One of the most influential books in my life was John Thorn and Pete Palmer’s The Hidden Game of Baseball. When I read it back in high school, I was fascinated by the idea that sports statistics could be used, not just for counting or determining success rates at certain discrete baseball tasks, but also to assign a value to a player’s contribution to his team. This was done by determining the values, positive or negative, of discrete baseball events and totaling them up for the player. The events were measured in terms of runs. For example, hitting a single might be worth 0.47 runs, while being caught stealing might be worth -0.28 runs.

In part inspired by the Hidden Game approach, I have devised a formula for a player’s fantasy value in term of fantasy points, which can be used to assess past player value based on actual stats, or predict future player value based on detailed projections.

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## Definitions: Absolute contribution, differential contribution, and value

Before getting to value, I need to talk about the related notions of absolute and differential contribution. A player’s absolute contribution is the number of fantasy points he scored when it made a difference to his team – that is, when his fantasy owner actually put him in the starting lineup. Differential contribution is the difference between a player’s absolute contribution and the points scored by the best alternative on the player’s team who didn’t start. Value will be defined as optimal differential contribution. Therefore, value is the optimal difference between a player’s contribution and the contribution of the best alternative to that player; that is, not necessarily the difference on any given team – which might have made bad decisions about which player to start – but the differential contribution on a hypothetical team, or set of teams, exercising very good judgment in making start/sit decisions.

Let’s start with the definition of absolute contribution:

A player absolute contribution (on the season) is the sum of his fantasy points scored for his team in games in which he started.

Simple, both conceptually and computationally. A player’s contribution is how many actual fantasy points he scored for his team, regardless of how well the fantasy owner played. If the owner benched Todd Gurley all year, Todd Gurley’s absolute contribution was 0. It is easy to see that a team’s total fantasy points on the year is equal to sums of the absolute contributions of each player on the team.

Differential contribution is defined as follows:

The differential contribution of a player x (on the season) is the sum of the fantasy points he scores in games in which he starts, minus the sum of fantasy points scored by the player who was not started in games in which player x was started, but would have been if x had not been the starter.

Wordy and counterfactual, but conceptually simple. We are still counting the fantasy points that a player actually scored and that counted towards his team’s total, but we also want to account for the opportunity cost of starting the player instead of the alternative. Just like absolute contribution, differential contribution is still relative to a particular player and roster, and does not account for how well or poorly the fantasy owner made choices. It therefore does not get at player value in any objective way. Notice also that a player can have a negative differential contribution. If you score points but get in the way of a player who scores more points, you have made a negative contribution.

Finally, the definition of value:

A player x’s value is the sum of the fantasy points scored by x in games in which he should be started, minus the fantasy points scored by the next best starting option in games in which x should be started.

## Fantasy value is player contribution – opportunity cost, assuming optimal usage

The basic idea is that a player’s value is the sum of production minus opportunity cost, when deployed optimally. This definition is objective, using the modal concepts of “should be started” (and “next best starting option,” which is basically the same thing) to ensure that a player’s value does not suffer due to bad usage by fantasy owners.

Note that “should be started” does not mean “should have been started with hindsight.” It assumes that there is a good method for making start/sit decisions, and it considers the opportunity cost to be the points scored by the best replacement, not by some other bench player who had a surprisingly good day. Like differential contribution, value can be negative if the opportunity cost exceeds the points scored.

While value is not difficult to understand conceptually, it is not at all straightforward to calculate. This is not because it is complicated to determine which player on a given roster should be started and which player is the next best option. This can be easily done by looking at a good set of weekly rankings. The tricky part is figuring out rosters. The calculation assumes a given roster, but of course a player’s value is going to vary from roster to roster, since start/sit decisions and next best player determinations are going to depend on who else is available. It also matters which players are available on waivers in a given league.

