On the WARpath: Understanding Performance Above Replacement (Fantasy Football)

I am going to let you in on a little secret: fantasy football is not about scoring the most points. By extension, it’s not about picking the players who will score the most points.

When you get down to brass tacks, fantasy football is about scoring more points than your opponent.

Now, don’t hear what I’m not saying. Scoring more points is certainly a good thing, and it’s highly correlated with scoring more points than your opponent. The important takeaway is a simple one: by definition, you win a week, and eventually your league, when you outscore your opponent. We’re all familiar with the fantasy team that finishes in the top three ‘points for’ and doesn’t make the playoffs. Scoring a lot of points doesn’t necessarily mean you will win; scoring more than your opponent guarantees that you do.

If we really want to precisely measure the fantasy impact of a specific player, we don’t want to consider the points they scored, we want the wins they added to your fantasy team. Why are these two measures different? Simply put, not all points are created equal. We’re all aware of positional advantages: it’s far more valuable to have a TE scoring 15.0 PPG than a WR posting that same amount. Further, there’s a diminishing return effect on points scored: on average, you get a big win probability bump when a player scores 20 points vs. 10 points, but not as big of a bump when that player scores 30 points vs. 20 points.

In this series, we’re going to dust off a well-known sports statistic, with the goal of popularizing the approach for the fantasy world. That statistic is Wins Above Replacement, or WAR for short. There are clear advantages to this approach, perhaps most simply that it gets everything down to one number: it allows us to compare player value across positions, rounds, years, and scoring formats.

WAR is a concept that applies to a variety of real-life sports: front offices, analysts, and die-hard fans love to crunch the numbers to measure value in different team. WAR has been popularized in fantasy football through the work Jeff Henderson on FantasyPoints.com and featured prominently by Pro Football Focus. WAR in fantasy football builds upon Major League Baseball’s WAR and value over replacement (VORP) concepts conceived by Bill James in the 1980s and a 2018 published article highlighting nflWAR reproducible metrics in the Journal of Quantitative Analysis in Sports.

Replacement Level Production

WAR, at its core, is asking a simple question. Imagine two perfectly average fantasy teams facing off. Now imagine that one of the teams gets Cooper Kupp, who was far above average this year (we’ll be using him in our examples for that reason). Clearly, the Kupp-led team has a higher win probability, but how much higher? How many expected wins did Kupp add?

To get at this question, we need to establish the concept of replacement-level players, or perhaps more apt for this setting, average players. The name is self-annotating: this is the level of production that we, the fantasy manager, could reasonably replace Cooper Kupp with if he got injured. Put another way, we want to examine how a team does with a perfectly average player vs. how they do with Kupp.

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There are a lot of ways to calculate replacement-level production, and I am going to choose the most straightforward. Imagine a standard fantasy lineup: QB/2RB/3WR/TE/FLEX. In this scenario, where we want an average team, it makes sense to get the average performance of the QB1-QB12 on the year, the average performance of the RB1-R24 (and double it) on the year, etc. For FLEX, I combine RBs and WRs and average from the 48th to the 72nd spot. Here are the results for your perfectly average Joe squad:

QB: 18.3 PPG
RB (total across two players): 23.5 PPG
WR (total across three players): 33.1 PPG
TE: 8.8 PPG
FLEX: 8.1 PPG
Total: 91.8 PPG

A quick sense check tells us these numbers seem fairly reasonable with Half-PPR scoring: the average RB scores just under 12.0 (divide the 23.5 PPG total by two) and so on. Now that we have our ‘replacement squad’, we can think about the impact of a single player on your fantasy team.

Expected Wins

We’ll be thinking about the distribution of outcomes for your team; specifically, we’ll be using the Normal distribution, which is a reasonably good fit and has a lot of nice properties that we can take advantage of. If interested, you can read more about the underlying statistical mechanisms down below in the footnotes.

First, it should be clear that Cooper Kupp represents an improved distribution of outcomes. Specifically, he scored 21.6 PPG in fantasy this year, much higher than the replacement level WR who scored 11.0 PPG (take the 33.1 PPG from the overall WR position and divide by three). That means that the bell curve of a team with Cooper Kupp should be shifted to the right:

This is all pretty intuitive. What’s interesting is when we consider the difference in score between your fantasy roster and your opponent. Again, this is what’s really important: if that number is positive, you win, and if it’s negative you lose (no matter how any points you scored). Let’s look at that distribution for an average team and one with Cooper Kupp; this is assuming a perfectly average opponent.

