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Hey! This repository deals with fantasy basketball. If you are unfamiliar with how it works, here are some useful links

Improving Z-scores for H2H fantasy

Fantasy basketball has a standard way of quantifying player value across categories, called 'Z-scoring', and it is used to make objective rankings of players. However, as far as I know, nobody has ever laid out exactly why Z-scores should work. They just seem intuitively sensible, so people use them.

I looked into the math and did manage to derive a justification for Z-scores. However, the justification is only appropriate for the "Rotisserie" format. When the math is modified for head-to-head formats, a different metric that I call "G-score" pops out as the optimal way to rank players instead. I wrote a paper to that effect a few months ago which is available here.

I realize that the paper's explanation is incomprehensible to anyone without a background in math. To that end, I am providing a simplified version of the argument in this readme, which hopefully will be easier to follow

1. What are Z-scores?

You may have come across Z-scores in a stats 101 class. In that context, they are what happens to a set of numbers after subtracting the mean (average) signified by $\mu$ and dividing by the standard deviation (how “spread out” the distribution is) signified by $\sigma$. Mathematically, $Z(x) = \frac{x - \mu}{\sigma}$

For use in fantasy basketball, a few modifications are made to basic Z-scores

  • The percentage categories are adjusted by volume. This is necessary because players who shoot more matter more; if a team has one player who goes $9$ for $9$ ($100\%$) and another who goes $0$ for $1$ ($0\%$) their aggregate average is $90\%$ rather than $50\%$. The fix is to multiply scores by the player's volume, relative to average volume
  • $\mu$ and $\sigma$ are calculated based on the $\approx 156$ players expected to be on fantasy rosters, rather than the entire NBA

Denoting

  • Player $p$'s weekly average as $m_p$
  • $\mu$ of $m_p$ across players expected to be on fantasy rosters as $m_\mu$
  • $\sigma$ of $m_p$ across players expected to be on fantasy rosters as $m_\sigma$

Z-scores for standard categories (points, rebounds, assists, steals, blocks, three-pointers, and sometimes turnovers) are

$$ \frac{m_p - m_\mu}{m_\sigma} $$

The same definition can be extended to the percentage categories (field goal % and free throw %). With $a$ signifying attempts and $r$ signifying success rate, their Z-scores are

$$ \frac{\frac{a_p}{a_\mu} \left(r_p - r_\mu \right)}{r_\sigma} $$

See below for an animation of weekly blocking numbers going through the Z-score transformation step by step. First the mean is subtracted out, centering the distribution around zero, then the standard deviation is divided through to make the distribution more narrow. Note that a set of $156$ players expected to be on fantasy rosters is pre-defined

BlockVisv2.mp4

Adding up the results for all categories yields an aggregate Z-score

2. Justifying Z-scores for Rotisserie

It is impractical to calculate a truly optimal solution for Rotisserie or any other format, since they are so complex. However, if we simplify the Rotisserie format, we can at least demonstrate that Z-scores are a reasonable heuristic for it

A. Assumptions and setup

Consider this problem: Team one has $N-1$ players randomly selected from a pool of players, and team two has $N$ players chosen randomly from the same pool. Which final player should team one choose to optimize the expected value of categories won against team two, assuming all players perform at exactly their long term mean for a week?

This problem statement makes a few implicit simplifications about Rotisserie drafts

  • The goal is to maximize the expected value of the number of categories won against an arbitrary opponent in a single week, where all players perform at their season-long means. Using weekly means instead of season-long totals is just a convenience, to align with the definition of Z-scores. And optimizing for victory against an arbitrary opponent is equivalent to optimizing for total score at the end of a season, since each category victory over any opponent is worth one point
  • Besides the player being drafted, all others are assumed to be chosen randomly from a pool of top players. This assumption is obviously not exactly true. However, it is somewhat necessary because we are trying to make a ranking system which does not depend on which other players have been drafted. It is also not as radical as it may seem, since real teams have a mix of strong and some weak players chosen from a variety of positions, making them random-ish in aggregate
  • Position requirements, waiver wires, injury slots, etc. are ignored. Drafters use their drafted players the whole season

The simplified problem can be approached by calculating the probability for team one to win each category, then optimizing for their sum

