Does inequality make NBA teams lose?

I

Some people believe that income inequality can hurt group performance. They think that people work better together when employees are more likely to see themselves as equal.

I don't know if that's true or not. But it's a coherent hypothesis, that makes sense in terms of cause and effect.

On the other hand, here's something that doesn't make sense: the idea that when salaries are more unequal, the result is that the total becomes lower. That doesn't work, right? You can tell the CEO, "if you paid your people more equally last year, the company would have done better." But you can't tell the CEO, "if you paid your people more equally last year, they'd have collectively taken home more money."  

Because, the relationship between total pay and individual pay is already known. The total is just the sum of the individuals. Equality can't possibly cause any additional pay, beyond adding up the amounts.

It would be like saying, "You shouldn't carry $50 bills and $1 bills in the same wallet. If you reduced inequality by carrying only $5 bills and $10 bills, you'd have more money."   

That would be silly.

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Well, that's almost exactly what's happening in a recent NBA study, by the same poverty researcher who wrote the baseball inequality article I posted about three weeks ago.

The author looked up individual player Win Shares (WS) for the 2013-14 season. He measured Win Share inequality within each team by calculating the Gini Coefficient for the population of players. He then ran a regression to predict team wins from player inequality. He found a strong negative correlation, -0.43. 

In other words, the more equal teams won significantly more games. 

The author suggests this might be evidence of the benefits of equality. On the more equal teams, the better performance might have been created by the "psychological and motivational benefits" of the weaker players having "better opportunity to develop and showcase their skills."

But ... no, that doesn't make sense, for exactly the same reason as the $10 bill example. 

Win Shares is really just a breakdown and allocation of actual team wins. The formulas take the number of games a team won, and apportions that total among the players. In other words, the team totals equal the sum of the individual totals, the same way the total amount in your wallet equals the sum of the individual bills. (*)

Last year, the Spurs won 62 games, while the 76ers won only 19. That can't have anything to do with equality. It's due to the fact that the Spurs had 62 win dollars in their wallets -- say, eleven $5 bills and seven $1 bills -- while the 76ers had only 19 win dollars -- say, a $10 bill and nine $1 bills. 

It's true that the 76ers players' Win Shares were more concentrated among their best players. In fact, the top 5 percent of their players accounted for more than half the team total. But that doesn't matter. They had 19 wins total because they had a total of 19 wins individually.

If you want the Philadelphia 76ers to win 50 games this year, find players who add up to 50 Win Shares. It doesn't matter if you find ten guys with 5 WS each, or one guy with 30 WS and ten guys with 2 WS. 

In fairness to the author, he does explicitly say that the correlation does not necessarily imply causation here. But the point is: he doesn't realize that he's looking at a relationship where correlation CANNOT POSSIBLY imply causation.

And that's what I found so interesting about the study. At first reading, it looks like such a strong finding, that equality may cause teams to win more ... but after a bit of thought, it turns out it's logically impossible!

The only other time I remember seeing that kind of logical impossibility was that study "proving" that listening to children's music makes you older, by retroactively changing your year of birth. And that one had been created deliberately to make a point.

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II

As an aside, another thing I found interesting: in his baseball article, the author argued against unequal pay for baseball players because, he believed, pay seemed to have so little to do with actual merit. But, here, by measuring inequality in Win Shares instead of dollars, he seems to be arguing against inequality of merit itself!

Well, that may be just a tiny bit unfair. Reading between the lines, I think the author thinks Win Shares are much more heavily based on opportunity than they actually are. He writes, "maybe top teams, by virtue of their abundance of success, are more willing to share the glory ... Lack of opportunity, by contrast, can lead to despair and diminished performance."

But, actually, the author never demonstrates that the bad teams have more inequality of opportunity (playing time). I suspect that they don't.

In any case, we can see that the 76ers high Gini isn't much caused by differences in opportunity. Even limiting the analysis to "regulars," players with 1,000 minutes or more, the effect remains. On the 76ers, the top two players had 53 percent of the regulars' total Win Shares. On the Spurs, it was only 29 percent.

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III

So why is it that the unequal teams tend to be worse? I think it's a combination of (a) the way the Gini coefficient measures inequality, and (b) the mechanism by which NBA performance creates wins. 

Suppose that on a good team, the five regulars have field goal percentages (FG%) of 59, 57, 55, 53, and 51 percent, respectively. On a bad team, the five players are at 49, 47, 45, 43, and 41 percent.

If you measure inequality on the two teams by variance, it comes out equal: a standard deviation of 2.8 on each team. But if you measure it by Gini coefficient, or a similar calculation of "proportion of total wealth," they're different. 

On the good team, the total percentage points add up to 275. The top player, with 59, has 21.5 percent of the total.

On the bad team, the total percentage points add up to 225. The top player, with 49, has 21.8 percent of the total. So, the bad team is equal by SD, but less equal by "percent of total."