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The chart on the left makes sense: when two average teams face each other, the ‘difference distribution’ is a bell curve centered at zero. You win when that difference is positive, which is the shaded green region, and that happens 50% of the time. That checks out; if you, an average team, play another average team, you have a 50% win probability.

From here, the Cooper Kupp chart follows intuition. Clearly, the ‘difference distribution’ is shifted right, with more density in positive territory. In fact, 72% of the distribution is greater than zero, meaning that you have a win probability of 72% in this case. This is also a good reminder of diminishing returns: imagine moving the bell curve another 10 points to the right. Less area would be captured under the curve and colored green than the first 10 point-move (you can tell this visually, but it’s technically because of the slope of the curve), and the win probability would not jump up as much.

Anyways, this tells you that if the perfectly average team gets Cooper Kupp, they see a jump in win probability to 72%. More generally, a WR who scores 21.6 points in a week adds 22% of win probability to an average roster. The final step is to convert this to wins added. Originally, the average team had an expected 0.50 wins, and now they have an expected 0.72 wins. Therefore, a WR who scores 21.6 points adds 0.22 wins above replacement. If the player scored a ridiculous amount of points that all but guaranteed you a win (say, 200 points) then your win probability would be 1.0 and thus the player added 0.5 wins above replacement.

Note that wins above replacement can be negative if a player scores under the average mark. By symmetry, a WR scoring 0.4 points (instead of 21.6 points) against an average of 11.0 points will have -0.22 wins above replacement (or subtract 0.22 wins above replacement, whichever you prefer). A player scoring exactly average will have zero WAR.

Let’s Go To WAR

The above process describes how to calculate WAR for a single week; we can do this for every player, and then find their season totals. Again, this is a more exact measure of player value than just points scored because it takes into account player position and scoring distributions. WAR is also intuitive – it’s helpful to think about how drafting the right player could add 1.3 wins to your season total – and allows us to value players across different positions by one common measurement.

In the next installment, we will dive deeper into some of the implications of WAR on the fantasy landscape (there are a lot). This entry was intended to introduce the concept and, hopefully, get you familiar with the idea. Still, I don’t want to leave you hanging, so here are the top 10 WAR players in 2021:

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It’s not surprising that Cooper Kupp is first on this list: drafting him added nearly 3.0 wins to your roster. Similarly, it makes sense that Jonathan Taylor came in second, and Deebo Samuel third.

A big surprise here for me is Derrick Henry in ninth, even above Joe Mixon, who scored far more total points on the year. How is it possible that Henry finished the RB16, but ninth overall in WAR? This is the magic of WAR: when Derrick Henry played, he dominated, often nearly guaranteeing you a win (close to 0.5 wins added). The WAR stacked up along the way. Mixon played in more games, but he had plenty of outings closer to average or even below average, thus adding zero or negative WAR to his total. Nothing against Mixon, who was the RB3 on the season and finished top-10 in WAR, but this metric shows just how insanely awesome Derrick Henry played when he was on the field.

Other results jump off the page as well. Mark Andrews was the 6th highest WAR-producer, despite scoring far fewer points than players below him; this is thanks to the positional advantage at TE. Josh Allen was number eight even though he scored the most total points of any player: naturally, this is because of a lack of positional advantage at QB. What does this mean for drafting?

We’ll dive deeper into WAR by position, team, and player, as well as all of the fantasy strategy that comes with it, in our next installment.

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The Normal distribution is an apt choice here because of the Central Limit Theorem: a fantasy roster is the sum of a bunch of random variables (individual player scores), which means that the distribution of fantasy scores is approximately Normal. Now, this isn’t a perfect representation of fantasy scoring, which is both bounded and right-skewed, since player scores have a floor of zero (well, usually). Still, it’s a decent approximation (although it will overvalue a ‘boom’ game where a player scores 30+ points) and it allows us to conveniently take the difference between two distributions (your roster and your opponent’s fantasy roster) and still maintain Normality.

Another nice feature of the Normal is that the variances sum: to find the distribution of the scoring difference, we just add the variance of the two teams playing. It’s true that correlations between players (either a stack on your roster, or a stack between rosters) will affect this variance, but it’s largely negligible and might even net out to zero in the long run.

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