Win Rates & Standard Deviations & Your Edge

[Double-A]

Whenever I say "bet" I am essentially implying limit.

In the original quote I responded to you saying that you can't reverse the Kelly to find an optimal bankroll (which made me implicitly assumed that you agree with the mathematical logic behind the Kelly) that we are looking for a minimum amount necessary which, in essence, is an "optimal" amount because it is the amount at which we can move up in limits safely according to the Kelly. If you're saying that we can not go from a given "bet" size back to bankroll size, consider that for a given set of inputs that we treat as constants (win rate and standard deviation), there is a formula with two variables (bankroll size and "bet" size). Thus, given a bankroll size, we can come up with a "bet" size because we have one unknown. In a similar fashion, if we are given a "bet" size, we can come up with a minimum bankroll size necessary because it is the only unknown in the formula. In essence, for a given set of inputs, there is always a 1:1 correspondence between bankroll size and "bet" size. If that whole original statement was meant to point out that we only have a finite number of "bet" sizes to choose from, then I rest my case because I thought the whole point of that original statement was saying that given that we can use the Kelly in a given scenario, we can't reverse it (bankroll and "bet" size).

Oh, we CAN solve for bankroll. I'm saying we shouldn't solve for (a poker) bankroll. Say, I do the math and go to Mr. Kelly to get it checked. "Mr. Kelly, I used your KC in reverse. Given these inputs that would be my optimal bankroll correct?" I think his response would be, "If you found a game like this then you need to go get some more money because with that edge we want to get our bet size up. WAY up."

"Sorry Mr. Kelly I can't do that. To increase my bet size I have to move up in limits where my win rate and/or standard deviation will be different. I just want to know what the smallest possible bankroll I can have for this limit is and still use your formula."

Mr. Kelly, "Why?"

Sorry for the narrative... low blood sugar.

I am not saying we can not use past data to predict future win rate and standard deviation. We use the past to predict the future in many aspects of our lives including poker. We assume an 80/0 is more likely to call our bets than a 9/6. If we beat a given limit over 10,000 hands at 5bb/100, we assume we can beat it over the next 10,000 hands. This will not always happen but we are essentially predicting what happens in the future based on past observations.

I'm fine with using past data to predict the future. I'm not fine with using past data to calculate our exposure to risk. Parroting heavily from Taleb (and others here). If we've beaten a given limit for 5bb/100 over 5,000 hands and someone bets us our entire net worth that we can't do that for the next 5,000, should we take that bet?

I think there would be many ways to do it that are logically sound but I guess I'll just throw out what I would probably do. Assuming that I have a large sample size, I would probably use the number that is at the bottom end of a 95% confidence interval while keeping standard deviation the same. For the smallest limit, I would probably use the number that is at the bottom end of a 99% confidence interval.

A more simple approach would probably be to just use a quarter Kelly but, as you have stated, I find it to be too arbitrary for my taste.

I'll have to double check, but I believe using the OP's numbers the 1/4 Kelly would recommend a bankroll of 20 BI's. If that's the case, I think it would have been more optimal to just skip the KC and tell him not to buy in for more than 5%.

It is safe if we use ridiculously conservative inputs or a quarter Kelly or whatever variants of the original Kelly Criterion you want to use...

I don't see how being ridiculously conservative will arrive us at an optimal anything. I think our disagreement may be over our concepts of optimal.

I get (and agree with) most of what you are saying WurlyQ. We both seem to agree that applying the KC to poker is impractical. Our debate seems to be centered on my reasons for it's impracticality. You seem to be saying that if we get conservative in our inputs that most of my reason will be null and void.

But to me, doing that negates the entire purpose of using the KC to begin with. You're using the KC, sure. But, you're changing the question it answers. Kelly was attempting to (and I believe he succeeded) attach value to information. Kelly uses a gambler that is receiving horse racing results from the future (not really but close). He still gets to place his bets at fair odds. But, no type of communication line is perfect so there could be SOME errors in the information he is receiving. How much of his bankroll should this guy be betting? The goal was to maximize his bankroll growth rate (because he's greedy, or the communication line might break, or whatever). The gambler doesn't have to worry about going broke. His bets can get infinitely smaller. To paraphrase, we know our chances of winning and the odds we're getting. How much should we be betting to maximize our bankrolls growth rate?