The Gini is more than just the top player, of course ... the formula it involves every member of the dataset. Using an online calculator, I found:

The Gini of the good team is 0.029. 
The Gini of the bad  team is 0.036.

So, by Gini, the bad team is less equal than the good team. (A higher Gini means less equality.)

Why does this happen, that the Gini is higher but the variance is the same? Because of the way the two measures differ. Variance stays the same when you *add* the same amount to every player. But not the Gini. The Gini stays the same when you *multiply* every player by the same amount. 

If you *add* a to every player instead of multiplying, the Gini drops. (And, if you *subtract* a positive number from every player, it increases.)

That's often what you want to have happen -- for incomes, say. If I make $50K and you make $10K, we're very unequal. But if you give both of us a $100K raise, now we're at $150K and $110K -- much more equal, intuitively.

The Gini confirms that. Before our raise, the Gini is 0.33. Afterwards, it's 0.08. (But if we use the variance instead, we look the same both ways.)

But for Win Shares, is the Gini-type of inequality really what we want? Are two players with 7 WS and 6 WS, respectively, really that much more equal, in an intuitive basketball sense, than two players with 2 WS and 1 WS? What about two players at 0.002 and 0.2 wins? In that case, one player has 100 times the wins of the other. But does "100 times" really give a proper impression of how different they are?

I don't think so. I think it's just an artifact of the way performance translates to wins.

What's wins? It's performance above replacement value. (Well, actually, WS is measuring above zero value, which is lower, but I'll call it "replacement value" anyway since the logic is the same.)  

So, to get wins, you start with performance, and subtract a constant. As we saw, when you subtract the same positive number from every player, the Gini goes up. It's a "negative raise" that makes employees less equal.

Suppose the average FG% is 50 percent. Suppose that 40 percent is "replacement level" that leads to exactly zero wins, the level at which a team is so bad it will never win a game. Conversely, 60 percent is the level at which a team is so good it will never lose a game. 

If the relationship is linear, it's easy to convert player FG% to Win Shares. Actually, I'll convert to "wins per 100 games," because the "out of 100" scale is easier to follow.

On the good team we talked about earlier, the players had FG% of 59, 57, 55, 53, and 51. That corresponds to W100 of 95, 85, 75, 65, and 55.

On the bad team, the players' FG% of 49, 47, 45, 43, and 41 translate to W100s of 45, 35, 25, 15, and 5.

See what happens? The FG% looks a lot more equal than the wins. On the bad team, the best player was only 20 percent better than the worst player in field goal percentage (49 vs. 41). But in wins ... he's 800 percent better! (45/5.)  On the good team, though, there's still enough performance after subtracting that the numbers look reasonably equal. 

The actual Gini coefficients:

FG%:  The Gini of the good team is 0.029. 
FG%:  The Gini of the bad  team is 0.036.

Wins: The Gini of the good team is 0.11.
Wins: The Gini of the bad  team is 0.32.

That's just how the math works. The Gini coefficient is very sensitive to where you put your "zero". If you measure zero as 0 FG%, inequality looks low. If you measure zero as zero wins (say, FG% of 40 percent), inequality looks higher. If you measure zero as replacement level (say, FG% of 43 percent), inequality looks even higher. And if you measure zero as an NBA average team (say, FG% of 50 percent), it's even more unequal -- the top half of the teams have 100 percent of the wins! (**) 

The higher the threshold that you call zero, the greater the inequality. 

In baseball, a player hitting .304 has only about 1% more hits than a player hitting .301. But he has 300% more "hits above .300". 

In the economy, the top 10% of families may have (say) 45% of the income -- but probably close to 100% of the new Ferraris. 

And a real-life example: In the NHL, over the last ten seasons, the Black Hawks have 13% more standings points than the Maple Leafs -- but 500% more playoff appearances.

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One last analogy:

Take a bunch of middle-class workers, and tax them $40,000 each. They become much more unequal, right? Instead of making, say, $50K to $80K, they now take home $10K to $40K. There's a much bigger difference, now, in what they can afford relative to each other.

But if you tax the same $40K away from a bunch of doctors, it matters less. They may have ranged from $200K to $300K, and now it's $160K to $260K. They're a bit less equal than before, but you hardly notice.

Measuring after the $40K tax is measuring "income above $40K," which is like measuring "FG% above replacement level of 40%" -- which is like measuring Win Shares.

So that's why bad teams in the NBA appear more unequal than the good teams -- because "Wins" are what's left of "Performance" after you levy a hefty replacement-level tax. Most of the players on the good teams stay middle-class after paying the tax -- but on the bad teams, while some stay middle class, more of the others drop into poverty.

It has nothing to do with the social effects of equality or inequality.  It's just an artifact of how the Gini Coefficient and basketball interact.


