Take 60 people and give each of them exactly 100 coins. Now let them trade, one pair at a time, with the fairest rule you can think of: flip a fair coin and the winner takes a stake from the loser. Nobody cheats, nobody is smarter, and the odds are always 50/50.
After enough trades, almost everything belongs to one person. This is the yard-sale model of wealth, and it is one of the starkest demonstrations that fairness at every step does not guarantee a fair outcome.
Two random people meet. The stake is a fixed percentage (say 20%) of the poorer person's wealth, so neither can bet money they don't have. A fair coin picks the winner, who gains the stake while the other loses it. Total wealth never changes. The model was set out in this form by Anirban Chakraborti in 2002, and the science writer Brian Hayes gave it the "yard-sale" name the same year, because the haggling feels like two neighbours swapping junk on a Saturday.
Because the stake is tied to the poorer side, a rich person risks very little against a poor one, and a poor person who wins gains a little. The bets are even in odds and in size. Nothing in the rules favours the rich.
Each bar is one person. Press Run and watch the bars. The two numbers that matter are the Gini coefficient (0 means everyone has the same, 1 means one person has everything) and the share held by the richest person.
One round is 60 trades. Tap a bar to see that person's wealth. Try a 50% stake, then a 5% stake: the result is the same, only slower.
Each person's expected wealth never changes: a fair bet has zero average gain. In maths this is called a martingale. And a standard result says a bounded martingale must settle down eventually. The only places a person's wealth share can stop wandering are the places where the game itself stops: owning nothing, or owning everything. As long as two people both still have something, the coin keeps moving them, so the wandering never stops in the middle.
Think of it as a ratchet in percentages. A person with 1 coin can lose at most that 1 coin, and a person who falls close to zero finds the stakes in every game shrinking with them, so they can barely climb back. Meanwhile a few lucky streaks lift someone well above the crowd, and every later win is a percentage of a bigger base for the people they trade with. Luck compounds, and the compounding is what turns a gentle spread into a pile.
Notice what is missing: skill, effort, and cheating. Run the model with identical people and you still get a winner, simply because somebody has to be luckiest. Real economies are far more complicated, and real wealth distributions are not literally this extreme. The model is a thought experiment about one ingredient, repeated multiplicative luck, not a forecast.
Slide the redistribution control above zero and run again. Each round, a small slice of everyone's wealth is pooled and shared equally, a crude stand-in for taxes and public services. Even a small slice changes the ending: instead of condensing onto one person, the system settles into a steady spread with a rich tail and a middle that survives. Researchers such as Bruce Boghosian and Christophe Chorro studied versions of the yard-sale model with taxes, and found that redistribution can replace total concentration with stable distributions. Models that add a built-in advantage for the already-wealthy can bring the concentration back, if the advantage is strong enough.
That is the lesson worth taking away. Equal opportunity at every step is not the same as an equal outcome, and a system that compounds luck needs something pulling the other way if it is not to end with one winner.