I acknowledge, and promise a future post about, the similarity between this fantasy value analysis and value-based drafting. This is a valuation method, not a drafting strategy. Some of the concepts of VBD would not carry over well to this analysis. For example, fantasy value is based only on weeks in which a player should be scoring points for his fantasy team, while VBD is based on all points scored, even if scored on the bench. VBD’s use of a baseline rank as the subtrahend in its equation means that the cutoff line between valuable and valueless players is set arbitrarily, and is either too inclusive, making bench players too valuable, or too exclusive, making occasional contributors valueless.

It seems obvious that the thing to do is to get the player’s value in a range of rosters, and average them. But what range of rosters? I can think of several ways to do this:

- Get a hold of a good sample of actual rosters, either from real leagues or mock draft leagues.
- Simulate a bunch of rosters.
- Run the analysis against the set of all players who could reasonably be considered against the player for start/sit decisions and next best options (call it the “opportunity cost set”).

For time and resource reasons, I’m going to use option 3 for now, leaving to the future the development of methods to use approaches 1 and 2.

## Case study: Deshaun Watson’s 2017 season

This will be a crude study, aimed more to show how the process works than anything else. Watson played in 7 games in 2017, and I will assume that the relevant set of quarterbacks to compare him to is the top 20 projected QBs other than Watson each week, based on fantasydata.com‘s projections. This is based on the assumption that Watson was drafted late in some redraft leagues, by an owner who drafted a higher QB, and in other leagues went undrafted and was added by an owner after week 2 or 3. This suggests a range of alternative options across leagues. In contrast, fantasy owners are not likely to have picked 2 top QBs like Rodgers and Ryan (in 1-QB leagues), so an analysis of Rodgers’ value should probably not include Ryan as an alternative starter.

###### Deshaun Watson.

Image source: Atlanta Falcons.

url: https://www.youtube.com/watch?v=fVHNp5iX-PM

We do our valuation by comparing his performance in each week against that of every quarterback in the top 20 who Watson was projected to outscore. We don’t care about quarterbacks projected to outscore Watson, because he should not have been a starter on any roster containing those QBs.

Weeks 1 through 3 are easy. Watson is not projected in the top 20 in any of them, so against any QB in the top 20, he is not a starter. The formula provides that if a player is not a starter, his fantasy value is 0. If it seems wrong to you that Watson’s 301 pass yard, 2 TD, 41 rushing yard performance should not add to his value, blame fantasydata.com’s conservative rankings. Remember, valuation depends on an analysis of who should be started, so a ranking that had Watson higher earlier would have resulted in a higher valuation.

In week 4, Watson was the number 20 ranked QB. He was therefore not a starter on any roster except the ones where the number 21 QB, Eli Manning, was the alternative. Watson’s value for week 4 will be 1/20th the difference between his fantasy points scored and Manning’s (because his value is 0 in 19 out of 20 rosters). As it happens, both players had great games, with Watson that week’s top points producer at 33.72, and Manning the QB4 at 27.72, a difference of 6. Watson’s value that week was therefore only 6/20, or 0.3 fantasy points.

In Week 5 Watson’s projection improved significantly, and he was ranked as QB14, ahead of Dalton, Rivers, Brissett, Cutler, Kiser, Goff, and McCown. His average value is therefore the total difference between his points and these players’, divided by 20. Watson was the QB1 again, scoring 35.54 points. The points scored by these other QBs were, respectively: 13.32, 20.82, 17.96, 6.48, 2.38, 7.72, and 13.46. The total between Watson’s points and the others is 166.64, which divided by 20 is 8.33.

In week 6, Watson was projected as the QB6, meaning that his value would be 0 only against the 5 higher ranked QBs – Brady, Brees, Ryan, Rodgers, and Cousins. The point differential between his 23.3 point (QB2) performance and the other 15 QBs were: 2.64, 8.42, 8.98, 9.06, 8.16, 20.86, 2.38, 16.46, 1.92, 9.32, 10.24, 17.36, 3.04, 11.42, and 21.94, for a total of 152.2. Dividing by 20 yields 7.61 as Watson’s week 6 value. Although his fantasy points were much lower in week 6 than week 5, his value is almost the same because he was the projected starter in many more roster situations.

