Everybody knows about income inequality. We know it’s probably not a good thing, c.p., and we know it’s getting worse. The rich get richer, and the poor get poorer. Such is life.
What If The Rich Deserve It?
It’s hard to say something that pisses people off more. On the right, they’re outraged for doubting it, and on the left they’re outraged for even hinting that it might be true. And on the face of it, one might agree with the liberals: A wealth distribution where the top 20% owns more than 80% of the wealth and the bottom 40% has nothing seems pretty much unpardonable. At least until you think about the last group project you did, where two of the five people alternated between doing nothing useful and getting in other people’s way. But nevertheless, I think the burden of proof should lie with the defenders of inequality.
Thus, the task is to construct an explanation for this inequality that doesn’t involve any nepotism, monopolism, or favoritism of any kind. If we can do that, then the inequality defenders have a real shot at a moral defense of their stance, regardless of any practical implications.
A Tennis Detour
Before we dive into the nitty-gritty details of real life, let’s look at a useful simplification: tennis. Doing away with all the vestigial and esoteric scoring wrinkles, tennis can be approximated as follows: Two players compete for points. The best of five points wins a game. The best of five games wins a set. The best of five sets wins the match. We’ll use this model to pull apart of some the dynamics of tennis-like games.
Now, a surprising fact about professional tennis rankings is that they are relatively static. This is particularly surprising when one watches a tennis match, and sees a moderately ranked player rack up a substantial number of points on Roger Federer (or so I’m told). It might seem that the porousness of Federer’s defense belies his head-and-shoulders number one status.
Until you crunch the numbers.
In this chart, we show how the likelihood of winning games, sets, and matches varies with the likelihood of winning points. That is, we consider how the broader victories in tennis depend upon the smaller constituent victories. We let the probability of winning a point range from 0.3 to 0.7, and we consider how these changes affect the other types of victories. As you can see, the probability of winning a match ranges from 0.00036 — when one’s probability of winning a single point is 0.3 — to 0.99964 — when one’s probability of winning a single point is 0.7.
In other words, even if you win nearly a third of the points, your opponent is roughly 2,800 times better than you.
Though this tennis demonstration may be unsurprising to you, it’s worth pinpointing exactly what causes the behavior. Tennis matches are won and lost based on the aggregation of many single points. For a quick test of your intuition, instead of a tennis point, consider an unfair coin that shows heads 70% of the time. Any single flip might show tails, but after 100 flips, it would be very surprising to see more tails than heads. Thus, heads is virtually guaranteed to win over tails.
A Generalization to Real Life
One of the most common complaints about a radically unequal distribution of wealth is that we’re all just human. Granted, some people may be smarter or work harder than others, but there’s just no way that a CEO making $10 million a year is 250 times “better” than a line worker making $40,000. Thus, such an income distribution is a priori unfair.
However, the tennis model outlined above provides a possible explanation. Certainly, the CEO is not 250 times better at any single task than a line worker. The worker probably could lead a project to completion using far less than 250 times the CEO’s budget, and when it comes to factory work, the CEO is almost certainly worse. But, just as a 70/30 point split in tennis virtually guarantees a match victory, a comparatively small skill differential may have tremendous impact on business victories. Thus, when the interests of the business are considered, a CEO might very well be worth 250 times the worker. In fact, with the value differential we saw in tennis, the CEO should be paid a salary of even more — $112 million.
Now, it’s one thing to draw a line between tennis and business, but we’ve left out all the details. After all, if one is just keeping tracking of points, the 70% tennis player ends up with only a little more than twice the points of the 30% player. Perhaps a CEO should only be paid twice as much as the factory worker? In order for the tennis model of value to apply within some other domain, two characteristics must be present:
- Iterativity: The behavior within the domain must break into sequences of tasks, where a given actor’s skill persists across many tasks. Victories must occur after some number of iterations. Tennis matches are won by successive victories in points, games, and sets.
- Winner takes most: The victor of each stage must collect the bulk of the rewards, regardless of how close the runners up were. (Clearly, the rewards in this context are the value created in whichever domain we’re considering — not the payment the victor receives in exchange. Jumping straight to payment would be begging the question). In tennis, the the tournament winner is the ultimate victor, whether he wins by a lot or a little.
The iterativity serves to guarantee that “skill will out”. That is, the aggregation of a large number of trials will reveal small differences in skill level, just as we saw in the unfair coin example. The winner taking most simply means that the winner creates much more value compared to the runners up, even if the runners up put forth almost as good efforts.
It’s fairly obvious that iterativity is present in most parts of life. Skill at many tasks is correlated with general intelligence, and success at almost everything is highly dependent on work ethic and perseverance.
The winner also often takes most in the real world. The search divisions at Google, Microsoft, and IAC represent virtually all the equity value in the US search industry today. Alta Vista, Lycos, Excite, Cuil, Duck Duck Go, and all the rest are worth virtually nothing. Even though the underdogs are likely no more than few dozen times “worse” than the victors, they are worth thousands or millions of times less. On an individual level, even if a candidate is only slightly better than the others, she typically gets the job, and the salary and other rewards that come with it. More generally, the winner takes most whenever there are increasing returns to scale in skill level.
Yeah, So What?
This is all just theory, and the back-of-the-napkin sort at that. But it’s possible that, due to iterativity and increasing returns to skill, the value created by different people and companies will diverge widely. Society doesn’t need more than a handful of search engines, so the dozens of runners up — while they may be close in terms of search quality — are virtually value-less. Thus, if one receives compensation in proportion to one’s value to society — a system most people find just — we can expect a highly unequal distribution of income and wealth.
However, there are large negative externalities associated with unequal distributions of wealth — eventually the mob decides that it wants more, and a revolution ensues, causing chaos and pain for everyone. Thus, it may be in society’s interest to underpay some people for the sake of stability. But what about the people who are underpaid relative to the value they create? How do you justly reward them and avoid a tyranny of the majority? It may sound crazy, but perhaps there really was a kernel of wisdom lying beneath all those medieval titles.