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* Actually, there's a bit of wiggle room in the particular version of WS the author used, the version from basketball-reference.com. That version doesn't add up perfectly, but it promises to be close, certainly close enough that it doesn't make a difference to this argument. 

** That's if you give the bottom teams zero. If you give them a negative, the Gini actually winds up at infinity. (The overall total has to be zero relative to the average, and you can't divide by zero.)  

Are CEOs overpaid or underpaid?

Corporate executives make a lot of money. Are they worth it? Are higher-paid CEOs actually better than their lower-paid counterparts?

Business Week magazine says, no, they're not, and they have evidence to prove it. They took 200 highly-paid CEOs, and did a regression to predict their company's stock performance from their chief executive's pay. The plot looks highly random, with an r-squared of 0.01. Here's a stolen copy:

The magazine says,

"The comparison makes it look as if there is zero relationship between pay and performance ... The trend line shows that a CEO’s income ranking is only 1 percent based on the company’s stock return. That means that 99 percent of the ranking has nothing to do with performance at all. ...

"If 'pay for performance' was really a factor in compensating this group of CEOs, we’d see compensation and stock performance moving in tandem. The points on the chart would be arranged in a straight, diagonal line."

I think there are several reasons why that might not be right.

First, you can't go by the apparent size of the r-squared. There are a lot of factors involved in stock performance, and it's actually not unreasonable to think that the CEO would only be 1 percent of the total picture.

Second, an r-squared of 0.01 implies a correlation of 0.1. That's actually quite large. I bet if you ran a correlation of baseball salaries to one-week team performance, the r-squared would probably be just as small -- but that wouldn't mean players aren't paid by performance. As I've written before, you have to look at the regression equation, because even the smallest correlation could imply a large effect.

Third, the study appears to be based on a dataset created by Equilar, a consulting firm that advises on executive pay. But Equilar's study was limited to the 200 best-paid CEOs, and that artificially reduces the correlation. 

If you take only the 30 best-paid baseball players, and look at this year's performance on the field, the correlation will be only moderate. But if you add in the rest of the players, and minor-leaguers too, the correlation will be much higher. 

(If you don't believe me: find any scatterplot that shows a strong correlation. Take a piece of paper and cover up the leftmost 90% of the datapoints. The 10% that remain will look much more random.)

Fourth, the observed correlation is fairly statistically significant, at p=0.08 (one-tailed -- calculate it here). That could be just random chance, but, on its face, 0.08 does suggest there's a good chance there's something real going on. On the other hand, the result probably comes out "too" significant because the 200 datapoints aren't really indpendent. It could be the case, for instance, that CEOs tend to get paid more in the oil industry, and, coincidentally, oil stocks happen to have done well recently.

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BTW, I don't think there's a full article accompanying the Business Week chart; I think what's in that link is all we get. Which is annoying, because it doesn't tell us how the 200 CEOs were chosen, or what years' stock performance was looked at. I'm not even sure that the salaries were negotiated in advance. If they weren't, of course, the result is meaningless, because it could just be that successful companies rewarded their executives after the fact. 

Furthermore, the chart doesn't match the text. The reporters say they got an r-squared of 0.01. I measured the slope of the regression line in the chart, by counting pixels, and it appears to be around 0.06. But an r of 0.06 implies an r-squared of 0.0036, which is far short of the 0.01 figure. Maybe the authors rounded up, for effect? 

It could be that my pixel count was off. If you raise the slope from 0.06 to 0.071, you now get an r-squared of 0.0051, which does round to 0.01. So, for purposes of this post, I'm going to assume the r is actually 0.07.

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A correlation of 0.07 means that, to predict a company's performance ranking, you have to regress its CEO pay ranking 93% towards the mean. (This works out because the X and Y variables have the same SD, both consisting of numbers from 1 to 200.)

In other words, 7 percent of the differences are real. That doesn't sound like much, but it's actually pretty big. 

Suppose you're the 20th ranked CEO in salary. What does that say about your company's likely performance? It means you have to regress it 93% of the way back to 100.5. That takes you to 95th. 

So, CEOs that get paid 20th out of 200 improve their company's stock price by 5 rankings more than CEOs who get paid 100.5th out of 200.

How big is five rankings?

I found a website that allowed me to rank all the stocks in the S&P 500 by one-year return. (They use one year back from today, so, your numbers may be different by the time you try it.  Click on the heading "1-Year Percent.")  

The top stock, Micron Technology, gained 151.47%. The bottom stock, Avon, lost 42.80%.

The difference between #1 and #500 is 194.27 percentage points. Divide that by 499, and the average one-spot-in-the-rankings difference is 0.39 percentage points.

Micron is actually a big outlier -- it's about 33 points higher than #2 (Facebook), and 52 points higher than #5 (Under Armour). So, I'm going to arbitrarily reduce the difference from 0.39 to 0.3, just to be conservative.

