The role of variance in billiards

[JarnoV]

Fiddling around with Mike Page's Fargo Ratings made me wonder whether anyone has done any analysis on the role of variance ("luck", if you will) in billiards. That is, how much inherent fluctuation there is in results of pool billiard games (or other disciplines, for that matter). Obviously there's more luck involved in say 9 ball than there is in chess, but compared to, say, poker, has much less variance. (Though poker is obviously a skill-game too. It just has a way larger variance.)

As an example, consider two players that have a Fargo Rating difference of 100 points. By Mike's system, this means that the likelihood of the worse player winning a frame (a game, I guess I should call it) is 1/3 and 2/3 for the better player. This means that in race to six wins, the most likely result is 6-3 for the better player.

If the variance is very high, it would mean that in practice some of the matches are won by the worse player and more by the better player, but in average it would come out as 6-3 for the better player. In contrast, if the variance is very low, most results of matches would be just around 6-3. Some matches would end 6-2 and some 6-4, but it would be very unlikely that the worse player wins.

What I'm looking for is some way to describe or quantify the variance of pool billiards. Let's take as an example some pro tournament. I think it's fair to say that there is luck involved. You can't win by luck, of course, but in match where two players are pretty equally skilled, the luck plays sometimes a significant factor. Two players can break the rack equally good, but the other one landing on the smallest ball in 9-ball and so forth.

I haven't thought this through yet so I'm not sure if I make any sense, but I was interested if anyone else has thought about this and perhaps wrote about it too. I'm at a point where I'm not quite sure what the exact question is that I'm looking for, but I feel like analysis of the variance could bring some interesting comparisons, say between disciplines or how the amount of frames needed in a match changes the variance.

Edit: I guess I could just use Fargo Ratings and determine the standard deviation of a single game based on real results and go from there.

 

[1on1pooltournys]

I've thought about it.....

It is too hard to measure the amount of luck because of the many different variables. Whereas in other games, say poker, the variables are generally always the same like the number of cards in a deck, the hands, and other things. Same goes for a lot of other gambling games.

I always just try to reinforce the notion that if I play good I will get good rolls....but that isn't always the case!! :wink::thumbup::grin:

 

[Beer:30]

My take on it would be look at the variance of the players rather than the entire game. Table position is what the individual makes of it. The higher the skill level the more games necessary to determine the winner. You might be able to model opening breaks to the point where you could determine some variables that are correlated with winning but it would be a very labor intensive endeavor. I'd be willing to bet that the higher skill level player would have a higher correlation to winning than most table position related variables. Then there would be the question of the results being significant or not.

Finding player variance would be pretty easy with some game sheets. Although I would be interested to see the results of the above investigation I doubt it's going to get done until there is more money in pool ..... or there is a bored statistics grad student.

 

[pocketspeed]

we'll probably never know the answer to your question simply b/c pool doesnt keep stats the way other sports do. i have always felt that luck really doesnt come into play when pool is being played at a high level. pple speak of "rolls" etc but isnt that just a way to excuse a bad shot. when you play youre going to get what you shoot. the better you shoot the better your results. no luck involved. experience also helps in shot selection and handling the most random aspect of the game-the break. but even the break can become a "skill shot" as well. i have always felt we make our own luck and i think the same is true in pool.

brian

 

[deebee53]

With a large enough sample size, 99% of all variation will be inherent in the system and thus accounted for in their rating.

Someone who is "lucky" will have a rating that accounts for it. They will either have to be consistently lucky or their rating will decrease.

In the long term, the results of their matches will determine their rank, regardless of the amount of luck involved within those matches.

 

[JarnoV]

With a large enough sample size, 99% of all variation will be inherent in the system and thus accounted for in their rating.

Someone who is "lucky" will have a rating that accounts for it. They will either have to be consistently lucky or their rating will decrease.

In the long term, the results of their matches will determine their rank, regardless of the amount of luck involved within those matches.

I think I understand what you're saying, but that's not what I'm looking for. I'm not worried that some have a higher rating than they "should have". I'm mostly interested in quantifying the amount of luck. Let's say we have relatively accurate Fargo Ratings (or any other for that matter) for the players. What I'm interested in is how wide or narrow are the results going to distribute around the most likely result.

But now that I've had time to think of it, I think I can do it myself. And if not, I'll come back asking stupid questions. ;-)

 

[dr_dave]

Good questions.

Maybe you and Mike can look at some of his current league and tournament data and come up with some variance results. That would be interesting to see.