You seem to be saying, we know our chances of winning and the odds we're getting. How small should our bankroll be to bet this amount?

Double-A don't want no small bankroll.

 

[Double-A]

I've been thinking about this question for awhile now. My main impression is that I would answer him like you answered me, which is reasonable -- go have fun! If it's his dream, then I don't want to stand in the way. It's $700 after all. Now if he said he was moving to Las Vegas with $100,000 to be a pro player and he had no experience with the games there, then I'd be concerned.

Rogue, I think that this is a really, really great answer. And, oddly enough because you quoted me in your response, one I hadn't considered. You're right, he's better off going at it with $700 than $100k.

I tried to back you into a corner and get you to say, "I'd tell him not to do it because he'll just wind up broke." But, you side stepped my lame attempt and got all utilitarian on me. Kudos!

With what I'm gathering about you personality, might I suggest NOT attempting to grind out a profit in full ring cash games. Especially not limit and super-especially not Omaha hi-lo.

 

[WurlyQ]

Oh, we CAN solve for bankroll. I'm saying we shouldn't solve for (a poker) bankroll. Say, I do the math and go to Mr. Kelly to get it checked. "Mr. Kelly, I used your KC in reverse. Given these inputs that would be my optimal bankroll correct?" I think his response would be, "If you found a game like this then you need to go get some more money because with that edge we want to get our bet size up. WAY up."

"Sorry Mr. Kelly I can't do that. To increase my bet size I have to move up in limits where my win rate and/or standard deviation will be different. I just want to know what the smallest possible bankroll I can have for this limit is and still use your formula."

Mr. Kelly, "Why?"

Sorry for the narrative... low blood sugar.

Our bankroll is fixed. By knowing what the minimum amount of money we need to play any given limit, we can play the game that nets us the highest expected return given that we have a higher expectation (absolute money wise) at the higher limit.

I'm not sure I quite get the scenario. Is it a scenario where our edge is big at a smaller limit and we are being advised to move up in limits? Assuming we have a higher expectation (absolute money wise) at the higher limit, then the correct move would be to move up given that our bankroll can handle the risk of the higher limit. We are effectively deciding on how much money we need to build up before we can move up. This is no different than standard bankroll management.

I'm fine with using past data to predict the future. I'm not fine with using past data to calculate our exposure to risk. Parroting heavily from Taleb (and others here). If we've beaten a given limit for 5bb/100 over 5,000 hands and someone bets us our entire net worth that we can't do that for the next 5,000, should we take that bet?

So we can use it to predict the future but not as it pertains to variance...? Also, what does your example illustrate? Even if we know our true win rate were 7bb/100, we wouldn't take that bet because of diminishing marginal utility of money...

I'll have to double check, but I believe using the OP's numbers the 1/4 Kelly would recommend a bankroll of 20 BI's. If that's the case, I think it would have been more optimal to just skip the KC and tell him not to buy in for more than 5%.

The whole point of the KC is that it allows people who have higher win rates to advance faster because their risk of ruin is smaller as compared to someone who has smaller win rates. Standard bankroll management is a very conservative approach for everyone disregarding their win rate.

I don't see how being ridiculously conservative will arrive us at an optimal anything. I think our disagreement may be over our concepts of optimal.

Optimal bankroll size for me is the amount at which we can move up to the higher limit because it allows us to increase our bankroll the fastest. We have a higher chance of maximizing money if we move up earlier do we not? The minimum amount needed puts a constraint on how early we can move up to manage risk.

I get (and agree with) most of what you are saying WurlyQ. We both seem to agree that applying the KC to poker is impractical. Our debate seems to be centered on my reasons for it's impracticality. You seem to be saying that if we get conservative in our inputs that most of my reason will be null and void.

But to me, doing that negates the entire purpose of using the KC to begin with. You're using the KC, sure. But, you're changing the question it answers. Kelly was attempting to (and I believe he succeeded) attach value to information. Kelly uses a gambler that is receiving horse racing results from the future (not really but close). He still gets to place his bets at fair odds. But, no type of communication line is perfect so there could be SOME errors in the information he is receiving. How much of his bankroll should this guy be betting? The goal was to maximize his bankroll growth rate (because he's greedy, or the communication line might break, or whatever). The gambler doesn't have to worry about going broke. His bets can get infinitely smaller. To paraphrase, we know our chances of winning and the odds we're getting. How much should we be betting to maximize our bankrolls growth rate?