On that basis, five rankings is the equivalent of 1.5 percentage points in performance.

How much money is that, in real-life terms, for a stock to overperform by 1.5 points?

On the S&P 500, the average company has a market capitalization (that is, the total value of all outstanding stock) of 28 billion (.pdf). For the average company, then, 1.5 points works out to $420 million in added value.

If you want to use the median rather than the mean, it's $13.4 billion and $200 million, respectively.

Either way, it's a lot more than the difference in CEO compensation.

From the Business Week chart, the top CEO made about $142 million. The 200th CEO made around $12.5 million. The difference is $130 million over 199 rankings, or $650K per ranking. (The top four CEOs are outliers. If you remove them, the spread drops by half. But I'll leave them in anyway.)

Moving up the 80 rankings in our hypothetical example is worth only a $52 million raise -- much less than the apparent value added:

Pay difference:      $52 million
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Median value added: $200 million
Mean value added:   $420 million

Moreover ... the value of a good CEO is much higher, obviously, for a bigger company. The ten biggest companies on the S&P 500 have a market cap of at least $200 billion each. For a company of that size, the equivalently "good" CEO -- the one paid 20th out of 200 -- is worth three billion dollars. That's *60 times* the average executive salary.

Assuming my arithmetic is OK, and I didn't drop a zero somewhere.

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So, I think the Business Week regression shows the opposite of what they believe it shows. Taking the data at face value, you'd have to conclude that executives are underpaid according to their talent, not overpaid.

I'm not willing to go that far. There's a lot of randomness involved, and, as I suggested before, other possible explanations for the positive correlation. But, if you DO want to take the chart as evidence of anything,it's evidence that there is, indeed, a substantial connection between pay and performance. The r-squared of less than 0.01 only *looks* tiny.

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Although I think this is weak evidence that CEOs *do* make a difference that's bigger than their salary, the numbers certainly suggest that they *can* make that big an impact.

Suppose you own shares of Apple, and they're looking for a new CEO. A "superstar" candidate comes along. He wants twice as much money as normal. As a shareholder, do you want the company to pay it? 

It depends what you expect his (or her) production to be. What kind of difference do you think a good CEO will make in the company's performance?

Suppose that, next year, you think Apple will earn $6.50 a share with a "replacement level" CEO. How much more do you expect with the superstar CEO?

If you think he or she can make a 1% difference, that's an extra 6.5 cents per share. That might be too high. How about one cent a share, from $6.51 instead of $6.50? Does that seem reasonable? 

Apple trades at around 15 times annual earnings. So, one cent in earnings means about 15 cents on the stock price. With six billion Apple shares outstanding, 15 cents a share gives the superstar CEO a "value above replacement" of $900 million.

So, for a company as big as Apple, if you *do* think a CEO can make a 1-part-in-650 difference in earnings, even the top CEO salary of $142 million looks cheap.

Apple has the largest market cap of all 500 companies in the index, at about 15 times the average, so it's perhaps a special case. But it shows that CEOs can certainly create, or destroy, a lot more value than than their salaries.

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So can you conclude that corporate executives are underpaid? Not unless you can provide good evidence that a particular CEO really is that much better than the alternatives. 

There's a lot of luck involved in how a company's business goes -- it depends on the CEO's decisions, sure, but also on the overall market, and the actions of competitors, and advances in technology in general, and world events, and Fed policy, and random fads, and a million other things. It's probably very hard to figure the best CEOs, even based on a whole career. I bet it's as hard, as, say, figuring baseball's best hitters based on only a week's worth of AB. 

Or maybe not. Steve Jobs was fired as Apple's CEO, then, famously, returned to the struggling company a few years later to mastermind the iPod, iPhone, and iPad. Apple is now worth around 100 times as much as it was before Jobs came back. That's an increase in value of somewhere around $500 billion. It was maybe closer to $300 billion at the time of Jobs' death in 2011.

How much of that is due to Jobs' actual "talent" as CEO? Was he just lucky that his ethos of "insanely great" wound up leading to the iPhone? Maybe Jobs just happened to win the lottery, in that he had the right engineers and creative people to create exactly the right product for the right time?

It's obvious that Apple created hundreds of billions of dollars worth of value during Jobs' tenure, but I have no idea how much of that is actually due to Jobs himself. Well, I shouldn't say *no* idea. From what I've read and seen, I'd be willing to bet that he's at least, say, 1 percent responsible. 

One percent of $300 billion is $3 billion. Divide that by 14 years, and it's more than $200 million per year.

If you give Steve Jobs even just one percent of the credit for Apple's renaissance, he was still worth 50 percent more than today's highest-paid CEO, 300 percent more than today's eighth-highest paid CEO, and 1500 percent more than today's 200th-highest-paid CEO. 