Thanks,
Dave

Fiddling around with Mike Page's Fargo Ratings made me wonder whether anyone has done any analysis on the role of variance ("luck", if you will) in billiards. That is, how much inherent fluctuation there is in results of pool billiard games (or other disciplines, for that matter). Obviously there's more luck involved in say 9 ball than there is in chess, but compared to, say, poker, has much less variance. (Though poker is obviously a skill-game too. It just has a way larger variance.)

As an example, consider two players that have a Fargo Rating difference of 100 points. By Mike's system, this means that the likelihood of the worse player winning a frame (a game, I guess I should call it) is 1/3 and 2/3 for the better player. This means that in race to six wins, the most likely result is 6-3 for the better player.

If the variance is very high, it would mean that in practice some of the matches are won by the worse player and more by the better player, but in average it would come out as 6-3 for the better player. In contrast, if the variance is very low, most results of matches would be just around 6-3. Some matches would end 6-2 and some 6-4, but it would be very unlikely that the worse player wins.

What I'm looking for is some way to describe or quantify the variance of pool billiards. Let's take as an example some pro tournament. I think it's fair to say that there is luck involved. You can't win by luck, of course, but in match where two players are pretty equally skilled, the luck plays sometimes a significant factor. Two players can break the rack equally good, but the other one landing on the smallest ball in 9-ball and so forth.

I haven't thought this through yet so I'm not sure if I make any sense, but I was interested if anyone else has thought about this and perhaps wrote about it too. I'm at a point where I'm not quite sure what the exact question is that I'm looking for, but I feel like analysis of the variance could bring some interesting comparisons, say between disciplines or how the amount of frames needed in a match changes the variance.

Edit: I guess I could just use Fargo Ratings and determine the standard deviation of a single game based on real results and go from there.

 

[12310bch]

Right on.

With a large enough sample size, 99% of all variation will be inherent in the system and thus accounted for in their rating.

Someone who is "lucky" will have a rating that accounts for it. They will either have to be consistently lucky or their rating will decrease.

In the long term, the results of their matches will determine their rank, regardless of the amount of luck involved within those matches.

A large sampling of individual ratings will produce a ," bell shaped curve, "
thus a standard deviation. In a large sampling , the effect of luck
disappears as it is shared equally by opponents.

 

[jay helfert]

I have always said that pool is about 90% skill and 10% luck, as opposed to poker which is about 50% skill and 50% luck. JMHO naturally.

 

[pocketspeed]

your results will follow the premise that "luck" is involved in the outcome of a game/match. is it really? isnt one person's luck another person's skill?

heres an example: out team was shooting in league. My son was playing their best player. he ran to the 8 (playing 8ball) but left himself impossible shape. bad luck or roll? or poor shooting? he played a bad safe leaving the cb behind the 8 but on a direct line with my son's last ball at the 1st diamond at the far corner pocket. the other team is all high fiving in the "great safe" just played. my son calmly grabs his jump cue goes airborn over the 8 pockets his last ball an shoots the 8 for the win (and also eliminating that team from being able win the session). now we are high fiving and amost came to blows when my son's opponent started making loud comments about an "effing lucking jump shot". did luck have anything to do with the outcome of the match? no. bad position and a bad safety left my son (who plays the jump shot quite well) a shot well within his skill set. he executes and wins. "luck" in pool is just an excuse for bad play or lack of necessary skills.

brian

 

[12310bch]

come to blows?????????????

your results will follow the premise that "luck" is involved in the outcome of a game/match. is it really? isnt one person's luck another person's skill?
brian

Only in the eyes of the loser. Skill and luck are two different, separate
concepts. Skill variable. Luck constant. The fact that the balls lined up so your son could demonstrate his shot was luck. Your son's shot was skill. Let the idiots think it was luck. Don't argue the point. They can't argue who
won. Be smug.

 

[JarnoV]

"luck" in pool is just an excuse for bad play or lack of necessary skills.

I hear what you're saying, but that's strictly speaking not what I'm talking about. You're saying that no one should be attributing a loss to bad luck, because they could have just played better. And I agree with that.

But to say that there is no luck involved is just plain wrong. Let me prove that to you.

Let's say two players you know play a match. You know their game and happen to know that player Chris is clearly better player than Daniel, especially in the setting they are going to play. Now, unknown to them, you're making a sizable bet on the who wins the match and you obviously pick Chris for your horse. But adding to this, you can decide whether the match is played first to win one game or first to win ten games.

