Clutch Shooting Percentage (CSP): A New Cue Sports Stat (sabermetric)

[Bob Jewett]

... when you put a big number in the calculator and hit the square root button, there is no formula inside the calculator to get the answer. That's right; there is no formula there either. What happens inside your calculator is something like the blind person taking small uphill steps until the steps are not uphill anymore: then an answer displays. ...

For those who wonder how such a thing is possible, since everything in mathematics has a formula....

It is not a formula at work that gets you the square root, but it is a well-defined procedure. Square roots are often calculated by something called the Babylonian method which has been used for a few thousand years. A simple example illustrates this method:

Let's find the square root of 2. Guess that it is 1 -- almost any guess will do. Take the guess and 2 divided by the guess and take their average for the second guess:

guess2 = (1 + 2/1)/2 = 3/2 = 1.5 --- much better since we are looking for 1.414+small.

(The reason that this works is that if the guess is too small, then 2/guess will be too large by the same ratio and their average is going to almost split the difference.)

Go from guess2 to guess3 in the same way, taking the average of the guess and 2 divided by the guess:

guess3 = (3/2 + 2/(3/2)/2 = (3/2+4/3)/2 = (9/6 + 8/6)/2 = 17/12 = 1.41666... nice!

One more time:

guess4 = (17/12 + 2/(17/12)/2 = (289+288)/408 = 577/408 = 1.414215+small

which is close to the actual value of 1.414213+small

In this procedure, the number of correct digits doubles each time you turn the crank. Stop turning the crank when you have an answer that is good enough for you.

This is called an iterative method of calculation. In general, if you don't have a formula (or not an easy formula) to find an answer but you can tell how bad a guess is, an iterative method leads you to a better if still somewhat imperfect guess.

In a similar way, a list of Fargo ratings that is not quite right based on all of the recorded match results can be put into The Machine and the crank given a turn and the resulting list of Fargo ratings is better. The clever part of this is to figure out a method like the Babylonians did for square roots that makes a pretty good answer nearly perfect with only a crank or two of the handle. Mathematicians and computer scientists have spent a great deal of time figuring out how to make The Machine as efficient as possible -- FargoRate is not the only problem that the method is applied to.

 

[PoolStats]

For those who wonder how such a thing is possible, since everything in mathematics has a formula....

It is not a formula at work that gets you the square root, but it is a well-defined procedure. Square roots are often calculated by something called the Babylonian method which has been used for a few thousand years. A simple example illustrates this method:

Let's find the square root of 2. Guess that it is 1 -- almost any guess will do. Take the guess and 2 divided by the guess and take their average for the second guess:

guess2 = (1 + 2/1)/2 = 3/2 = 1.5 --- much better since we are looking for 1.414+small.

(The reason that this works is that if the guess is too small, then 2/guess will be too large by the same ratio and their average is going to almost split the difference.)

Go from guess2 to guess3 in the same way, taking the average of the guess and 2 divided by the guess:

guess3 = (3/2 + 2/(3/2)/2 = (3/2+4/3)/2 = (9/6 + 8/6)/2 = 17/12 = 1.41666... nice!

One more time:

guess4 = (17/12 + 2/(17/12)/2 = (289+288)/408 = 577/408 = 1.414215+small

which is close to the actual value of 1.414213+small

In this procedure, the number of correct digits doubles each time you turn the crank. Stop turning the crank when you have an answer that is good enough for you.

This is called an iterative method of calculation. In general, if you don't have a formula (or not an easy formula) to find an answer but you can tell how bad a guess is, an iterative method leads you to a better if still somewhat imperfect guess.

In a similar way, a list of Fargo ratings that is not quite right based on all of the recorded match results can be put into The Machine and the crank given a turn and the resulting list of Fargo ratings is better. The clever part of this is to figure out a method like the Babylonians did for square roots that makes a pretty good answer nearly perfect with only a crank or two of the handle. Mathematicians and computer scientists have spent a great deal of time figuring out how to make The Machine as efficient as possible -- FargoRate is not the only problem that the method is applied to.

Yeah, it's just an approximation algorithm like Newton's Method or the bisection method for finding roots. The key concern in all approximation methods is the starting value; a better picked starting value will lead to a better approximation.

And the reason why someone would care what the square-root button is doing on your calculator is because it because it has been peer-reviewed and verified as sound before being implemented. Imagine someone invented the calculator for the first time and said all the numbers were good, but didn't provide any details on how the algorithms worked.... would you trust it? The reason why we trust calculators is because the iterative processes has been long since verified and established.

I feel like FargoRate is probably a good approximation method, but without formal details on how the calculations are preformed, we'll never know. So it's blind trust at this point.

