H-scores

H-scoring is a framework introduced in the second paper for dynamic player selection. In short, for each candidate player, it optimizes for future draft pick strategy and estimates performance based on those strategies. This allows the algorithm to understand general drafting strategy, including the idea of punting (strategicially sacrificing) some categories and over-performing in the rest. It also understands how to work around position requirements, which enforce that e.g. a team must have at least one point guard. See the optimization section for some mathematical detail on how it works without the academic rigor of the paper.

Because of its ability to adapt to drafting circumstances, H-scoring arguably offers a more compelling and logical starting point for draft strategy than G-scoring. H-scores are featured on the website as the main way of evaluating players.

Parameter inputs

The H-scoring algorithm has three input parameters: \(\omega\), \(\gamma\), and the number of iterations which are configurable by the user.

\(\omega\) and \(\gamma\) control how the H-scoring algorithm thinks about the landscape of player statistics that it will have to choose from in the future. Roughly, when ω is high, the algorithm punts more. The default values were configured based on what worked well in testing.

The H-scoring algorithm runs for the specified number of iterations. Additional iterations increase the precision of H-scoring, at the cost of longer computation. In practice thirty iterations, the default, works reasonably well.

H-score table

The H-score table for candidate evaluation lists players in order of their H-score rank, along with category-level detail.

Top Each Category H-scores for the first pick, 2024-25 season

H-scores change as the algorithm runs. Rendering the results for every iteration would slow down the site, so the table is refreshed once every fifteen iterations. Also, only the top thirty players by H-score are shown until the last iteration, at which point all players are shown.

One might note that Giannis Antetokounmpo ranks highly by H-score. Fantasy veterans will be familiar with Giannis for being undervalued by static ranking systems like overall Z-score, because so much of his value is contingent on punting Free Throw %. H-scoring understands this punting strategy, and evaluates Giannis more appropriately.

Category-level H-scores

Category-level H-scores are an in important part of the H-scoring process. Unlike with G-scores, they are not direct reflections of the candidate player's characteristics. Instead, they show what the H-scoring algorithm expects the average win rate against all opponents will be, assuming the candidate player is taken. H-scoring calculates those expectations based on not just the characteristics of the candidate player, but also on previously chosen players and potential future picks. The statistics of future picks are estimated based on H-scoring's preferred strategy for future picks.

Because other factors are taken into account, the categorical strengths and weaknesses presented in the H-score table are often quite different from those of the candidate players. For early picks, the most important factor is the strategy for future draft picks. For example, Shai Gilgeous-Alexander's row above for the first pick in the draft shows a very low probability of winning the Rebound category, despite SGA being a decent rebounder himself. This is because H-scoring's preferred strategy with SGA involves deprioritizing rebounds with future picks.

In later draft rounds, the importance of previously chosen players increases and the importance of the strategy for future picks decreases. Also, the strategy for future picks tends to become more stable across players, since the direction of the team is already decided. So categorical H-scores tend to become consistent across players.

alt text — Top H-scores for a round seven pick in a mock draft, with relatively stable scores across categories. Each Categories, 2024-25

Most of the time, the algorithm punts one or two categories, reflected by low H-scores in those categories. For the categories it does not punt, it tries to be above average without going overboard.

Overall H-scores

The overall H-score is both the metric that H-scoring is trying to optimize with its future draft pick strategy, and the one used to rank players. It is based on the category-level H-scores.

Different kinds of categories deal with fantasy points differently, and necessitate different formulations for the overall H-score. In "Each Category", teams rotate opponents weekly and keep all the fantasy points they win. For that format, the H-score is defined average expected win rate across categories.

In "Most Categories", teams also rotate opponents, but instead of keeping all of their fantasy points, they get wins for every opponent they get a majority of fantasy points against. For that format, the average expected win rate is a poor proxy for success. A team that wins five out of nine categories 100% of the time and always loses the others is better for that format than one that wins each category 60% of the time, despite having a lower average expected win rate (56% vs 60%). For that reason, switching the format to Most Categories switches the definition of the overall H-score to the probability of winning a majority of categories (assuming they are independent for the sake of making the calculation less intensive).

