Confusing Math with Morality

This essay is in response to Counterintuitive problem: Everyone in a room keeps giving dollars to random others. You’ll never guess what happens next. Dan Goldstein attributes the problem to Uri Wilensky, of Northwestern, who formulates it thusly:

Imagine a room full of 100 people with 100 dollars each. With every tick of the clock, every person with money gives a dollar to one randomly chosen other person. After some time progresses, how will the money be distributed?

Suppose we make this a social policy; let’s call it Random Redistribution of Wealth, or RRW. Goldstein then makes two claims:

  1. “More or less equally” is an intuitive answer to Wilensky’s question
  2. RRW is an example of an innocuous policy fostering inequality

I believe, and I’ll attempt to show, that he’s incorrect on both counts. But before that, let’s cut right to the chase. Here’s how the distribution of 45 people that started with $45 looks like after 5000 iterations (video by Dan Goldstein):

The end result might be surprising, infuriating, or both. It seems that even in statistical models, we have those pesky 1%-ers. It’s time to march on Gauss Street and redistribute that wealth! To arms, comrades!

The will of the capitalist is certainly to take as much as possible. What we have to do is not to talk about his will, but to enquire about his power, the limits of that power, and the character of those limits — Karl Marx

On the Intuition of Distributions

Jokes aside, I don’t quite understand why Goldstein makes the claim that a uniform distribution (right) is intuitive. Because things happen randomly? I’m sure we all remember randomly rolling die in middle school and eventually plotting the normal — or Gaussian — distribution (left). In fact, my initial gut feeling was that we would end up with a Gaussian distribution and had no real reason to even consider a uniform one.

As it turns out, we end up converging to the geometric distribution where the probability mass \(f(x,p)=(1-p)^x\) and where \(p=0.01\) in the 100-person case, or \(p=0.\overline{222}\) in the 45-person case. A geometric distribution looks like this:

Finally, there’s a pretty interesting reason why our wealth ends up being skewed exponentially. An anonymous comment on the original article clears it up: “the exponential (a.k.a. geometric) distribution is the maximum entropy distribution with a fixed average.” A bit of research validates that claim; if we take a look at Theorem 3.3 in Keith Conrad’s Probability Distributions and Maximum Entropy, it becomes clear that the geometric distribution has maximal entropy over \((0,\infty)\).

Statistics: RRW is not Unequal

Next, let us tackle the question of whether or not RRW is unequal. Goldman’s video is highly misleading; it seems to imply that there is an end state where the distribution of wealth is highly skewed to favor certain agents over others. This is, however, incorrect. The distribution is indeed skwewed (and remains skewed), but it is in constant flux: one instant, I may be the wealthiest while the next, someone else takes that spot. In fact, it’s guaranteed that every agent will, at some point, be both the richest, as well as the poorest.

Goldman tackles this conundrum in his addendum:

There’s some confusion in the comments below and on other sites that we thought we’d address. The point is not that some people become rich and never lose their top position. This runs infinitely and will contain every possible sequence of good and bad luck for every person. The richest will become the poorest, everyone will experience every rank, and so on. The interesting thing is that this simple simulation arrives at a stationary distribution with a skewed, exponential shape. This is due to the boundary at zero wealth which, we imagine, people don’t consider when they think about the problem quickly.

But the question lingers: how is this policy unequal? Does a certain mathematical shape imply you should or shouldn’t implement a social policy? That seems odd. I’d like to posit that not only is RRW not unequal, it’s the very epitome of equal!

  • RRW is perfectly random and fair, not favoring one agent over another
  • Every agent has the exact same expected value
  • Every agent will, at some point, be both the richest, as well as the poorest

Goldman’s example of an “innocuous policy that leads to inequality”, what we called RRW, is complete nonsense. All this policy does is add entropy to the system, causing wild swings in wealth and systemic uncertainty. At the very least, it has no moral valence; and at the very best, it’s perfectly equal and we ought to favor it.

Philosophy: RRW is Unequal

But “best” and “good” and “ought” are terms of philosophy, not mathematics. And whether or not we implement a social policy should be contingent on its moral valence, and not (the rather arbitrary) shape of its distribution. With that in mind, it can be argued that RRW is, in fact, unequal — although not at all for the reasons Goldman claims.

In What is Equality? Part 2: Equality of Resources, Ronald Dworkin makes a distinction between justified inequality of resources and unjustified inequality of resources. Dworkin borrows from Rawls, but the gist of his sentiment is summarized by SEP: “Unequal distribution of resources is considered fair only when it results from the decisions and intentional actions of those concerned.” Conversely, provisions, gifts, bribes, and pure luck are conducive to an unfair (or unjust) inequality of resources.

For example, a state lottery is unequal — and creates an unjust inequality of resources — because it’s random. So RRW is not unequal because it converges to an exponential distribution, but rather because it’s purely driven by a stochastic process.

It’s trivial to come up with a social policy that always converges wealth to a uniform distribution. The problem is that unless this convergence is driven by what Dworkin calls ‘ambition-sensitive’ forces, this social policy is just as bad as RRW — even though it might seem that it’s fair.

In short, the shape of the distribution of RRW does not matter. What matters is the process.