Salaries and merit

People like to say that salaries should be based on merit. Employees should be rewarded by their ability and performance, and not by connections, or family, or race, or class, or luck.  

In a previous post, I argued that it's something we say, but don't really mean.  We don't really act as if we believe that "merit" is what counts.  In fact, outside of sports and school, we don't seem to want to measure merit at all.

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The classic example is teachers.  We love to talk about how valuable teachers are and how important education is, and almost everyone has a story about a great teacher and what a difference he or she made to their life.  But, even so ... we don't make much effort to separate the better teachers from the worse ones.  

Sure, that's not necessarily easy to do.  Ratings are arbitrary, and subject to all kinds of manipulation and favoritism.  We can't let the students vote on merit, because they'll just reward the teachers who give the highest grades and the least homework.  How do you judge "good" and "bad", anyway?  Isn't it mostly subjective?

All of those issues are legitimate, but not insurmountable. The students themselves know who the best and worst teachers are, independent of the grades they give, right?  In my high school, we all pretty much agreed on who were the best.  A good analogy is MLB broadcasters.  Here's one poll of the best and worst.  You probably agree with it, for the most part, right?  (Except that Vin Scully should really be #1.)

There is a movement to pay teachers based on merit, but mostly from policy wonks.  It's not something that parents seem to care about, or students.  And even when it is, it's almost all about improving education. That's important, sure. But, what about from an ethical standpoint?  If we really believe in paying for merit, shouldn't the argument be that better teachers should be paid more on the principle of simple fairness?  Tellingly, the Wikipedia article on teacher merit pay doesn't mention that argument at all.

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There are some situations where we *do* want to recognize merit.  It's obvious that someone who works 40 hours deserves --  "merits" -- twice as much pay as someone who works the same job for only 20 hours.  They do twice the work, so they deserve twice the pay.

But, what about when one employee can do twice the work in the same amount of time?  

In my field, software development, there's an adage that the difference between the best and worst programmer is 100 times the productivity: 10 times between best and average, and 10 times between average and worst.  Let's say it's really only 5 times instead of 10, and let's say we fire all the bad programmers.  That still leaves a 5x difference between best and worst.

Should a fast programmer make 500 percent of the salary of an average one?  If you believe in merit, then, you have to say yes.  Don’t you?  

You could even argue that the difference should be *more* than five times.  If you had to hire five slow employees to replace the one fast one, you'd need five times the desks, the heating bill, the parking lot, and lots of other fixed costs. Furthermore, critical situations often arise when something needs to be done very fast.  The single employee provides you that capability, but the five slow ones will just run into each other.  (Brooks's Law: "Adding manpower to a late software project makes it later.")

But no company could get away with that kind of salary differential.  There'd be a riot.  

Some of the average guys would feel underappreciated and quit.  Others would instead spend their time coming up with rationalizations that the fast programmers are producing lower-quality work, and perhaps even sabotaging them.  The managers would enviously rebel against their underling programmer making three times their salary (even though they'd have to grudgingly admit he's worth it).  The suits in head office would say, why is that geek with no social skills making more than me, when I have an MBA and a thousand dollar suit?  And, when the superstar programmer got older and slowed down, you'd have to find a way to cut his salary in half, without humiliating him.  

I’d argue that merit pay doesn't fail because merit is too difficult to measure.  It fails because merit CAN be measured, and we don't like the results.  

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You're a teacher.  You've been doing it for a few years, and you're proud of your career choice.  The parents like you, and people show a lot of respect when they find out you're a teacher.  You have a satisfyingly high level of status in the community.

But, now, someone figures out a way to accurately evaluate all the teachers.  When they’re done, it turns out that you’re only 30th percentile.  Seven out of ten of your colleagues are better than you.  

Now, you look and feel like a bit of a failure.  Even if you kind of knew, before, that you weren't as wonderful a teacher as some of your colleagues, you didn't really have to face it. You could think, hey, I’m good in my own way.  I may not be able to hit home-run explanations like Barry Bonds, but I use time-tested approaches, so I don’t strike out much either!  

But now that "Teacher WAR" shows you’re barely above replacement, your rationalizations don't work any more.  Also, your pay gets cut.

Even the *good* teachers don’t necessarily benefit from evaluations.  When the world gets a reminder that some teachers are worse than others, everyone’s status takes a hit.  Then the mediocre teachers quit, better ones replace them, there's more competition, and suddenly you’re middle of the pack.  And what happens when you have a bad year? Whatever it is you’re doing right, you have to keep it up in order to keep your place in the pecking order.  In fact, you probably have to get substantially better.  It's lose/lose for almost everyone.  

Except: if they only rate the best.  Then, nobody else really gets hurt.  Sure, you didn't win, and that shows that you're not the very, very best, but that’s no real shame.  And, for all the world knows, you might still be the second best, or in the top 1%.  