The table above is based on the same dataset as the Each Category version. The numbers, and the order of players, are different because they use the Most Categories objective instead of the Each Category objective. With Most Categories scoring, the algorithm is even more incentivized to punt, and tends to do so to a more extreme degree. Players that benefit strongly from punting like Giannis also end up ranking better (third instead of fifth).

Rotisserie is another degree more complicated than Most Categories. See the Roto section for details

H-score details tab

The main H-score table gives only indirect insight into the strategies that H-scoring wants to use with each candidate player. The H-score details tab explains the strategy for individual players more directly. It is broken down into two parts: expectations, and future strategy.

Expectations

The first row is the same as that from the main H-scoring table, included for convenience.

The table below breaks down the components of the team, in terms of G-scores vs. the average of other teams. 'Current diff' represents the G-score differential for the draft so far, including players already drafted in the current round and excluding the candidate player. Teams that have not made their pick for the round are filled in with an estimate of the statistics of their next player. So in this case above, 'Current diff' represents other teams' first two picks vs. Giannis, with estimates for other teams that have only drafted one player. 'Future player diff' is the expected difference between future picks made by the drafter and those made by other teams, based on the strategy adopted by H-scoring. In this case the G-score for Free Throws is heavily negative because the algorithm wants to punt it with future picks. 'Current diff' plus the candidate player plus 'Future player diff' equals the total differential versus other teams, which H-scoring uses to calculate win probabilities.

The ranks show how the candidate player ranks as a pick in this situation, both by H-score and G-score. This can also be seen through the H-score or G-score main tables, and is included here as a convenience.

Future strategy

The future strategy tab shows H-scoring's strategy for future picks.

The category weightings displayed in the first row are based on H-scoring's internal model of how drafting works. It assumes that the drafter will use those weights exactly for candidates going forward, and it also assumes that those weights will have a certain influence on the aggregate statistics of future picks. One might note that in the case above, the weight for Free Throws is surprisingly high, despite the obvious fact that the algorithm is deprioritizing the category heavily. The reason for this is that in general, H-scoring does not think it needs to adjust weights all that much in order to skew the available candidates to the categories it wants.

The flex position allocations show how the algorithm expects to use its flex spots, which can take players of multiple positions. This is relevant because the algorithm understands that different positions have different statistical tendencies. In the example above, the algorithm is leaning heavily on taking Power Forwards and Centers with its flex spots, likely because they tend to have poor Free Throw rates, and that synergizes with the strategy of punting Free Throws.

The algorithm also has some leeway in how it arranges players already taken in terms of position, freeing up different positions to take with future draft picks. The roster assignment row shows what the algorithm is thinking in this regard. In the example above, it is choosing to categorize Daniels as a SF, likely because it does not want to take more SFs in general.

Optimization

Warning- math

H-scoring needs some way of making decisions about category prioritization and position allocation. It makes these decisions through an optimization process, specifically gradient descent. Each iteration of the algorithm is an iteration of gradient descent. The properties of the underlying function which H-scoring performs gradient descent on is what teaches it the strategy of punting some categories and competing in others.

The gradient of a function is derivative over multiple dimensions. As an extremely simple example, the gradient of \(x - 3y\) is \(1\) in the x direction and \(-3\) in the y direction. The gradient gives a hint at which direction the function can be minimized or maximized. The idea of gradient descent is to step in the direction of the gradient (or the opposite direction) to try to find a point which minimizes or maximizes the function.

In order for gradient descent to work, the function to be mazimized or minimized must be differentiable. This motivates the structure of the model behind H-scoring.

Roughly, the function for H-scoring has three components: category strength expectations, category-level victory probabilities, and the outer-level objective function. The decisions made by the algorithm impact the category strength expectations, which in turn impact category-level victory probabilities, which in turn impact the outer-level objective function. The total gradient relative to an input decision is the gradient of all three steps relative to the previous, multiplied together.

Category strength expectations: weights

The heart of the H-scoring algorithm is its treatment of future draft picks. Essentially, it assumes that it will be able to choose from a small slate of available players whose statistical profiles are random, conditioned on the scores being equal in terms of total G-score. It assumes that it will choose the best player available based on its choice of category weights. Using some mathematical estimations, it can calculate the expected deviation from the average for each category based on the category weights. The math behind this is quite complicated...

