In May 2014 I wrote a post on modelling the Lorenz Curve, which is an income or wealth curve whose curvature is expressed as the Gini Coefficient. In this model an ideal society has a straight-line curve and the more unequal a society is, the greater the curvature.
The post shows how a pure, free-market economy naturally gives rise to the highest possible gini coefficient, approaching 1 over time. This is the case even if the population involved has no ulterior profit motive and all participants play by the same, equally applied rules. The post provides a program, written in JavaScript which simulates the process, but the program doesn't run, it's just a listing.
Informally, the algorithm works as follows. There are 100 players. At first each player is given the same amount of cash: $10. $1 is randomly taken from the pool of money, and so the player who owned it now has $1 less; then another $1 is picked randomly from the remaining pool and the player who owns that one is now given the previously taken $1. So, usually, one player loses $1 and another player gains $1 (unless the same player gets picked for both steps).
Intuitively you would think that the probabilities would even out. As a player loses money, they are less likely to have money taken from them (and given to them), but likewise, as a player gains money, they are more likely to have money taken from them (and also given to them).
This is not what happens. Instead as players gain money, they are more likely to gain in subsequent transactions. This is because the probabilities change between transactions, in favour of previous winners. For example, consider a situation near the end game where one player has $1 and the remaining player has $99,999. Although 99.999% of the time, the dominant player will have $1 removed, 99.999% it will be returned with another 0.001% of an opportunity that it goes to the lesser player. However, in the 0.001% of the time that the lesser player's $1 is removed, it becomes impossible to receive that $1, and in subsequent plays, they now have a 0% of winning.
In the real world, this corresponds to the way in which larger players, who occupy more of the market, are more likely to be chosen to trade with: thus increasing their market share. In this version, the javascript is embedded in the article itself and thus it can be played live. You can see a Lorenz Curve being mapped out in realtime as it becomes more extreme. A variant allows you to generate interest with a given probability (interest works by leaving the 'loser' with the original $1 they had), but it has no effect on the overall outcome: the richest get richer while the poorest lose everything.
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