### Spontaneous Inequality

A fun problem making the mathematical rounds: if you give everybody some number of dollars (say you give 45 people \$45 each), and at every tick of the clock everyone with money chooses one random person \$1, how will the money eventually end up distributed? You might think it’ll stay approximately equal—”approximately” only because of the randomness—but in fact the randomness creates a rather unequal distribution:

Now you might wonder if the inequality will keep increasing until one person has everything. Jordan Ellenberg shares the insight over on his blog that this process is a random walk on a network where the nodes are distributions and two nodes are connected if they differ only by the location of just \$1. So instead of reaching a stable state, we should expect this process to wander around and hit every state eventually.

But we can still ask how unequal the distribution will be on average. Is the screenshot above typical, or unusually unequal? One way of measuring the inequality of a distribution is called its Gini coefficient (pronounced like “Genie” coefficient), a number that ranges between 0 (everyone has the same amount) and 1 (one person has everything). I re-ran the original simulation for 100,000 iterations and found that after about 5,000 steps the Gini coefficient tends to hover in the 0.4 to 0.5 range, so the distribution doesn’t get much more unequal than the example pictured above.

A friend of mine wanted to know what would happen if the players each gave a fixed fraction of their wealth at each time step, say 10%. Would that prevent a big pileup of wealth in any one player’s hands? It wasn’t hard to change the simulation to find out, and sure enough, the distribution seems to stay much more equal:

This made sense: the 10% “tithe” forces the richest players to give up a big chunk of their wealth, but doesn’t impact the poorest players as much. I expected, then, that the bigger the tithe percentage, the less inequality once the distribution stabilizes.

I was wrong: the Gini coefficient of inequality is higher for higher tithe percentages:

I don’t know what’s going on here. My intuition was telling me that the new process was still a random walk on the network of all possible distributions (just with slightly different connections between nodes), but that would have implied that the range of typical Gini coefficients would be the same for each tithe percentage. Anyone have a clue?