An article in the November 2019 issue of Scientific American, “The Inescapable Casino” or “Is Inequality Inevitable” caught my attention and has led me to do some research on wealth inequality in the United States.  In particular, the article claims that wealth naturally trickles up in free-market economies.  That is, the growing separation between the very wealthy and the rest of society is shown mathematically to be a result of free-market economics.  If this is true, then free market democracies need to pay closer attention to the theoretical mathematics that is driving this conclusion for otherwise the end game is inevitable oligarchies.


The Gini coefficient is a commonly used measure of wealth inequality.  It ranges from 0 to 1.00 where 0 would be absolute wealth equality and 1 a perfect oligarchy.  The Gini coefficient has been rising in most nations, growing in the U.S. from .79 in 1989 to .86 in 2016.  The Scientific American article is mainly devoted to explaining a recent model developed by physicists and mathematicians that reproduces to high degree of accuracy the empirical wealth distributions of various nations including the United States.  This model is named “the affine wealth model” (called thus because of its mathematical properties) and involves billions of computer simulated transactions that are conditioned by three mathematical parameters.  The article claims that this model reproduces the current distribution of wealth in the United States to within less than a fifth of a percent.

It is not possible for me to reproduce the affine wealth model on my computer – the mathematics and programming are not something I can do.  However, I was able to transfer enough data from the Federal Reserve Bank’s Survey of Consumer Finances for U.S. wealth 2016 and to calculate the Gini coefficient for the U.S. since it is a relatively straight-forward calculation once the data is in place.  I came up with the value .86, which exactly agrees with the official number that economic and governmental agencies have released.  I accepted that Gini coefficients in prior years were less, and thus that wealth inequality as measured by this index was growing in our country.  Now I had to understand why a mathematical model was saying that growing inequity was inherent in our free-market economy.

The affine wealth model starts with a simpler model called the Yard Sale, which does not reproduce national wealth distributions but shows that wealth naturally moves up and does not trickle down.  Here is Scientific American’s description of the yard sale model:  The yard sale, a simple mathematical model developed by physicist Anirban Chakraborti, assumes that wealth moves from one person to another when the former makes a “mistake” in an economic exchange.  If the amount paid for an object exactly equals what it is worth, no wealth changes hands.  But if one person overpays or if the other accepts less than the item’s worth, some wealth is transferred between them.  Because no one wants to go broke, Chakraborti assumed that the amount that can potentially be lost is some fraction of the wealth of the poorer person.  He found that even if the outcome of every transaction is chosen by a fair coin flip, many such sales and purchases will inevitably result in all the wealth falling into the hands of a single person – leading to a situation of extreme inequality.”

Thus if you attempt to computer simulate the economy using the yard sale model, wealth becomes concentrated at the top even if every transaction is decided by a fair coin flip.  This was surprising to me, and as I attempted to figure out why this may be the case, I came up with a thought experiment that made the result somewhat plausible.  At the same time, I was skeptical that the yard sale model, even modified, could simulate the free market economy.   Before I outline my doubt, here is the thought experiment: Suppose there are only two people (Heads and Tails) in the economy and they each have $1,000 to start (equality).  They flip a fair coin to move $10 back and forth between the two of them.  It would not be strange after 1 billion throws for there to be 500,000,100 heads and 499,999,900 tails from this fair coin, but Heads has taken all the money.   Thus, in brief, oligarchy is the mathematical outcome of the yard sale model because of the nature of probability.

Now here is a critique of the yard sale model, which was not discussed in the SA article.  The yard sale model seems to ignore wealth production.  Standard economic theory assumes that free market exchange produces wealth by allowing individuals to specialize and become very efficient at producing certain items, and then exchanging for the other items that they would be inefficient in producing.  If the production of wealth is multi-leveled and ongoing, the yard sale model is not realistic because both agents in the exchange may be gaining in wealth.

With this background, I now look at the three ways the yard sale model has been modified to produce the affine wealth model as described in the SA article: (1) Have each agent in the transaction take a step toward the mean wealth in the society after each transaction.  The size of the step was some fraction of the agent’s distance from the mean.  This transfers wealth from those above the mean to those below it and improves the model at matching empirical wealth distributions; (2) Factor in a wealth-attained advantage by changing the fair coin flip in favor of the wealthier agent by an amount proportional to a new parameter times the wealth difference divided by the mean wealth.  This refinement improved agreement between the model and the upper tail of actual wealth distributions; (3) Include negative wealth in the model with a third parameter that shifts the wealth distribution downward.  This is necessary since nearly 11% of the U.S. population was in net debt.  These 3 modifications yield the affine wealth model, which reproduces the 2016 U.S. empirical wealth distribution to within less than a fifth of a percent.

The critique of the yard sale model made above does not seem to apply to the affine wealth model because the three parameters collectively yield a model that reflects reality.  Perhaps the second refinement, which serves as a proxy for a multitude of biases favoring the wealthy, includes the notion that they are the ones who benefit the most from wealth production.

Real economies are very complicated, and if the affine wealth model continues to be verified as an accurate description of empirical national wealth distributions that show increasing Gini coefficients, then elected officials of those nations will have to implement policies that will correct inequalities inherent within the economy, and those officials, in turn, will have to rely on ethical mathematicians and data scientists who can handle these problems.

(In order to view the spreadsheet that contains the data and calculation of the Gini coefficient, click on this link: )