After a week 7 bye, Watson was only projected as the QB16 for his matchup against Seattle in week 8. That means that his value for the week was the average of fifteen 0s and the difference between Watson’s fantasy points (32.78, another QB2 performance) and those of Matt Stafford (18.02), Josh McCown (18.28), Case Keenum (17.52), Jacoby Brissett (15.92), and Trevor Siemian (7.92). This is a total difference of 86.24 points, making his value for the week 4.31.

By the way, Watson’s week 8 performance wonderfully illustrates one reason why fantasy points are a superior method of expressing fantasy value, compared to rankings. Watson outscored the 5 QBs listed by an average of 17.25 points, but they finished the week as the QBs 5, 4, 6, 10, and 23 respectively, an average of only 7.6 slots behind him. If Siemian’s 23rd place, 7.92 point performance is excluded, the remaining 4 QBs finished, on average, a bare 4.25 slots behind Watson in the rankings, but a whopping 15.35 fantasy points behind him. Fantasy games are won based on points and not ranks, and ranking-based value leaves out important information.

And that was it for Deshaun Watson’s season. He was injured before taking the field for week 9, so his value for the rest of the season was 0. That means his total value on the season, under the assumptions made here, was 20.55.

This brings up three questions: (1) What does that number mean? (2) Why is the number so low given his huge games? and (3) What does that portend for his 2018 value?

### What does the number mean?

It should be clear that this analysis is a crude first pass at fantasy player value. But if the analysis was done correctly, a player’s value should be the average points gained by a team, across a range leagues and rosters, of having that player on the roster. A player who sits on a team’s bench all year will have a value of 0, reflecting that he has neither scored any points for your team nor blocked a higher scorer. A player who is occasionally usable and does well, like Deshaun Watson in 2017, will have a low score. An every-week starter at the top of his position will have a high score. An overhyped player who is started often and does not do well will have a negative score.

### Why was Deshaun Watson’s value so low?

Obviously, Watson’s value was low in part because he played less than half a season. But even his value per game (2.94) is extremely low considering his amazing production. The answer, quite simply, is that according to fantasypros.com’s weekly rankings, he should not have been started in many situations last year. If the rankings had been more bullish on Watson’s projections, he would have been rated more valuable.

And of course, if a fantasy owner ignored the rankings and went with her own judgment, Watson’s differential contribution – his subjective value to the team – would have been higher. For example, if an owner drafted Dalton and Watson and her personal rankings had her starting Watson every week until he got injured, Watson’s value on the season would have been the sum of differences between Watson’s production in those weeks and Dalton’s, which is 168.86 – 93.52 = 75.34. His per-game value would have been a very impressive 10.76.

### What can be inferred about Watson’s 2018 value?

The simple answer is that Watson’s 2018 value can be expected to be much higher, even on a per-game basis, even if his point production substantially regresses. The reason is that Watson will be drafted high by a team that should start him almost every week. His opportunity cost set will therefore not be Rodgers, Wilson, and the rest of the top QBs, but in most cases a late-drafted QB or waiver option that Watson should consistently outscore by a significant margin.

## Conclusion

Fantasy value should be measured in fantasy points, because points are what win games and championships. Value is based only on points scored in starts, because points scored by bench players don’t win games or championships. Value is not based on absolute point contribution, but rather on the difference between the starter’s points and the next best starting option’s points, to account for opportunity cost. And in order to be an objective, fantasy value needs to be based on the assumption that a player is used properly by a team owner looking to maximize her points.

A case study of Deshaun Watson shows that he had a surprisingly low fantasy value (20.55 fantasy points) if we assume that start/sit decisions should be made based on fantasypros.com’s rankings. This is because he was ranked relatively low every week except Week 6, when he had a good week, but not anywhere near as good as his top 3 weeks. Because the analysis weighs fantasy value by startability and deducts for opportunity costs, Watson’s great weeks did not register much of an impact. Because Watson will probably be startable in many games in 2018, his fantasy value should improve even if his efficiency regresses.

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