You seem to be saying, we know our chances of winning and the odds we're getting. How small should our bankroll be to bet this amount?

Double-A don't want no small bankroll.

I think my previous answer about moving up = faster money maximization and the minimum amount necessary controlling the risk answers this but I could be wrong.

WurlyQ- Not trying to ignore your last post (I will make a concerted effort to give you a direct response), but could you please explain what our inputs will be for:

Basic Kelly Criterion Formula: f* = (bp-q)/b

f* is the fraction of the current bankroll to bet

b is the odds received on the bet

p is the probability of winning

q is the probability of losing, which is 1 - p

Or, how you would modify it?

Yes it's a cut and paste.

b is the odds received on the bet
p is the probability of winning
q is the probability of losing, which is 1 – p
WR is win rate
L is the limit (or bet size)
BR is bankroll
SD is standard deviation

1. WR = b*p*L + (1 – p)*(-L) (how much we expect to make on a given flip)
1a. =L(b*p – (1 – p)) = L[(b + 1)*p – 1]
1b. WR/L = (b + 1)*p – 1

2. f* = (b*p – q)/b (the original formula)
2a. f* = [b*p – (1 – p)]/b = [(b + 1)*p – 1]/b = WR/L/b = WR/(b*L) (from 1b)
2b. f* = L/BR
2c. L/BR = WR/(b*L)
2d. BR = b*L^2/WR

3. SD = [E(X^2) – E(X)^2]^.5 (SD formula)
3a. SD = [p*L^2 + (1 – p)*(-L)^2 – [p*L + (1 – p)*(-L)]^2]^.5
3b. SD = [L^2*(p + 1 – p) – [L*(2p – 1)]^2]^.5
3c. SD = [L^2 – [L*(2p – 1)]^2]^.5
L^2 >>> [L*(2p – 1)]^2 because 1 >>> (2p – 1) for most practical purposes because edges are not that large (we are making SD larger than it should be by removing this component so we are taking the safe side)
3d. SD = [L^2]^.5 = L
For getting b odds, SD = L*b^.5 given that E(X^2) >>> E(X)^2 which is taking the conservative side because we are making SD larger.
3e. SD = L*b^.5

4. Plugging 3e into 2d we get BR = SD^2/WR

This is the formula that Rogue had at the beginning. I don't know if he was looking at the same source as myself but the math makes sense and uses the conservative side of approximations. The result makes sense because we are basically coming up with a size based on our win rate and standard deviation which are the deciding factors of risk of ruin.

 

[Double-A]

Our bankroll is fixed. By knowing what the minimum amount of money we need to play any given limit, we can play the game that nets us the highest expected return given that we have a higher expectation (absolute money wise) at the higher limit.

How is our BR fixed? I don't understand that statement. We won't know the minimum amount of money we'll need to play ANY given limit. Our knowledge of our WR and SD only apply to our current and past limits.

Are our opponents not going to get better as we move up?

I'm not sure I quite get the scenario. Is it a scenario where our edge is big at a smaller limit and we are being advised to move up in limits? Assuming we have a higher expectation (absolute money wise) at the higher limit, then the correct move would be to move up given that our bankroll can handle the risk of the higher limit. We are effectively deciding on how much money we need to build up before we can move up. This is no different than standard bankroll management.

The point of my scenario (yes it was cheese/yes I regret it now) was to illustrate that we aren't using the KC for it's intended purpose and would probably be better off using something else. I didn't do it very well. The drive behind the KC was to attach value to information. The more accurate the info. (signal/noise) the more valuable it is. We're assuming that the info. is 100% and back solving for a bank.

I just don't like it. I think anything we come up with will be too small because of other variables (risk) that we haven't put into the equation. You're trying to convince me that we'll be okay because we'll use conservative estimates (some random number between x and y percentage of our WR).