I think that’s why, in many professions, you'll see awards given to the best in the field, while nobody else gets rated or mentioned.  (Google any job description followed by "awards," and, usually, you'll find something.)  The awards aren't just to benefit the winners ... they’re also there to raise the status of the profession in general.  When a local employee wins a national award for best mechanic, all mechanics gain a little bit of additional respect.  ("Geez, I thought that mechanics were just people too dumb to graduate high school, but this guy seems really smart!  Now that I think about it, my guy might be pretty good too.")

In the everyday world, the equivalent to an award is a promotion.  Instead of saying, "Bob, you fix cars only half as fast as Joe, so we're going to pay you only half as much," we say, "Bob, as you know, Joe is the very best mechanic here, so we're promoting him to senior technician."  You get to recognize the best without having to formally call out the worst.

Another benefit is we tend to compare ourselves to our peers, rather than people at different positions in the hierarchy.  So, after the promotion, Bob is less likely to become resentful, because he's less likely to compare himself to Joe.  Also, if Joe is doing a different job, Bob doesn't get reminded every day that Joe is twice as good.

But ... it's a promotion to a different job.  You're taking the best performers out of the environment where they've excelled, and moving them to something unproven.  In software in the government, they take the best programmers, and move them to something that’s very different -- like “project leader,” which is a middle-management job.  Sure, you need strong technical expertise to do that job, but almost any decent programmer can do it competently.  When you have a programmer who’s five times as good as average, why put him in a job that’s he’s less suited for?  It’s like telling Mike Trout, "You’re so good at baseball that we’re promoting you to bench coach."

And there are only so many promotions to go around.  We've all seen lots of low-paid workers who are excellent at what they do, and lots of their colleagues who aren't that good.  I’m pretty sure the range of their salaries is much narrower than the quality and quantity of their output -- Walmart cashiers, for instance, seem to all be within a 25% range. 

So if you're good, and you want to be rewarded on merit, you have to count on promotions.  But how many of the best can be promoted?  Walmart has 2.2 million associates, and promotes 170,000 of them annually.  That’s less than 8 percent.  That means you're going to have a lot excellent cashiers making only slightly more money than average cashiers.  

If we really cared about merit, we'd be outraged by that.

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In Canada, we have "pay equity" legislation, designed to close the gap between men’s and women’s earnings, by reclassifying male-dominated and female-dominated jobs in such a way that their pay scales become more equal.  

The rationale is "equal pay for work of equal value."  As in, "a secretary is just as valuable as a carpenter and should be paid no less."

But ... we only seem to care that the pay in question becomes literally equal.  We demand that equals be paid equally, but we don’t really care that non-equals are paid non-equally.

If you really believe in equal pay for work of equal value, the faster programmer should be earning much more than the slower programmer, and the better teacher should be earning much more than the worse teacher, and the industrious janitor should be earning much more than the average janitor. But we don’t take to the street to say, "Jan is working twice as hard at the ice cream parlor as Marsha, but being paid the same!  Stop the exploitation!"

When we do accept pay differences, we don’t seem to care much about whether they're proportional to ability.  We may agree that doctors are more "meritorious" than, say, welders. But, how much more?  Should a doctor be paid twice what a welder makes?  Ten times?  Twenty times?  Nobody cares, really, do they?  

The rare times that we do care, it’s almost always on the side of making everyone more equal.  Much is made of the ratio between CEO and worker pay, how it's too high.  Why do we think CEOs make too much?  For egalitarian reasons that have nothing to do with merit.  I have no idea, honestly, what the ratio should be, how much more valuable a fast-food CEO is worth than the guy who flips the burgers.  Is it 100 times? 1000 times?  10,000 times?  If we did care about equal pay for equal value, it would matter, wouldn't it?

It certainly matters in sports.  We know Mike Trout is a bargain at 600 times the US minimum wage, but overpaid at 6000 times the minimum wage.  Why don’t we have the same argument for CEOs?  I think it's because we don’t care about merit for corporate bigwigs; we're just uncomfortable that they make millions. Maybe I'm wrong -- maybe we really do believe CEOs make more than they deserve on merit.  But, if that's the case, why does hardly anyone argue numbers about how much they're actually worth?

I stumbled onto this article that profiles Marie Sanders, a single mother who works for McDonald's.  After more than two years of full-time work at Mickey D's, she still earns only $7.75 an hour and has trouble making ends meet.  

There's nothing in the article about merit.  The article never asks: "Is Ms. Sanders worth more than $7.75?  Is she worth the $15 she implies she deserves?  How were her job evaluations?"  If the writer *did* ask that, there would be outrage.  "How dare you imply that she's not competent enough to deserve a living wage!  Don't big corporations owe their workers a decent enough paycheck to allow them the dignity of feeding their family?"

