Agents trade fairly. Wealth condenses anyway. Explore the mechanism, the mathematics of what stabilizes it, and the physics that explains why.
dw/dt = g·w(1−w) − r·w, from Boghosian & Börgers (2024) [12]. w is the wealth share held by oligarchs.
fixed point w* = 0.600 · supercritical
establishedIn 2002 Anirban Chakraborti proposed a minimal model of an exchange economy: pick two agents at random, and let one transfer to the other an amount equal to a small fixed fraction β of whichever agent is poorer [1]. A fair coin decides direction. Brian Hayes gave the model its name that same year, in a piece contrasting it with an earlier rule where the transfer is proportional to the wealth of whoever loses the trade — a rule he called "theft and fraud," since a rational agent under that rule refuses to trade with anyone poorer [2]. The yard-sale rule has no such perverse incentive: every trade has zero expected value for both parties, so nothing discourages trading with the poor.
establishedRun this rule for a while and, without exception, the wealth ends up almost entirely in one agent's hands — wealth condensation. This is not a near-tie broken by chance; it is close to certain. Each agent's wealth changes multiplicatively, by a factor of \(1\pm\beta\) per trade. Pair a win with a loss and the factor is \((1+\beta)(1-\beta) = 1-\beta^2 < 1\): over any matched pair of favorable and unfavorable trades, a poor agent loses ground on net, purely from the arithmetic of multiplicative change, not from any bias in the coin [13]. Because total wealth is conserved, whatever one agent loses in expectation across the population, another must gain. Formally, each agent's wealth is a martingale with an absorbing barrier at zero, and Christophe Chorro showed that convergence to total oligarchy follows almost immediately from the martingale convergence theorem [10]. An earlier, related observation — that redistribution-free multiplicative exchange condenses wealth — goes back to Jean-Philippe Bouchaud and Marc Mézard [3].
establishedBruce Boghosian gave the model a continuum treatment borrowed from the kinetic theory of gases: instead of tracking individual agents, track the wealth density \(f(w,t)\) over a population, and let pairwise exchanges act like molecular collisions [4][5]. The base model becomes a nonlinear, nonlocal Fokker–Planck equation, and Boghosian, Johnson, and Marcq proved an H-theorem for it, showing entropy production is one-signed — the distribution moves in one direction, toward condensation, with no way back [6].
establishedAdd a flat wealth tax: each agent loses a fraction χ of wealth per unit time, pooled and returned to everyone in equal shares. This is a basic-income mechanism funded by a wealth tax, and it adds a term to the Fokker–Planck equation that pulls every agent's wealth toward the population mean μ [7]:
This is exactly the mechanic behind the χ slider in the simulator: each tick, every agent's wealth is nudged a fraction χ of the way toward the mean.
establishedBoghosian and Christoph Börgers isolated a simpler, aggregate version of this competition for teaching purposes: let w be the fraction of society's wealth held by oligarchs, let g measure how effectively existing wealth begets more wealth ("g" for grab), and let r be a flat tax rate on the oligarchs' holdings [12]:
This has fixed points at \(w=0\) and \(w = 1 - r/g\). For \(r>g\) the zero point attracts and oligarchy dies out; for \(r
my adaptationThe simulator's "progressive" toggle scales each agent's own tax rate by \((w_i/\mu)^p\) before pooling and redistributing. This captures the qualitative result above — same threshold behavior, smaller realized share for the rich — but it is my construction, not a literal implementation of the specific \(P(w)\) function analyzed in [12].
establishedReal wealth doesn't just sit still — it buys lobbyists, lawyers, and better information, which make the next trade more likely to go the wealthy agent's way. Boghosian and coauthors modeled this "wealth-acquired advantage" (WAA) as a small bias in the coin flip, increasing continuously with the wealth gap between the two trading agents [7]:
The ζ slider in the simulator uses β in place of \(\sqrt{\gamma\Delta t}\) as the discrete-time stand-in for that infinitesimal scaling — a reasonable analogue for a fixed-tick simulation, not a rescaled reproduction of the continuum limit. Combining WAA with redistribution gives a population of oligarchs whose wealth share c(t) obeys its own logistic-type equation [7][8]:
with fixed points \(c=0\) and \(c = 1-\chi/\zeta\): subcritical (no oligarchy) for \(\zeta<\chi\), supercritical (partial, stable oligarchy) for \(\zeta>\chi\) — another transcritical bifurcation, at \(\zeta=\chi\) this time rather than \(r=g\). This is the reference curve plotted against the simulated Gini coefficient on the Simulator tab. Boghosian and Börgers report an estimated \(\zeta/\chi \approx 1.39\) for the U.S. economy circa 2016, comfortably in the supercritical regime [13], drawing on the parameter fits in [9].
establishedJie Li, Boghosian, and Chengli Li extended the model further by allowing agents to hold negative wealth (debt), shifting the steady-state distribution by a third parameter κ. The resulting three-parameter "affine wealth model" fits the Lorenz curves of real wealth data to within about one-fifth of one percent for U.S. wealth 1987–2019, and about half a percent for fourteen European countries circa 2010 — a notably tight fit for a three-parameter model [9]. This extension is not implemented interactively here; the simulator clamps wealth at zero rather than allowing debt.
synthesis / scope noteThe agent-based simulator is a faithful discrete implementation of the base trade rule and the flat-redistribution mechanism. The ζ (WAA) rule and the c(t) reference curve are discrete- and finite-population stand-ins for continuum results that are proven in the \(N\to\infty\), \(\Delta t\to 0\) limit — at small N or few trades per tick, expect the simulated curve to fluctuate visibly around the mean-field prediction rather than track it exactly. That gap between finite-size simulation and mean-field theory is itself a standard feature of statistical mechanics, not a bug specific to this model. The Phase-Line Explorer's \((g,r)\) model and the Simulator's \((\zeta,\chi)\) model are two different levels of description from the same research program — aggregate oligarch share versus full wealth distribution — and are not meant to be read as the same two numbers under different names.
live debateThe affine wealth model's fit to real Lorenz curves is close enough to be genuinely striking, but a tight statistical fit doesn't by itself establish that wealth-acquired advantage and taxation are the actual causal mechanisms behind real-world inequality, as opposed to a good curve-fitting family among several that could match the same data. Work continues on tightening the mathematics of this specific model — for instance, rigorous bounds on how fast the Gini coefficient approaches its limiting value [11], and alternative probabilistic derivations of the same condensation result [10] — which is a sign the convergence theory, at least, is still actively being refined rather than fully closed.