B. Category differences

The difference in category score between two teams tells us which team is winning the category and by how much. By randomly selecting the $2N -1$ random players many times, we can get a sense of what team two's score minus team one's score will be before the last player is added. See this simulation being carried out for blocks below with $N=12$

SimulationVisv2.mp4

You may notice that the result looks a lot like a Bell curve even though the raw block numbers look nothing like a Bell curve. This happens because of the surprising "Central Limit Theorem", which says that when adding a bunch of random numbers together, their sum always ends up looking a lot like a Bell curve. This applies to all the other categories as well

C. Properties of the category difference

The mean and standard deviation of the Bell curves for category differences can be calculated via probability theory. Including the unchosen player with category average $m_p$

  • The mean is $m_\mu - m_p$
  • The standard deviation is $\sqrt{2N-1} * m_\sigma$ (The square root in the formula comes from the fact that $STD(X + Y) = \sqrt{STD(X)^2 + STD(Y)^2}$ where $STD(X)$ is the standard deviation of $X$)

D. Calculating probability of victory

When the category difference is below zero, team one will win the category

The probability of this happening can be calculated using something called a cumulative distribution function. $CDF(x) =$ the probability that a particular distribution will be less than $x$. We can use $CDF(0)$, then, to calculate the probability that the category difference is below zero and team one wins.

The $CDF$ of the Bell curve is well known. The details of how to apply it to this case are somewhat complicated, but we can cut to the chase and give an approximate formula

$$ CDF(0) = \frac{1}{2}\left[ 1 + \frac{2}{\sqrt{\pi}}* \frac{- \mu }{ \sigma} \right] $$

We already know $\mu$ and $\sigma$ for the standard statistics. Substituting them in yields

$$ CDF(0) = \frac{1}{2}\left[ 1 + \frac{2}{\sqrt{(2N-1) \pi}}* \frac{m_p – m_\mu}{m_\sigma} \right] $$

And analagously for the percentage statistics

$$ CDF(0) = \frac{1}{2} \left[ 1 + \frac{2}{\sqrt{(2N-1) \pi}} * \frac{ \frac{a_p}{a_\mu} \left( r_p – r_\mu \right) }{r_\sigma}\right] $$

D. Implications for Z-scores

The last two equations included Z-scores. Adding up all the probabilities to get the expected number of categories won by team one, with $Z_{p,c}$ as player $p$'s Z-score for category $c$, the formula is

$$ \frac{1}{2}\left[9 + \frac{2}{\sqrt{(2N-1) \pi}} * \sum_c Z_{p,c} \right] $$

It is clear that the expected number of category victories is directly proportional to the sum of the unchosen player's Z-scores. This tells us that under the aforementioned assumptions, the higher a player's total Z-score is, the better they are for Rotisserie

3. Modifying assumptions for Head-to-Head

"Head-to-Head: Each Category" is deceptively similar to Rotisserie, in the sense that winning one category against one opponent is worth one point. Yes, head-to-head matchups are one at a time rather than simultaneous, but that doesn't matter when the goal is just to do as well as possible against an arbitrary opponent. The main substantive difference between the two formats is that head-to-head matchups occur over a single week, rather than over an entire season. This is important because it means that players don't necessarily perform at their season-long averages for any given matchup. Instead, their performances are somewhat random, depending on how they happen to perform during the week of the matchup.

For Rotisserie, we handled uncertainty about which other players would be chosen by assuming they were chosen randomly. To extend this for head-to-head, we can assume that their performances are randomly chosen as well. Effectively, the revamped assumption is that we are randomly choosing player/weekly performance combos from a set of top players and their performances for a season, rather than just choosing a player and taking their average. Below, see how metrics for blocks change when we look at every weekly performance of the top $156$ players, instead of just their averages

ComparisonVis.mp4

Although the mean remains the same, the standard deviation gets larger. This makes sense, because week-to-week "noise" adds more volatility, which is reflected in the additional $m_\tau$ term. Note that the new standard deviation is $\sqrt{m_\sigma^2 + m_\tau^2}$ rather than $m_\sigma + m_\tau$ because of how standard deviation aggregates across multiple variables, as discussed in section 2B