Actually, Ms. Sanders sounds intelligent and competent.  I bet she’s well above average among her peers.  If that's true, does that bolster her case for $15 an hour?  I think it does, and I think most everyone would agree.

But if her excellence is an argument for paying her more, that's the same thing as saying her co-workers' averageness (or even mediocrity) is an argument for paying them less.  If Marie should earn more than Andy, the reasonably competent fry cook, that means Andy should earn less than Marie. 

How much less?  Maybe even ... $7.75?  If not that low, then, what?  $10?  OK, say $10.  But, now, what if there's someone even worse than Andy, someone who's just marginally competent?  

I think we don't want to bite the "merit" bullet and face that some workers are worth a lot less than others.  We're uncomfortable if Marie earns the $15 she's worth, and Andy earns the $7 he's worth.  Instead, we're happier if Marie and Andy both make $11 -- even if Marie is a lot better at her job.  

We say we want salaries to be merit-based, but what we really seem to want is for salaries to be equal.  

Luck, careers, and income inequality

According to this (anonymous) blog post from "The Economist," increased income inequality is partly due to random chance becoming more important in picking winners and losers.  

The post (and accompanying article) cites a recent paper on "herd effects," where a song or book becomes popular kind of randomly, because people tend to flock to whatever other people have already flocked to.  So, you can have two songs of equal merit, but one is a smash hit just because, for random reasons, its early adopters had better social networks.

Furthermore, there's experimental evidence.  In a recent academic study, researchers sent job applications to two different employers.  The resumes were identical except that, in one of them, the applicant said he had been unemployed for several years, while in the other, the applicant said he had a current job.  

Overall, the "out of work" applicants got around half the callbacks of the currently employed applicants.  Presumably, the employers believed that those who were out of work were less likely to be suitable candidates than those who weren't.  

The author writes,

" ... the implication is troubling: someone who ends up unemployed through bad luck, and for some idiosyncratic reason doesn't quickly land a job, finds his chances of reemployment diminish until he’s part of the long-term unemployed. ... 

"... this research opens a new and troubling dimension on  inequality. Unlike deficiencies of skill, it’s hard to tell society's  losers they should go back to school to become luckier. "

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Certainly, there a huge, huge amount of luck involved in career success. There has to be: there's already luck in hockey, golf, and chess, and real life is infinitely more random and complicated than those.

For starters, there's when and where you were born, what kind of schools you went to, what books you happened to read, what kind of influences you had from your friends and parents, how good your teachers were in various subjects, chance remarks from strangers about what careers are likely to be lucrative, and many more.

Even in the more concrete sense, there are many, many small random things that can make a big difference.  What company you choose to join for your first job, what your first manager thinks of you, how your bosses think of your performance amidst the noise of team accomplishments, whether your co-workers give you credit or blame behind your back ... and you can probably think of hundreds more.

Basically, where you wind up in your life, in your career or any other sense, is overwhelmingly the result of so many random events that it's basically ... well, it's kind of a miracle that we are where we are now instead of somewhere else.  If we had to live our lives over again, from any given point, we'd probably be in different jobs, doing different work, with a different spouse, maybe living in a different city, earning a different income.  

In that light, I find it a bit perplexing that the authors chose these two specific examples as evidence of how luck factors into it.  They seem pretty minor, don't they?  I mean, the unemployed applicants still got half as many calls as the employed ones ... that seems like a pretty small career stopper as opposed to, say, if their first manager didn't like them and put them into a dead-end position and never promoted them, among hundreds of other possible events.

As for the "herding" argument ... wouldn't it be the case that there's LESS herding luck now, with the internet?  It used to be that you only knew songs you heard on the radio.  Now, you can find whatever you want on YouTube, on demand. There's more opportunity to find a niche.  Isn't that established wisdom?  Isn't that what "The Long Tail" is about, that, these days, there are many different markets that never existed before?

In terms of "herding," I'd bet that luck is much LESS a factor than it used to be.  

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Also: as much as luck is a huge, huge part of where any individual winds up, I don't think it's a big factor in the overall distribution.

In baseball, luck is, what, around 30 percent of the variation in team W-L percentage?  

Now, suppose you expanded the major leagues to include 30 little league teams.   Now, luck is a much smaller factor, isn't it?  The "real" teams always beat the kids, so, now, whether or not you're over .500 is completely a matter of talent.  Luck is still important within your group, but, overall, I'd bet the r-squared drops from 30 percent to ... I dunno, maybe 5 percent. 

Something similar happens with careers.  Two accountants, Andy and Beth, graduate from the same school with the same talent.  Andy winds up in a job where the culture matches his talents, while Beth does not.  Just by that random happening, Andy eventually winds up a CFO making $1 million a year, and Beth winds up in some middle-manager type position, making $70,000.  

