# Fantasy, in Theory: The Law of Power

Winning isn't a matter of having good players. It's a matter of having great ones.

It's week 13, which means for many teams the fantasy season is coming to a close. Any postmortem conducted at this point will probably point to several potential causes— mistaken start/sit decisions, probably a decent helping of schedule luck.

But for most teams, the root cause of an early exit will probably be attributed to “not enough good players”.

That's fine as far as explanations go, but I can provide a better one: “not enough great players”. Because in fantasy football, good players are nice enough to have, but it's the great ones who determine fates.

# The Law of Power

Most of you are probably familiar with the concept of a “normal distribution”, that ubiquitous bell shape that is the most famous of curves. In a normal distribution, there is a range of possible outcomes, but most of them pile up around the middle, while outliers become rarer and rarer the further out you go. And yes, the normal distribution crops up a lot in fantasy football.

For instance, consider weekly scores. At the individual player level, they tend to skew a bit, (after all, if a player averages 10 points per game, he can more easily score 20 points over his average than 20 points under his average). But for fantasy teams as a whole, weekly scoring falls into a normal distribution.

Schedule luck also follows a normal distribution. My favorite measure of schedule luck is a comparison of a team's actual winning percentage to its all-play winning percentage. Such a comparison will also be normally distributed, with most teams being at or quite near to 0 schedule luck, but a few outliers down along the tails with +/- 1 or 2 wins due to vagaries of the schedule.

The normal distribution, however, is not the only distribution which is normal. No, the distribution that really rules fantasy is called the power law distribution, which looks more like a ski slope than a bell.

When you look at player production over any specific timeline, you're going to see that power law distribution crop up. The higher the production threshold, the fewer players qualify. For instance, consider PPR scoring so far this season.

Right now, Antonio Brown paces all skill players with 249.5 fantasy points in PPR, and there are five players with between 200 and 250 points. Meanwhile, there are 29 players with between 150 and 200 points. There are 46 players with between 100 and 150 points, 51 players with between 50 and 100 points, and 297 players with between 0 and 50 points.

Individual games exhibit a similar trend. Among quarterbacks, there have been 4 games of 35-40 points, 10 games of 30-35 points, 32 games of 25-30 points, 47 games of 20-25 points, and 92 games of 15-20 points. Every time we step down in production, games at that level become much more common.

This probably isn't surprising. The more productive the threshold, the rarer the player that meets it. But the implications are big. The higher you get up in the distribution, the larger the gaps between performances become.

Consider receiver. You might think that having a pair of really good receivers would help you make the playoffs. And certainly, they would. A team that had Marvin Jones Jr (WR11 in PPR) and Michael Thomas (WR12 in PPR) is essentially starting two “WR1s” on a weekly basis, which is a strong advantage.

But it's not nearly as strong of an advantage as just owning Antonio Brown would have been. Jones and Thomas combined have 327.6 points in PPR. Antonio Brown could beat that total even if he was paired with Tyler Lockett, who is WR61 on the year.

Again, WR1 + WR61 > WR11 + WR12. Thanks to power law distributions, that's how math works in fantasy football. And it's not just a matter of Antonio Brown being uniquely amazing. At running back, RB11 (Leonard Fournette) and RB12 (Lamar Miller) have scored fewer points than RB1 (Todd Gurley) and RB43 (Adrian Peterson).

In two-quarterback leagues, a team with QB11 (Philip Rivers) and QB12 (Ben Roethlisberger) is actually worse off than a team with QB1 (Russell Wilson) and QB27 (Jameis Winston)... even if the latter team just took a zero in every game Winston missed.

At tight end, TE11 (Jared Cook) and TE12 (Austin Hooper) have scored the same number of combined points as TE1 (Travis Kelce) and TE50 (Dion Sims).

If you had a fantasy team with Philip Rivers, Leonard Fournette, Lamar Miller, Michael Thomas, Marvin Jones Jr, and Jared Cook, you don't have a shortage of good players. You've got nothing but good players at every position. And yet you still would be an underdog for the title to a team with a couple great players surrounded by random guys off the street.

That's the cruel nature of power law distributions. It's not about consistently getting decisions right. It's about occasionally getting decisions spectacularly right. And given how difficult it is to know in advance just which players are going to have the kind of breakaway seasons that dominate the distribution, there's not a lot we can do but make our choices and hope for the best.

But then again, practical advice was never this column's strong point. I don't really know how to ensure you get a distribution-breaking superstar. But for those owners who are still alive after this week, I give my sincerest hopes that you already have one or two on your roster somewhere.