy multiplicative exchange a continuously evolving power law wealth distribution arises. Received: 13 August 1997 / Revised: 31 December 1997 / Accepted: 26 January 1998     

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.     

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.     

Characteristic times in stock market indices

In this study we analyze the Standard and Poor's 500 index data of the New York Stock Exchange for more than 32 years. Using a simple random walk model we demonstrate that the proper variable to look at is the logarithmic return. In the statistical analysis we have done fittings to the Lévy distribution using either the index data as such or pre-processing it with ARCH, GARCH or IGARCH methods, which tend to remove the time-dependent variance. For short times the truncated Lévy distribution is found to fit the data quite well. Since this is not a stable distribution, the scaling behavior observed for short times should brake down for longer times. We demonstrate that the characteristic time where this cross-over starts is of the order of one day.     

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.     

f-Gintropy: An Entropic Distance Ranking Based on the Gini Index

We consider an entropic distance analog quantity based on the density of the Gini index in the Lorenz map, i.e., gintropy. Such a quantity might be used for pairwise mapping and ranking between various countries and regions based on income and wealth inequality. Its generalization to f-gintropy, using a function of the income or wealth value, distinguishes between regional inequalities more sensitively than the original construction.     

Ideal-gas-like market models with savings: Quenched and annealed cases

We analyze the ideal-gas-like models of markets and review the different cases where a ‘savings’ factor changes the nature and shape of the distribution of wealth. These models can produce similar distribution of wealth as observed across varied economies. We present a more realistic model where the saving factor can vary over time (annealed savings) and yet produce Pareto distribution of wealth in certain cases. We discuss the relevance of such models in the context of wealth distribution, and address some recent issues in the context of these models.     

Microscopic origin of non-Gaussian distributions of financial returns

In this paper we study the possible microscopic origin of heavy-tailed probability density distributions for the price variation of financial instruments. We extend the standard log-normal process to include another random component in the so-called stochastic volatility models. We study these models under an assumption, akin to the Born-Oppenheimer approximation, in which the volatility has already relaxed to its equilibrium distribution and acts as a background to the evolution of the price process. In this approximation, we show that all models of stochastic volatility should exhibit a scaling relation in the time lag of zero-drift modified log-returns. We verify that the Dow-Jones Industrial Average index indeed follows this scaling. We then focus on two popular stochastic volatility models, the Heston and Hull-White models. In particular, we show that in the Hull-White model the resulting probability distribution of log-returns in this approximation corresponds to the Tsallis (t-Student) distribution. The Tsallis parameters are given in terms of the microscopic stochastic volatility model. Finally, we show that the log-returns for 30 years Dow Jones index data is well fitted by a Tsallis distribution, obtaining the relevant parameters.