# Random Exchange Leads to a Skewed Distribution

## Giving Randomly

Professor Uri Wilensky, developer of the popular NetLogo agent-based simulation model, likes to ask his students to develop a simulation that shows what happens when 100 people in a room give one dollar to another person chosen randomly (other than themselves) repeatedly. After playing this game a few thousand times, wealth is not distributed evenly as one might expect but rather follows an upward sloping line with an exponential curl at both the richest and poorest ends.

Here are the averaged results of 50 runs of a 100-person simulation when every person starts with \$200 and every simulation runs for 5,000 rounds:

This simulation shows that a completely random exchange in a fixed-wealth society eventually leads to a stable, upward sloping distribution with skewed, exponential curls. The shape of this distribution is similar to that of Rugged Island, presumably because a similar, random process is at work in both situations (though in this case the chances for gain and loss are exactly the same and the amount gained or lost in each round is very small).

After 5,000 rounds, the richest person here, with \$377, might think he or she is smarter or better than the poorest person, who has \$24. But it is only random luck that has caused this disparity. Over time, the richest person will have a string of bad luck, the poorest will have a string of good luck, and they will exchange places in the distribution.

Here are the values of several common measures of wealth inequality for this distribution:

Random Exchange,
5,000 Rounds
Range  (wealth of richest person minus poorest person) \$353
Gini Coefficient 0.20
Hoover Index (Robin Hood Index) 0.14
Coefficient of Variation 0.35
Quartile Coefficient of Dispersion (Q3–Q1)/(Q3+Q1) 0.23
Decile Coefficient of Dispersion (D9–D1)/(D9+D1) 0.56
20:20 Ratio  (ratio of wealth owned by the top 20% to the bottom 20%) 2.93
Palma Ratio  (ratio of wealth owned by the top 10% to the bottom 40%) 0.61

##### Notes

1. ^

This 5,000-round simulation uses slightly more than 500,000 random numbers between 1 and 100 to determine the random recipient of each dollar given away (5,000 rounds x 100 givers/round = 500,000 random numbers needed plus some extra when the recipient turns out to be the same as the giver). For each run of the simulation, we use a randomly scrambled data table containing 2 million truly random integers between 1 and 100 generated by Random.org. This helps ensure that the results are not merely an anomalous artifact of a possibly flawed random number generator.

2. ^

Statistician John Angle determined that, in such a situation, personal wealth is distributed as a mixture of negative binomial probability functions (closely related to a gamma probability density function). Also, see the discussion by economist Thomas Lux and the paper by Jonathan Silver, Eric Slud, and Keiji Takamoto.

A succinct equation can produce an approximation of the graph above: Let X = the Rank of the person (from poorest to richest) – 50.5. Then
Wealth = 200 + 2*X + 100*( e^((X–50.5)/4) – e^(–(X+50.5)/4) )

The first of the summed terms is the initial average wealth. The second term produces the upward sloping line in the middle of the distribution. The last two terms produce the two exponential curls at the ends. Note that “*” means “multiply”, “^” means “raise to the power of”, and “e” is the Euler mathematical constant (the base of the natural logarithm and approximately equal to 2.71828).