If you're rational and don't hate money, you'd obviously pick the match to be to ten games. Why? Because the probability that Chris wins the whole match in a race to one compared to race to ten is much lower. In a race to one game, Chris is still a sizable favorite, but he's even bigger when it's race to ten. Even if he's a good player, he can miss some balls and that might make Daniel win the match if it's played to one game. But if their skill difference is sizable, then in a race to ten games, Daniel basically has no chance at all.

This difference in the probability of Chris winning the match is what I mean by variance or luck.

If you look at match itself, you can surely say that in a race to one game when Chris missed a ball that it was not bad luck, it was his skills failing. But if you look at the offer from the point of view of the person betting the match and deciding whether it's played to one or ten games, it is obvious that it's way more rational to decide that they play to ten games.

 

[pocketspeed]

I hear what you're saying, but that's strictly speaking not what I'm talking about. You're saying that no one should be attributing a loss to bad luck, because they could have just played better. And I agree with that.

But to say that there is no luck involved is just plain wrong. Let me prove that to you.

Let's say two players you know play a match. You know their game and happen to know that player Chris is clearly better player than Daniel, especially in the setting they are going to play. Now, unknown to them, you're making a sizable bet on the who wins the match and you obviously pick Chris for your horse. But adding to this, you can decide whether is played first to win one game or first to win ten games.

If you're rational and don't hate money, you'd obviously pick the match to be to ten games. Why? Because the probability that Chris wins the whole match in a race to one compared to race to ten is much lower. Even if he's a good player, he can miss some balls and that might make Daniel win the match if it's played to one game. But if their skill difference is sizable, then in a race to ten games, Daniel basically has no chance at all.

This difference in the probability of Chris winning the match is what I mean by variance or luck.

If you look at match itself, you can surely say that in a race to one game when Chris missed a ball that it was not bad luck, it was his skills failing. But if you look at the offer from the point of view of the person betting the match and deciding whether it's played to one or ten games, it is obvious that it's way more rational to decide that they play to ten games.

yes i see your point, but then the variance you speaking of would be more along the lines of statistical probablities rather that random occurances. I am a d player (on good days). if i play an open player statisically i should lose any lengthy race, however also statisically in any lengthy race i should win a game or two.

you can always look on paper and see statisically who should win-be the better player or team, but upsets do occur which is why the game is played. this variance in the expected outcome could be based on may factors (better player sick, mentally distracted by real life and cant focus etc...) which some pple may interpret as luck "wow he was lucky to catch him on a bad day) but in fact wasnt luck at all. you just played better on that given day even if statisically you should have lost. i think it would be difficult to quantify this.

brian

 

[Tramp Steamer]

There is a line in a Steven Seagall movie (the one with the terrorists on the train) where the main badguy states that luck is 'the residue of success'. I tend to agree with that. The next time you are playing someone who seems to be getting all the rolls, look at how they are playing and you wll probably see that they are pretty much mistake free. By the same token, when someone is playing poorly (unsucessfully) they can't get any luck at all. So there you have it.
Now, if you will excuse me, I have some unsucessful One Pocket to play this afternoon. :wink:

 

[12310bch]

.
Now, if you will excuse me, I have some unsucessful One Pocket to play this afternoon. :wink:

"It seems to me that you have only one question to ask; ' Do I feel Lucky?'
Well, do you Tramp?" I guess that if your going to play you, "just gotsta' know." :killingme:

 

[CreeDo]

It's hard to quantify luck, that's why it's luck. It's a deviation (sometimes a violent one) from the statistical norm. A guy can catch 6 horrible rolls in one night or he can keep lining up 9-ball combos. He can get mildly lucky one night and incredibly lucky the next night and there's no way to measure, chart, or predict it.

There's also the issue of figuring out what was luck and what was skill. The player probably won't inform you. You sure won't be able to tell just by looking at the stat sheet. Maybe he was just kicking really badly or breaking really well on a given night. They say the better you play, the more lucky you get.

I think the best you can do is chart someone's playing over a long time and then compare that to a specific game or set or tournament. If they claim they were playing "normally" and you take their word for it, how they perform in that tournament vs. how they usually perform could be chalked up to luck, good or bad.

-----

I think if your goal is to figure out the luckiest games, you can just look at the rules to see what will give the most luck.

