Random Events Can Produce a Highly Unequal Distribution


Intuitively, we might assume that random events might cause a lot of motion hither and yon, but would not change anything significantly, like a leaf floating on waves that goes up and down but always ends up close to where it started. However, random events can actually cause a uniform wealth distribution to become highly unequal. Here are three examples:

Coin Flips

Imagine there are 100 people in a room and each one has $200. At the sound of a bell, each person flips a coin and if it lands heads up, they win that round of the game and gain a dollar. If the coin comes up tails, they lose a dollar. After playing 5,000 rounds of this (rather dumb) game, how rich would each of the people in the room be?

One might think that the wins and losses would balance out for every participant and each would still have $200. But that is not what actually happens. Instead, it is likely that, at most, only one or two participants would end up with exactly $200. Instead, most would have a few dollars more or a few dollars less. Those who were really lucky, having flipped a significantly larger number of heads than tails, would have $250 or maybe even $300. Those who were especially unlucky would have $150 or even less. It is conceivable that someone might flip 5,000 heads in a row and end up with $5,200. But this is extremely unlikley.[1] The overall results of a binary random event like this typically adhere to the principles of probability and produce a bell-shaped “normal distribution,” as shown below.[2]

Coin flips of 5,000 rounds produces a normal distribution

In this coin-flip game, the curve peaks at $200 indicating that this is the most likely level of wealth, with about 0.57% of the participants ending up with this exact amount; $200 is also the average (mean) level of wealth among the 100 participants. Other levels of wealth are less likely the further they are from this mean. The chances of ending up with exactly $50 or $350 is just 0.06%.

The standard deviation of the curve is determined by the nature of the game and the number of rounds played. In this case, it is 70.711.[3] For the normal distribution, the values within one standard deviation from the mean account for 68.27% of the total, so 68 of the 100 participants would likely end up with between $130 and $270. A normal distribution also prescribes that the wealth of 95.45% of participants would end up being within two standard deviations of the mean, that is, between $60 and $340.

This distribution is very different than the initial distribution in which every participant had the same wealth ($200). The mean is still the same ($200), but initially the standard deviation was 0 and now, after 5,000 rounds of the game, it is 70.711.

If we sort the participants by their final wealth and plot their wealth versus each participant’s wealth rank order, we get the “quantile function” or “inverse” of the normal distribution which is called the “probit.” This curve (shown below) looks like an upwardly-sloping straight line with exponential curls at both the richest and poorest ends.

Coin flips of 5,000 rounds produces a probit distribution

Half the participants end up with more than the initial $200 and half less. On this curve, it is easier to see that 68% of the participants have wealth between $130 and $270 and 95% have wealth between $60 and $340.

Note how unequal this distribution is compared to the initial distribution (shown in green) in which every participant had the same wealth ($200). Through a totally random coin flip process, some people have become relatively rich and others relatively poor. In this game, the more times the coin is flipped, the more unequal the distribution becomes.[4]

Giving Randomly

Professor Uri Wilensky, developer of the popular NetLogo agent-based computer simulation model, likes to ask his students to develop a simulation that shows what happens when people in a room give one dollar to another person chosen randomly (other than themselves) repeatedly. When there are 100 people who play this game 5,000 times,[5] wealth is not distributed evenly as one might expect but rather follows a probit curve (which hasppens to have the same mean and standard deviation as the coin flip game).

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:

Random Exchange Results Graph after 5,000 Rounds

This simulation shows that a completely random exchange in this fixed-wealth society eventually leads to a probit distribution. After 5,000 rounds of this random exchange simulation, the richest person, with $377, might think he or she is smarter or better than the poorest person, who has just $24. But it is only random luck that has caused this large 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 random exchange 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

Wealth Quintiles

Random Exchange Wealth Quintiles Graph As shown in this graph of wealth held by each quintile of the population (20 of 100 people) after 5,000 rounds, the richest 20% (the top quintile, shown in blue) owns 30% of all wealth. The next wealthiest 20% (shown in purple) owns 24%. The middle quintile (in green) owns 20%. Together, these three quantiles have 74% of all the wealth, leaving 26% for the poorest 40%. The poorest 20% (in red) has 10% of the total wealth.

The Distribution on Rugged Island

Like the two examples above, wealth on Rugged Island is shaped only by random events – many large and small monetary losses caused by illness, accidents, and natural disasters randomly striking households at unpredictable times over 50 years. This process is much more complex than a simple coin flip or random dollar exchange and produces a much more complex wealth distribution.

Below is a histogram of wealth on Rugged Island in Year 50. Clearly this is not a normal distribution – compared to a normal curve it is shifted to the right, has a long tail to the left, and has no tail on the right side.[6]

Still, this graph shows that wealth on Rugged Island is also highly unequal. Though Rugged Island began in Year 0 with a uniform wealth distribution – every household blessed with $205,000 in wealth – after 50 years, 20 households have more than doubled their wealth while others have sunk to a negative wealth of less than –$300,000. This severe inequality was caused solely by a series of random events.

Rugged Island histogram is clearly not a normal distribution.


Notes

1. ^

The probability would be 1/(0.5^5000) (where “^” means “raised to the power of”) which is approximately one chance in 10^1505 (1 followed by 1,505 zeroes).

2. ^

When the number of participants is large, the event is binary (only two possible outcomes), and each participant is independently subjected to a large number of this same random event), the central limit theorem establishes that the results will tend to a normal probability distribution.

In this graph, the average results of 50 runs of the coin flip simulation (shown in blue) and the normal distribution (in red) differ slightly since the simulation is based on a finite number of random occurrences and the normal curve assumes an infinite number.

3. ^

For a binomial distribution, the standard deviation is equal to the square root of the multiplication product of three values: the number of events, n (5,000 in this case), the probability of one outcome, p (0.5 that a fair coin flip will produce a heads), and the probability of the other outcome, q (also 0.5 that a fair coin flip will produce a tails). This value is then multiplied by the gain from a successful outcome (in this case the gain=2 since heads= + $1 and tails= – $1). So overall, the standard deviation = ( (5000 * 0.5 * 0.5)^0.5 ) * 2 = 70.711.

4. ^

Here is how the probit curve looks for other numbers of coin flips:

Probit distribution after various numbers of rounds.

After 100 coin flips, the distribution is still pretty flat with the richest person having $226 and the poorest person having $174 – just $52 less. But after 20,000 coin flips, the richest person has $564 and the poorest person has negative –$164 which is $728 less than the richest.

5. ^

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.

6. ^

Also, using the Alexander-Darling test, the Rugged Island data produces a value of p less than 0.0000001, far below the minimum value of 0.05 that would indicate this is a normal distribution. In contrast, for both the Coin Flip and Random Exchange games p=0.99999 providing a very high confidence that they produce normal distributions.