 

[mikepage]

Yeah, it's just an approximation algorithm like Newton's Method or the bisection method for finding roots. The key concern in all approximation methods is the starting value; a better picked starting value will lead to a better approximation.

And the reason why someone would care what the square-root button is doing on your calculator is because it because it has been peer-reviewed and verified as sound before being implemented. Imagine someone invented the calculator for the first time and said all the numbers were good, but didn't provide any details on how the algorithms worked.... would you trust it? The reason why we trust calculators is because the iterative processes has been long since verified and established.

I feel like FargoRate is probably a good approximation method, but without formal details on how the calculations are preformed, we'll never know. So it's blind trust at this point.

To say Newton's method or FargoRate's iterative approach is an "approximation method" is like an unfinished sentence that loses a key meaning. The "methods" are not approximation methods; it is individual steps of the method are approximations.

In both cases there are iterative steps that when things are working lead to successively better approximations. You keep doing this until you are satisfied. This is important because it means that you can find the solution you are looking for --whether it is the square root of 5672843 or today's optimum Fargo Ratings--pretty much to arbitrary accuracy.

What beginning at a different point does is make the journey to the same right answer a little different. It doesn't, as I fear someone might think from what you say above, change the answer.

We get the same Fargo Ratings whether we start the process with everybody rated the same or with everybody rated as they were yesterday before the latest data was added.

There are some starting points that are so bad that the iterative process blows up or gets into a big oscillating thing. There are a lot of tricks of the trade to settle things down. Once again, you either know it didn't work, or you get the answer you were looking for pretty much to the accuracy you want. And again, this is all stuff of the same level of concern to the person looking up a player rating that the person hitting the square-root button has on the calculator about what's going on inside.

 

[mikepage]

There are different maximum likelihood estimators based on the type of distribution i.e. normal, Poisson, Bernoulli. So it would help if they said what distribution they were using. But, we think Fargo is a good system anyhow and are not too concerned about it

Bernoulli: i.e., when you and I play a game, there is some chance I WIN, and if I don't win I LOSE. The different games don't depend on one another.

 

[Bob Jewett]

.. The key concern in all approximation methods is the starting value; a better picked starting value will lead to a better approximation. ...

As has been pointed out to you, this is false. You can choose for your guess of the square root of 2 the number 500 instead of 1. The Babylonian method will achieve the same correct answer but just take more cranks of the handle.

(The number of extra steps needed will be nearly equal to abs(log2(guess)) since for very large errors, the error is reduced by only a factor of 2 in each step. Negative guesses may have issues.:wink

 

[PoolStats]

To say Newton's method or FargoRate's iterative approach is an "approximation method" is like an unfinished sentence that loses a key meaning. The "methods" are not approximation methods; it is individual steps of the method are approximations.

In both cases there are iterative steps that when things are working lead to successively better approximations. You keep doing this until you are satisfied. This is important because it means that you can find the solution you are looking for --whether it is the square root of 5672843 or today's optimum Fargo Ratings--pretty much to arbitrary accuracy.

What beginning at a different point does is make the journey to the same right answer a little different. It doesn't, as I fear someone might think from what you say above, change the answer.

We get the same Fargo Ratings whether we start the process with everybody rated the same or with everybody rated as they were yesterday before the latest data was added.

There are some starting points that are so bad that the iterative process blows up or gets into a big oscillating thing. There are a lot of tricks of the trade to settle things down. Once again, you either know it didn't work, or you get the answer you were looking for pretty much to the accuracy you want. And again, this is all stuff of the same level of concern to the person looking up a player rating that the person hitting the square-root button has on the calculator about what's going on inside.

I don't mean to downplay approximation methods as they are highly accurate; but they are not exact. The whole field of Numerical Analysis is approximations whereby you are minimizing ε, or your error. There is always exists some ε away from the exact value, regardless how small.

It goes without saying that I'm in no way downplaying FargoRate. Numerical Analysis has allowed rockets to reach outer space. I took several classes in numerical analysis, and while there is rigorous proofs in NA justifying the method, it is not a "pure" math like Number Theory (my research area in school). That does not imply it is any less beneficial to society; just the opposite, most pure math benefits itself and not much else, usually for a long time, if at all, ever.

In mathematics one has integration and numerical integration; they are two separate beasts. I suppose the term used in mathematics to call these methods "approximation algorithms" doesn't do justice to it's accuracy, but in the field of mathematics, that's exactly what they are.

Also to clarify, that is what I meant by a better starting value, that iterative processes can grow exponentially or oscillate between two numbers indefinitely. And yes, there are techniques to stabilize these unfortunate occurrences.