- There's more room for good luck in a game with slop than a game with called shots.
- More room for luck in a game with a hard break vs. a game with a soft break.
- More luck in games where it's possible to win on the break or with an early shot vs. a game where all the balls must be run out before the gamewinning ball.
- More luck in a game with less congestion vs. more congestion
- More luck when you can shoot into any pocket vs. only one
- More luck when fewer balls are needed to win vs. many balls

 

[Beer:30]

When players are good enough at a specific game that they can step up to the table and have a significant chance of running out you run into a situation where it's suddenly not the best player that has the chance to run out but the player who is at the table. Once this level is reached by both players you need longer and long sets to weed out the better player. The game doesn't allow enough back and forth within a single game to let a superior player elevate within a single game. 14.1 is a much better game for finding true speed in a single game (at least with respect to some skills) because the game is much longer and the better player has a chance to overcome any significant advantages imposed by "lucky" setup.

By "lucky" I'm talking specifically about favorable ball orientation and not things like lucky or bad rolls. I'll go as far as to say that the break is the most pivotal point in the game where randomness or "luck" is concerned.

(Qualification on table condition: Non-uniform playing surface does impose a randomizing effect on play. Conditions such as bad or dirty cloth will throw a shoe into the predictable ball rolling machinery as will dead rails. On a bad table you may need even more games to find a statistically significant better player.)

 

[pocketspeed]

When players are good enough at a specific game that they can step up to the table and have a significant chance of running out you run into a situation where it's suddenly not the best player that has the chance to run out but the player who is at the table. Once this level is reached by both players you need longer and long sets to weed out the better player. The game doesn't allow enough back and forth within a single game to let a superior player elevate within a single game. 14.1 is a much better game for finding true speed in a single game (at least with respect to some skills) because the game is much longer and the better player has a chance to overcome any significant advantages imposed by "lucky" setup.

By "lucky" I'm talking specifically about favorable ball orientation and not things like lucky or bad rolls. I'll go as far as to say that the break is the most pivotal point in the game where randomness or "luck" is concerned.

(Qualification on table condition: Non-uniform playing surface does impose a randomizing effect on play. Conditions such as bad or dirty cloth will throw a shoe into the predictable ball rolling machinery as will dead rails. On a bad table you may need even more games to find a statistically significant better player.)

there may be in fact no such thing as a statistically better player. 2 players of near equal skill (knowledge, experience, ability to execute) will be effected but "human" elements that cant be taken into statisical consideration- one player just had a huge fight with SO, is having back spasms, was up all night with the baby ect... all of which have a determental effect on play on a given day.

also randomness is really irrelevant. good or bad layout after a break, poor playing circumstances it doesnt matter. in pool as in life there are very few ideal circumstances. it is up the player to utilize his skill to make the best of the situation in front of him.

brian

 

[mikepage]

[...]

As an example, consider two players that have a Fargo Rating difference of 100 points. By Mike's system, this means that the likelihood of the worse player winning a frame (a game, I guess I should call it) is 1/3 and 2/3 for the better player. This means that in race to six wins, the most likely result is 6-3 for the better player.

If the variance is very high, it would mean that in practice some of the matches are won by the worse player and more by the better player, but in average it would come out as 6-3 for the better player. In contrast, if the variance is very low, most results of matches would be just around 6-3. Some matches would end 6-2 and some 6-4, but it would be very unlikely that the worse player wins.

Here are probabilities for particular game scores for your two players.

Better Player - Worse Player- your example

6-0 8.8%
6-1 20.5%
6-2 27.3%
6-3 27.3%
6-4 22.8%
6-5 16.7%
5-6 8.3%
4-6 5.7%
3-6 3.4%
2-6 1.7%
1-6 0.6%
0-6 0.1%

where, here, 3-6 3.4% means the probabliity that after playing 9 games, the weaker player is ahead with this score. Note the is 8 times more likely to be 6-3 in favor of the better player.


The analysis here is exactly the same for pool and for chess. That is, if two players have p=2/3 and 1/3 for chess, then their probabilities of finishing 6-3 or 3-6 or the weaker player winning, etc is exactly the same. The analysis from there really doesn't know or care what kind of events the probabilities refer to.

In other words, the match-score statistics for two players with equal ratings is the same as the match score statistics for two player flipping coins is the same as the match score statistics for two players playing chess.

Ron Shepard's excellent treatise,

http://www.sfbilliards.com/Shepard_apapp.pdf,

examines exactly this situation.

Check out the discussion around pages 67-72 or so.

 

[12310bch]

one question

MIke , correct me if I'm wrong ( as if you wouldn't):wink: but I roughly
calculate the probability of winning 6-2 at about 24 percent. I just don't see it as being the same as winning 6-3, even with rounding.