4. Formulating G-scores

Most of the logic from section 2 can also be applied to Head-to-Head: Each Category. The only difference is that we need to use metrics from the pool of players and performances, as laid out in section 3, rather than just players as we did in section 2. The mean is still $m_\mu$ as shown in the example of blocks above. Therefore all we need to do is replace $m_\sigma$ with $\sqrt{m_\sigma^2 + m_\tau^2}$, which yields

$$ \frac{m_p – m_\mu}{\sqrt{m_\sigma^2 + m_\tau^2}} $$

And analagously for the percentage statistics,

$$ \frac{\frac{a_p}{a_\mu} \left( r_p – r_\mu \right) }{\sqrt{r_\sigma^2 + r_\tau^2}} $$

I call these G-scores, and it turns out that these are quite different from Z-scores. For example, steals have a very high week-to-week standard deviation, and carry less weight in G-scores than Z-scores as a result.

Intuitively, why does this happen? The way I think about it is that investing heavily into a volatile category will lead to only a flimsy advantage, and so is likely less worthwhile than investing into a robust category. Many drafters have this intuition already, de-prioritizing unpredictable categories like steals relative to what Z-scores would suggest. The G-score idea just converts that intuition into mathematical rigor

5. Head-to-head simulation results

Our logic relies on many assumptions, so we can't be sure that G-scores work in practice. What we can do is simulate actual head-to-head drafts and see how G-score does against Z-score.

The code in this repository simulates a simplistic version of head-to-head fantasy basketball, via a $12$ team snake draft. It doesn't include advanced strategies like using the waiver wire or punting categories, but for the most part it should be similar to real fantasy. For more detail, check out the code or the paper.

The expected win rate if all strategies are equally good is $\frac{1}{12} = 8.33\%$. Empirical win rates are shown below for Head-to-Head: Each Category 9-Cat, which includes all categories, and 8-Cat, a variant which excludes turnovers

G-score vs 11 Z-score Z-score vs. 11 G-score
9-Cat
2021 $15.9\%$ $1.5\%$
2022 $14.3\%$ $1.3\%$
2023 $21.7\%$ $0.4\%$
Overall $17.3\%$ $1.4\%$
8-Cat
2021 $10.7\%$ $2.9\%$
2022 $12.0\%$ $1.5\%$
2023 $15.4\%$ $0.9\%$
Overall $12.7\%$ $1.8\%$

When interpreting these results, it is important to remember that they are for an idealized version of fantasy basketball. Still, the dominance displayed by G-scores in the simulations suggests that the G-score modification really is appropriate.

To confirm the intuition about why the G-score works, take a look at its win rates by category against $11$ Z-score drafters in 9-Cat

G-score win rate
Points $77.7\%$
Rebounds $70.8\%$
Assists $81.6\%$
Steals $25.7\%$
Blocks $44.9\%$
Three-pointers $77.3\%$
Turnovers $16.2\%$
Field goal % $34.9\%$
Free throw % $40.6\%$
Overall $52.2\%$

The G-score drafter performs well in stable/high-volume categories like assists and poorly in volatile categories like turnovers, netting to an average win rate of slightly above $50\%$. As expected, the marginal investment in stable categories is worth more than the corresponding underinvestment in volatile categories, since investment in stable categories leads to reliable wins and the volatile categories can be won despite underinvestment with sheer luck.

Simulations also suggest that G-scores work better than Z-scores in the Head-to-Head: Most Categories format. I chose not to include the results here because it is a very strategic format, and expecting other drafters to go straight off ranking lists is probably unrealistic for it.

Another possible use-case is auctions. There is a well-known procedure for translating player value to auction value, outlined e.g. in this article. If the auction is for a head-to-head format, it is reasonable to use G-scores to quantify value rather than Z-scores

Addendum: Further improvement

Any situation-agnostic value quantification system is suboptimal, since a truly optimal strategy would adapt to the circumstances of the draft/auction.

In the paper, I outline a methodology called H-scoring that dynamically chooses players based on the drafting situation. It performs significantly better than going straight off G-score and Z-score. However, it is far from perfect, particularly because it does not fully understand the consequences of punting. There is a lot of room for improvement, and I hope that I, or someone else, can make a better version in the future!