Yes, Andy outearns Beth because of a lucky break.  But, the difference is contained within the accountants' income distribution, in the same way luck in our expanded MLB is contained within the teams in a single age level.

Luck matters more, obviously, when the differences in talent are small.  Within the subgroup of accountants, luck is going to matter a lot.  Within the subgroup of burger-flippers, luck is going to matter a lot.  But within the COMBINED group of accountants and burger-flippers, luck isn't that big a deal. You're not going to have many burger flippers, with average talent, getting lucky and outearning a typical accountant.

You can probably see this for yourself with a little introspection.  For my own part, I can see my career, and life, turning out radically differently based on a few key random events and decisions.  But ... I don't think I'd be poor.  I don't think I'd be making minimum wage.  I like programming, and I'd find a programming job somewhere, and even with the worst luck, I'd be making a decent living.  

Also, I don't think I'd be ultra-rich.  Even with extreme good luck -- say, 3 or 4 SDs above average -- I doubt I'd be a charismatic, celebrated CEO of a big company.  That requires skills and personality traits that I just don't have.

There's a lot of luck in individual baseball batting performance, but that doesn't mean you'll see Ozzie Smith hitting 60 home runs.  And there's a lot of luck in career outcomes too, but that doesn't mean you'll see lots of Bill Gateses working minimum wage.

At least, not because of the kind of luck the authors are talking about here.  

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But, fundamentally ...  I wonder about the entire premise of the argument, that the prevalence of luck causes inequality to increase.  I wonder if it's the other way around.

A "system that rewards ability", as the author puts it, is an important and desirable goal.  A job, or promotion, or salary, or other reward, should go to the person who does the job best, not the one with the right resume, or the right skin color, or the right DNA.

Nor, by extension, to the one who happened to obtain the best roll of the dice.  

But, why does it follow that a system that rewards ability must also be more equal?  I think that's the wrong conclusion. It's understandable ... we see that some people get rewarded for being lucky, and we think, well, if only those people didn't get to be so rich, randomly, we'd all be more equal!  

But, I think that's backwards, at least in the hiring context. The more merit matters, the *bigger* the differences in salary.  Because, an employer is only going to pay you the expected value of your production.  

Let's use a simplified sports analogy.  

A team is trying to sign players, who are $100 on average ... but, some are worth more, perhaps much more, and some less.  

Suppose you have no scouts.  So, you have no idea of the players' relative talent (assume, also, that neither do the other teams, or the players themselves).  

That means you have to hire randomly.  What are you going to pay?  Obviously, $100 per player.  You have complete income equality among your players, even though the guys on the team got there by 100% luck.

Later, you hire some mediocre scouts.  You're able to tell the better players from the worse ones, to a certain extent, based on "whether they look like ballplayers".  So, you hire some at $130, and some at $70.   Less equality.

Finally, you hire some sabermetricians and some expert scouts, and now you can estimate each player's value within $10.  You hire the superstar at $400, and a mediocre mop-up man at $5.  Even less equality.

Isn't that a better description of how real life works?  You can't be a well-paid employee unless you can prove to an employer that you're worth it.  The more merit-based the system, the better chance you have of getting full value for what you're worth.

If you don't like that analogy ... try this.  There are a hundred brand new Honda Civics on the lot, and they all cost the same.  But, now, suppose you have more information about their merit -- somehow, it becomes possible to predict reliability.  That blue Civic over there will run trouble-free for 20 years. But that red one ... the transmission is going to blow at 80,000 miles.  

Now, the top 1% of Civics will sell for a lot more than the bottom 1%.  Right?

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In general: the closer you look for merit, the more differences you find.  The more differences you find, the wider the distribution of the sum of those differences.   So, the more you hire by merit, the more spread you get in salaries.

Is that a problem?  Not for me.  I'm strongly in favor of hiring by merit ... and I don't think income inequality is necessarily always a bad thing.  So, to me, there's no serious trade-off. But, if, you like both merit and equality, I think you have to choose which is the lesser evil, in this context.

As a society, I think we've made our consensus trade-off ... because, there are many, many unspoken ways in which we deliberately ignore merit.  For instance, we pay our teachers by seniority and experience, rather than by how good teachers they are.  We try to avoid situations where long-term employees get demoted or has their pay cut, even when their performance drops.  We discourage comparisons of which employees are better workers than their colleagues in the same position.

We give lip service to the idea that we should be paid and rewarded by merit, but we don't actually believe it as much as we say we do.  We claim we can't have nepotism because then we're being unfair to candidates with more talent, but then we believe we CAN favor older, worse teachers, without being unfair to younger teachers with more merit.  

Like a lot of other things in our society, we start with the outcome we want and work backwards to rationalize it.  We hate nepotism, and we hate firing mediocre teachers.  Merit is just an incidental consideration.