 

[mikepage]

Another mathematical mystery... hard pass.

It's 2020 friend, nobody is going to accept that a simple formula should be closed. The only reason Fargo is getting away with it is because American Pool isn't currently popular enough for enough people to care.

FargoRate doesn't have any simple formula that is closed. We don't hide the ball. A data scientist who does machine-learning type stuff would --based on what we describe-- be able to come up with the same ratings from the same match data. And we only include public match data that is available for all to see.

I work pretty hard to try to help people understand in an honest way what we do. Here are some equations from a white paper I put out a decade ago. I could whip stuff like this out whenever someone asks a question, but I don't think it aids in understanding.

If your objection is

 

[Patrick Johnson]

I could whip stuff like this out whenever someone asks a question, but I don't think it aids in understanding.

Sometimes you need to whip it out to establish your credentials.

pj <- always works for me
chgo

 

[PoolStats]

Bernoulli: i.e., when you and I play a game, there is some chance I WIN, and if I don't win I LOSE. The different games don't depend on one another.

Not to rehash old things, but this has bugged me for a long time. My only gripe with this is that it will be accurate in alternate break; however, in a winner breaks format, the winner of one rack gains the initiative in the next rack, so the subsequent games are not independent of one another. This is important, especially among the higher end of the fargo. If I'm not mistaken, this would skew the ratings of those that played more winner breaks formats and won the match a bit higher than they should be. I don't know the entire process of FargoRate good enough to derive an estimate at which the dependent games alters the overall rating.


You may be able to use our Break & Run % stats to arrive at some estimate on the effect of the probabilistic outcome in winner break formats, though, I must confess, we simply do not have enough data on many players to reach a sound conclusion.

http://poolst.at/br10
http://poolst.at/br9

Another organization with more compressive b&r stats would prove effective until we get there at least. And, of course, breaking and running isn't the only component in winning a game given the initiative.

 

[straightline]

Needs to be weighted with: why are they having to shoot out of a hole? .

 

[Straightpool_99]

Ronnie was being slightly disingenuous, I think. Sometimes he seems to make the white ball go where he might have a good chance with the next try. Or maybe he just gets lucky a lot.

The best position player of all time, IMO, and even the best cue ball control overall. Of course this is snooker, in the game of pool Efren rules supreme.

Ronnie O'Sullivan definitely benefitted from being instructed by Ray Reardon, who helped him with his tactical game. So he is being disingeneous, he is a true student of the game. Of course in snooker, the culture is one of modesty. It is frowned upon to be boastful, though there have been some players like that, they usually got a lot of flak for it. Ronnie tends to downplay his own ability and knowledge. https://www.eurosport.co.uk/snooker...on-transformed-my-game_sto7832296/story.shtml

 

[mikepage]

Not to rehash old things, but this has bugged me for a long time. My only gripe with this is that it will be accurate in alternate break; however, in a winner breaks format, the winner of one rack gains the initiative in the next rack, so the subsequent games are not independent of one another. This is important, especially among the higher end of the fargo. If I'm not mistaken, this would skew the ratings of those that played more winner breaks formats and won the match a bit higher than they should be. I don't know the entire process of FargoRate good enough to derive an estimate at which the dependent games alters the overall rating.


You may be able to use our Break & Run % stats to arrive at some estimate on the effect of the probabilistic outcome in winner break formats, though, I must confess, we simply do not have enough data on many players to reach a sound conclusion.

http://poolst.at/br10
http://poolst.at/br9

Another organization with more compressive b&r stats would prove effective until we get there at least. And, of course, breaking and running isn't the only component in winning a game given the initiative.

The real problem here is not winner vs alternate breaks. It's that the game sucks when too often a winner is chosen without any player-player interaction. That's a game flaw. The problem you are identifying is just as large in an alternate break format.

Suppose SVB and Gorst are equally skilled players and equal breakers. First imagine they play 10-Ball races to 11 on a tight enough table that they run out less than 10% of the time or for which a roll out is required every game. They will split sets and maybe the most common score will be 11 - 8 or 11 - 7 one way or the other. It won't matter whether they do winner or alternate break. After 100 races, the game score might be close to 925 to 925.

Now instead have them play winner breaks bar-box 9-Ball with a template rack. There will be very few close scores and a bunch of 11 - 0 scores one way or the other. They will split the sets and after 100 sets the game score might be 700 to 700.

With alternate break, there will be lots of 11-10 scores one way or the other and rarely will the loser have less than 9. The score after 100 races might be close to 1025 to 1025. The games-not-independent runs still exist in the alternate break format. They just involve both players.

We call this the run-length issue. We need to stop being stupid about the roll of the break in our game.