But the basic intuition is that there are two mechanisms at play

The more weight the algorithm assigns a category, the higher its picks's expected value for that category will be
The more specific kind of player the algorithm is looking for, the more overall value needs to be sacrificed to find that kind of player

This allows the algorithm to understand that it can prioritize or deprioritize categories with future picks, with some cost to overall value.

Category strength expectations: position allocation

The algorithm can also influence expected category strengths by deciding how it expects to divvy up future draft slots in terms of positions.

Position allocations are binary, and therefore their effects cannot be differentiated. In order to avoid costly mized-integer optimization, H-scoring treats position allocation as a small sub-problem and solves it independently. Before each round of gradient descent, the algorithm estimates how much it wants a player of each position with future picks, by multiplying the gradients relative to categories (which encode how much the algorithm has to gain from improving in those categories) by the average value of a player of that position. It assumes flex spots are slightly more valuable than the best base position. The algorithm is then allowed to assign previously chosen players to various slots, to free up future position slots for the positions it wants to take players in.

This kind of problem is called an assignment problem, because slots are being assigned to players. Its reward structure can be encoded into a matrix as shown:

There are fast algorithms available for assignment problems, such as the Hungarian algorithm (though H-scoring actually uses a faster variant).

After the sub-problem is solved, the algorithm will have a strategy for how what positions it wants to prioritize with future picks- e.g. two guards, one shooting guard, and one center. It then has a degree of freedom in how it allocates the guards between SG and PG, which it optimizes jointly with the decision on how to weigh categories. All of these deviations are added together to produce a model for decision-based adjustments to categorical strengths.

Category-level victory probability

Victory probabilities can be estimated with Normal CDFs based on team-level category averages, thanks to the Central Limit Theorem. Normal CDFs are differentiable; their gradients are Normal PDFs.

That means that the algorithm implicitly "cares" about categories according to a Normal PDF of category strength. Normal PDFs are thick in the middle and thin on the side, so the algorithm naturally cares most about categories for which it has neutral strength.

This makes intuitive sense and explains why the algorithm behaves like it does, punting some categories and trying to compete in others without going overboard. If a team is already bad at a category, it might make sense to abandon that category and punt it, since the team is likely to lose it anyway. That creates a snowball effect- the team gets worse and worse at the category. If a team is extremely strong in a category, it doesn't need the extra help there, and prioritizing it is not important. That does not create a snowball effect, since as the team gets weaker in the category, H-scoring starts to prioritize it more. That means the algorithm is incentivized to put categories into one of two buckets- punting, or competing without going overboard.

Outer-level objective function

The outer-level objective function is the function that the algorithm is trying to directly maximize. For H-scoring, it is the overall H-score.

For Each Category, the objective function is just the sum of probabilities of winning each category. That is relatively simple to optimize, since the overall gradient is just the sum of all the individual gradients.

For Most Categories scoring, the objective function is the probability of winning a majority of categories, which is more complicated. Fortunately, this operation turns out to be differentiable, so it can be optimized via gradient descent. The gradient is the same as for Each Categories except multiplied by a 'tipping point' probability, which is the likelihood that any given category will end up being decisive.

Limitations

H-scoring has numerous limitations. Some of the most major are

H-scoring is reliant on a single set of projections which may differ from the beliefs of other managers. Assuming its projections are correct, the algorithm can become overconfident and assess its own team as being so strong that the only way to improve it is to "un-punt" a category. This can lead to late round picks which run counter to the build of a team. The website does have a way to mitigate this, to a degree- see the page on the Bayesian strength adjustment
The optimization process for H-scoring only considers one strategy profile. It does not consider how robust players are to different strategy profiles, which may be relevant because circumstances can change during a draft, and the algorithm might switch strategies drastically
The internal logic of H-scoring does not understand that other drafters may also be trying to punt categories. This will lead to inaccurate projections of other teams, and therefore inaccurate projections of expected win rates
H-scoring does not model category variance based on players. Instead, it assumes that week-to-week variance is the same for all matchups. This is not always accurate, especially when a team is punting a category
H-scoring's model for what sorts of players will be available in the future is simplified, and may fail to properly account for individual players with exceptional profiles
H-scoring does not take into account the effect of streaming players, trading, etc. These all may add additional strategic considerations