Follow the Money

Abstract

The rich get richer and the poor get poorer. You've heard that before. It is a maxim so often repeated, and so often confirmed by experience, that it begins to sound like a law of nature, as familiar and irresistible as gravity. And indeed perhaps there is some physical or mathematical rule governing the distribution of wealth in the world. No such general principle is going explain the specifics of who gets rich and poor, but it might illuminate the overall statistics. This idea goes back at least a century to the work of the Italian economist Vilfredo Pareto, who tried to show that the income distribution in all cultures and countries has the same mathematical form. In recent years the topic has been taken up with renewed enthusiasm by a small band of "econophysicists," who apply principles of statistical mechanics to questions in economic theory. The essence of their approach is to study an economy as if it were a many-body physical system such as a gas. Just as random collisions between gas molecules give rise to macroscopic properties such as temperature and pressure, random encounters between individuals in an economic system might determine large-scale phenomena such as the distribution of wealth. Some of the computational models for exploring these issues are remarkably easy to build and run. It takes just a few minutes' effort and a few lines of code. On the other hand, it's also remarkably easy to make subtle mistakes of implementation, as I'll have occasion to mention below. And the big challenge is not building the models but interpreting the results-deciding which kinds of random encounters might represent events in a real economy. The Price Is Right The economy being simulated in these models is a rather special one, based on pure, free-market trading. The exchange of assets is all that ever happens here; there is no production of new wealth, and no consumption either. Leaving out so much of the real economy is an obvious weakness, but there is a compensating advantage: What remains is a closed system. In the model, wealth is a conserved quantity, like energy or momentum. Because the total amount of wealth never changes, one person can get richer only if another grows poorer.

Self-organization and time-stability of social hierarchies

“…The other version of this model rule is a "multiplicative" one. Here, the amount of strength gained or lost is proportional to the strength of one of the individuals involved in the fight, such that effect of fighting accumulates multiplicatively [27,[31][32][33]. Defeating a strong opponent produces a large increase in strength, whereas defeating a weak opponent 3 produces a small increase in strength.…”

“…Here, we implement a formulation of the multiplicative rule in which the amount of strength won or lost is proportional to the pre-fight status of the losing individual ("loser scheme"). In another formulation that has been used in several econophysics models [32][33][34][37][38][39], the amount of strength won or lost is proportional to the prefight status of the weaker individual, regardless of who wins or loses ("poorer scheme"). The loser scheme formulation is more realistic in the context of dominance hierarchies, because upset victories, in which the lower-strength individual in the pair wins, produce large rewards for the winner and large penalties for the loser.…”

“…And, if so, what are the typical structures of hierarchy, and what are the characteristic times of formation and evolution of the said structures? "Winner-loser" models are a class of mathematical models that have been used to study the self-organization of social hierarchy in biology [18][19][20][21][22][23][24][25][26][27][28][29] and economics [30][31][32][33][34]. In these models, individuals are characterized by a property, such as "strength", "resource holding potential", or "wealth", that determines the individual's position in society (in the following, we use "strength" as a generic term for this property).…”

Assortative exchange processes

“…c(m, t) = 0 in the region m > m * (t) = (80t) 1/4 , is tiny. To appreciate this we take into account a very sharp decay and simplify (10) in the m > m * (t) region to dcm dt c 2 m−1 from which we deduce a double exponential decay…”

“…Exchange processes arise in numerous natural phenomena such as droplet growth via evaporation and recondensation [1], island growth [2] and phase ordering [3][4][5]. Exchange processes have been applied to social sciences, e.g., to modeling segregation of heterogeneous populations [6], studying the distribution of wealth through asset exchange [7][8][9][10][11][12], mimicking growth of urban populations [13] and aggregation behaviors in job markets [14]. Exchange processes are also used as toy microscopic models which are simple enough to allow the derivation of the macroscopic 'hydrodynamic' equations and explore other fundamental aspects of non-equilibrium statistical mechanics, see e.g.…”