Complex systems series · № 2

The yard-sale model

Every trade is a fair coin flip. One player still ends up with everything.

Here are 100 households, each starting with $100. Two at a time, they trade: the stake is a fraction of the poorer trader's wealth — nobody can lose what they don't have — and a fair coin decides who takes it. No skill, no cheating, no head start. Expected value of every single trade, for both sides: exactly zero.

Press Run and let the fair coin do its work.

The median goes broke

At the default 20% stakes, the median household is under $1 by roughly 20,000 trades. Let it run to 100,000 and the richest ten households own 99% of everything, with a Gini index around 0.97 — more unequal than any country that has ever been measured. Every one of those trades was a fair coin flip.

The trap is multiplicative. Lose a 20% stake and then win one, and you're at 1.2 × 0.8 = 0.96 of where you started — a fair coin shrinks you either way, because gains and losses compound against a moving base. Once you're the poorer trader in most of your trades, every flip risks a fifth of everything you have, while the richer side risks pocket change. Wealth is a ratchet: being rich is safe, being poor is dangerous, and the coin doesn't care.

This is the difference between a game that's fair per trade and one that's fair per lifetime. The expected value of each flip is zero, but follow any single household through time and its typical trajectory decays. The average is propped up by a handful of lottery-winner paths — which is exactly what the final bar chart looks like.

The two-percent thumb on the scale

Now drag the tax slider to 2% — each round, everyone pays a small share of their distance above the average, and everyone below gets topped up. The collapse stops. The Gini settles near 0.56 and the median household holds around $48 — forever. Not equality; a middle class. Turn it back to zero and watch the middle class evaporate again.

That's the model's sharpest claim: in a multiplicative economy of fair trades, oligarchy is the equilibrium and a middle class is maintained, not natural. The thumb on the scale isn't distorting the game — it's the only thing standing between a fair game and its own end state.

What the model actually proves

Be careful what you take from this. The yard-sale economy is zero-sum — real trades create value for both sides, real economies grow, and skill exists. The model deliberately deletes all of that, which means it can't tell you how much real inequality is earned.

What it proves is the converse: extreme inequality needs no explanation in talent, effort, or merit at all. Pure luck plus compounding is sufficient to produce a distribution more skewed than any real economy's. So the shape of a wealth distribution, by itself, is evidence of almost nothing about the people in it — the same one-way street from micro to macro as our segregation model, where a pattern that looks like intent can be built entirely out of something else.