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rotosource wrote:In a couple of threads the issue of valuation in the percentage categories has come up, so I wanted to try to establish this for future discussions – it is impossible to come up with a simple, additive valuation system for percentages (FG %, FT %) that will follow the same rules that a valuation system for absolute categories (Points, Rebounds, Assists, etc.) will follow. Not just difficult – actually impossible.

Let’s first establish what we would expect from a valuation system – and by that I mean a process by which we would assign a number value to each player for each category that would express the contribution of that player in that category in a rotisserie scoring fantasy league.

(1) We should expect that if a player value A in a given category is greater than player value B in that same category, then a team made up entirely of player A should perform better in that category than a team made up entirely of player B. The converse should also be true - if a team made of Player A alone performs better than a team of Player B alone, then A's value should be larger than B's.

This aspect of a rating system is pretty basic. Player values have to correlate to performance, or what good are they? In case anyone is fretting about the fact that one man teams are weird, just pretend that we instead have a full team made up of similar players.

(2) Values should be additive. Define a team’s value in a given category to be equal to the sum of the values of that team’s players in that category. If Team Alpha’s value in a given category is greater than Team Beta’s value in that same category, then Team Alpha should perform better than Team Beta in that category. The converse should also be true - if Team Alpha performs better than Team Beta, the Team Alpha's value should be greater than Team Beta's.

People might be able to argue with this requirement, but the simple fact is that I don’t know anyone who has actually drafted using player values which were not intended to satisfy this requirement. The whole point of creating player values is to determine how much a player contributes to team success (or failure) in a given category, and addition of player values is the usual method, as it works quite well for categories like points, rebounds, blocks, turnovers, assists and steals – the vast majority of the categories. The fact that these categories themselves are additive is the reason that they work – a team’s blocks is equal to the sum of the blocks of each player. Percentages don’t work that way – which is why an additive system won’t work – but I’m getting ahead of myself.

There are other requirements we would expect as well, but the critical ones are (1) and (2) listed above.

We define a team’s performance in a percentage category to be the standard formula: (total team makes)/(total team attempts). This is also true for a team of one person. If this quotient is larger for Team Alpha than for Team Beta, then Team Alpha is said to perform better than Team Beta in that category.

I’m assuming there isn’t a lot of argument so far – when we’re looking for a player valuation system that covers the percentages, we’re pretty much looking for something that satisfies the above requirements. Let us take four imaginary players and their season-long statistics in a given percentage category:

Player A: 190 makes, 200 attempts

Player B: 450 makes, 500 attempts

Player C: 300 makes, 600 attempts

Player D: 40 makes, 100 attempts

Let us assume that Player A has a value av in this category, Player B has a value bv in this category, Player C has a value cv in this category and Player D has a value dv in this category.

According to our performance rules, a team of Player A would beat a team of Player B, as 95% > 90%. Therefore, av > bv, because of requirement 1.

According to our performance rules, a team of Player C would beat a team of Player D, as 50% > 40%. Therefore, cv > dv, because of requirement 1.

We know that av + cv > bv + dv as a consequence of these two statements because this is one of the axioms of real numbers – which our values purport to be.

Therefore Team Alpha, made up of players A and C, should perform better than Team Beta, made up of players B and D, because requirement 2 dictates that Team values correspond to overall team performance, and av + cv and bv + dv are defined to be the Team values for Teams Alpha and Beta respectively. However,

Team Alpha’s percentage is (190 + 300)/(200 + 600) = 61.25%

Team Beta’s percentage is (450 + 40)/(500 + 100) = 81.67%

so Team Beta would beat Team Alpha in this category, which is a contradiction. Therefore, there are no such real values av, bv, cv, and dv that satisfy our requirements. If anyone thinks they have a way to estimate value in one of the percentage categories, then please indicate what av, bv, cv, and dv should be in the above example.

Just because we cannot get a perfect valuation system for the percentages doesn’t mean we should stop trying. There are a few ways to calculate values for percentages that work in most cases. You can calculate player value in a percentage category by first determining player value without reference to attempts, via the following formula:

(Player percentage – league-wide percentage)/(standard deviation in percentage), and then multiply this value by a ratio of (player attempts)/(average player attempts). This one works OK. I personally like to estimate the difference between the number of makes and the number of makes which would be expected given the player's number of attempts and a percentage equal to the league average. Then I estimate values from that in the same way one would for points or any other category, because I have reduced the category to an aggregate number that can properly be added or subtracted, and so fits into an additive system.

Note - these systems will work most of the time, but they won't always predict relative performance accurately. They can't.

Don't get into this one unless you have quite a bit of time:

Chrisy Moltisanti wrote: This one has given me my 'favorite' results so far Linkmamorris wrote:The (%s) formula for weighted standard deviation is:

= (sumproduct(fgpct, fgpct, fga)/sum(fga) - [weighted average] ^2) ^0.5

I almost feel like saying "remember kids, don't try this at home".

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geodbear wrote:I've been toying around with a way to compare players' percentages taking into account attempts, and tried the following.

First I assumed 12 teams, 13 players and 10 active roster spots. I figure out the average number of attempts a player takes per game, call it A. Then compute the average percent, call it P. Suppose a Player takes X attempts per game with a percentage of Y. So for a team of active 9 players, with average attempts and percentage, the Player affects the team percentage to be

(9*A*P+X*Y)/(9*A+X)

Now I want to figure out what the percentage, call it Z, that the Player would have to have if he shoots an average number of shot attempts per game to have the same affect as he would shooting X attempts with a percentage of Y. So assuming A attempts with Z percentage, I have the team percentage as

(9*A*P+A*Z)/(9*A+A)=(9*P+Z)/10

and set that equal to the first expression to get

(9*A*P+X*Y)/(9*A+X)=(9*P+Z)/10

Solving for Z,

Z = 10*(9*A*P+X*Y)/(9*A+X)-9*P

If I compute Z are all players, I can rank players by their Z percentage. Would it be reasonable to then use the standard deviation method of valuation on the new percentage? Any thoughts on how accurate that might be?