Statistical ensembles for money and debt

We build a statistical ensemble representation of two economic models describing respectively, in simplified terms, a payment system and a credit market. To this purpose we adopt the Boltzmann–Gibbs distribution where the role of the Hamiltonian is taken by the total money supply (i.e. including money created from debt) of a set of interacting economic agents. As a result, we can read the main thermodynamic quantities in terms of monetary ones. In particular, we define for the credit market model a work term which is related to the impact of monetary policy on credit creation. Furthermore, with our formalism we recover and extend some results concerning the temperature of an economic system, previously presented in the literature by considering only the monetary base as a conserved quantity. Finally, we study the statistical ensemble for the Pareto distribution.     

Statistical mechanics of money

In a closed economic system, money is conserved. Thus, by analogy with energy, the equilibrium probability distribution of money must follow the exponential Boltzmann-Gibbs law characterized by an effective temperature equal to the average amount of money per economic agent. We demonstrate how the Boltzmann-Gibbs distribution emerges in computer simulations of economic models. Then we consider a thermal machine, in which the difference of temperatures allows one to extract a monetary profit. We also discuss the role of debt, and models with broken time-reversal symmetry for which the Boltzmann-Gibbs law does not hold. The instantaneous distribution of money among the agents of a system should not be confused with the distribution of wealth. The latter also includes material wealth, which is not conserved, and thus may have a different (e.g. power-law) distribution. Received 22 June 2000     

Rigorous Proof of the Boltzmann–Gibbs Distribution of Money on Connected Graphs

Models in econophysics, i.e., the emerging field of statistical physics that applies the main concepts of traditional physics to economics, typically consist of large systems of economic agents who are characterized by the amount of money they have. In the simplest model, at each time step, one agent gives one dollar to another agent, with both agents being chosen independently and uniformly at random from the system. Numerical simulations of this model suggest that, at least when the number of agents and the average amount of money per agent are large, the distribution of money converges to an exponential distribution reminiscent of the Boltzmann–Gibbs distribution of energy in physics. The main objective of this paper is to give a rigorous proof of this result and show that the convergence to the exponential distribution holds more generally when the economic agents are located on the vertices of a connected graph and interact locally with their neighbors rather than globally with all the other agents. We also study a closely related model where, at each time step, agents buy with a probability proportional to the amount of money they have, and prove that in this case the limiting distribution of money is Poissonian.     

Statistical mechanics of money: how saving propensity affects its distribution

We consider a simple model of a closed economic system where the total money is conserved and the number of economic agents is fixed. Analogous to statistical systems in equilibrium, money and the average money per economic agent are equivalent to energy and temperature, respectively. We investigate the effect of the saving propensity of the agents on the stationary or equilibrium probability distribution of money. When the agents do not save, the equilibrium money distribution becomes the usual Gibb's distribution, characteristic of non-interacting agents. However with saving, even for individual self-interest, the dynamics becomes cooperative and the resulting asymmetric Gaussian-like stationary distribution acquires global ordering properties. Intriguing singularities are observed in the stationary money distribution in the market, as functions of the marginal saving propensity of the agents. Received 2 May 2000     

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.     

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.     

Rigorous Results for the Distribution of Money on Connected Graphs

This paper is concerned with general spatially explicit versions of three stochastic models for the dynamics of money that have been introduced and studied numerically by statistical physicists: the uniform reshuffling model, the immediate exchange model and the model with saving propensity. All three models consist of systems of economical agents that consecutively engage in pairwise monetary transactions. Computer simulations performed in the physics literature suggest that, when the number of agents and the average amount of money per agent are large, the limiting distribution of money as time goes to infinity approaches the exponential distribution for the first model, the gamma distribution with shape parameter two for the second model and a distribution similar but not exactly equal to a gamma distribution whose shape parameter depends on the saving propensity for the third model. The main objective of this paper is to give rigorous proofs of these conjectures and also extend these conjectures to generalizations of the first two models and a variant of the third model that include local rather than global interactions, i.e., instead of choosing the two interacting agents uniformly at random from the system, the agents are located on the vertex set of a general connected graph and can only interact with their neighbors.     

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.     

Model of wealth and goods dynamics in a closed market

A simple computer simulation model of a closed market on a fixed network with free flow of goods and money is introduced. The model contains only two variables: the amount of goods and money beside the size of the system. An initially flat distribution of both variables is presupposed. We show that under completely random rules, i.e. through the choice of interacting agent pairs on the network and of the exchange rules that the market stabilizes in time and shows diversification of money and goods. We also indicate that the difference between poor and rich agents increases for small markets, as well as for systems in which money is steadily deduced from the market through taxation. It is also found that the price of goods decreases when taxes are introduced, likely due to the less availability of money.     

Boyle temperature as a point of ideal gas in gentile statistics and its economic interpretation

Boyle temperature is interpreted as the temperature at which the formation of dimers becomes impossible. To Irving Fisher’s correspondence principle we assign two more quantities: the number of degrees of freedom, and credit. We determine the danger level of the mass of money M when the mutual trust between economic agents begins to fall.     

Completeness of the classical 2D Ising model and universal quantum computation

We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.     

Optimal dense coding with thermal entangled states

We study optimal dense coding with thermal entangled states of a two-qubit Heisenberg XX chain and a two-qutrit system. For a two-qubit Heisenberg XX chain, the dense coding capacity is a function of temperature and external magnetic field. Only in the case of an external magnetic field being less than the coupling constant, the optimal dense coding can be realized with thermal entangled states. For a two-qutrit system, we consider the dense coding capacity taking into account of nonlinear coupling constant and an external magnetic field. We find that the nonlinear coupling constant must be less than 0 for dense coding. For the two models, we give the conditions that the parameters of the models have to satisfy a valid dense coding.     

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.     

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.     

Grounding language in action and perception: From cognitive agents to humanoid robots

In this review we concentrate on a grounded approach to the modeling of cognition through the methodologies of cognitive agents and developmental robotics. This work will focus on the modeling of the evolutionary and developmental acquisition of linguistic capabilities based on the principles of symbol grounding. We review cognitive agent and developmental robotics models of the grounding of language to demonstrate their consistency with the empirical and theoretical evidence on language grounding and embodiment, and to reveal the benefits of such an approach in the design of linguistic capabilities in cognitive robotic agents. In particular, three different models will be discussed, where the complexity of the agent's sensorimotor and cognitive system gradually increases: from a multi-agent simulation of language evolution, to a simulated robotic agent model for symbol grounding transfer, to a model of language comprehension in the humanoid robot iCub. The review also discusses the benefits of the use of humanoid robotic platform, and specifically of the open source iCub platform, for the study of embodied cognition.     

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.