I'll just ask you to keep punching numbers in until we come up with something that equals 40 BI's at 100 6max. Which is what I think I'd need to take a shot at that level (yes, I'm apparently that bad [at 6max]). Then you could use the KC to justify the math behind some random number that I came up with to make me feel comfortable. A 1/8 Kelly wager or whatever...

So we can use it to predict the future but not as it pertains to variance...? Also, what does your example illustrate? Even if we know our true win rate were 7bb/100, we wouldn't take that bet because of diminishing marginal utility of money...

You're right. Not a good example. I was ignoring utility... Rogue nailed me on this earlier. Or maybe I'm not... I dunno. I'm coming from the perspective where money has no utility (to me) except for having enough of it to buy into the highest limit game that I can beat, tomorrow.

Oh, We can use past data to predict the future. (Although, I think current data is more useful. I don't bury my head in weather history to see if I think it will rain today.) I just don't want to only use past data when deciding how much of our current capital we should EXPOSE to variance.

The whole point of the KC is that it allows people who have higher win rates to advance faster because their risk of ruin is smaller as compared to someone who has smaller win rates. Standard bankroll management is a very conservative approach for everyone disregarding their win rate.

I think that is sorta the point of KC. If by "advance faster" you mean "bet more". But that kinda implies that we're betting "some thing" that we can beat a given limit. I don't think that we are... maybe we are. I guess we're betting a certain percentage of our BR that we can be successful when we move up. And, we'll lose that bet when we have to move down.

But, our risk of ruin won't necessarily be smaller than someone with a lower win rate. They could have a bigger bank... now I'm being a nit. Sorry.

I see your point here. I guess we could use conservative inputs for the KC to quickly find a level that we couldn't beat (assuming we could beat the first level or infinitely small levels). Then we could move down to a level that we could beat, input even more conservative values, study/improve our win rate, and build a bigger bankroll (than we previously moved up with) to take a shot.

 

[Double-A]

Optimal bankroll size for me is the amount at which we can move up to the higher limit because it allows us to increase our bankroll the fastest. We have a higher chance of maximizing money if we move up earlier do we not? The minimum amount needed puts a constraint on how early we can move up to manage risk.

We might go too far too fast and cost ourselves. Consider two players w/ same abilities, BR's, and opponents. Both use KC to determine which limits they will play. They crush every limit up to 10/20. Then one decides to move up and the other ditches the KC to stay where he's at. While player one goes off to slowly lose a percentage of his BR at 15/30, player two stays home and builds his at 10/20. By the time player one's KC tells him to move down, player two could have 1.5-2 times the bankroll he needs to move up.

Depending on how slowly player one lost (or more ominously, how long he played before he hit that unpredictable, worse than anything he has ever experienced, losing streak) player two could have enough money to comfortably stay where he's at with much less of a chance (than player one) of having to move down.

Or he could move up. He'd have much less of a chance of having to move back down than player one had. He might even have enough to withstand player ones losing streak and stay at 15/30.

This is the formula that Rogue had at the beginning. I don't know if he was looking at the same source as myself but the math makes sense and uses the conservative side of approximations. The result makes sense because we are basically coming up with a size based on our win rate and standard deviation which are the deciding factors of risk of ruin.

Thanks for posting the math. It will take me some time to digest it.

 

[WurlyQ]

Real quickly, I know I come off as a condescending crackhead sometimes during some of the longer threads I get involved in but do know that I am doing my best to consider and think about the points you make.

Now on to the more interesting stuff...

How is our BR fixed? I don't understand that statement. We won't know the minimum amount of money we'll need to play ANY given limit. Our knowledge of our WR and SD only apply to our current and past limits.

Are our opponents not going to get better as we move up?

We are faced with a scenario where we have a bankroll, win rates and standard deviations for all the available limits, and we're trying to find the limit we can play. Therefore we have an equation where bankroll, win rates and standard deviations are constant and we're trying to figure out what limit to play (or bet size to make). Stated in another way, at any given point in time where we're trying to decide what limit we should play, our bankroll as well as our past win rates and standard deviations are known constants.

I'm not sure what you're saying with the bolded parts. If we don't have a WR and SD for a given limit, we can not apply the KC.

The point of my scenario (yes it was cheese/yes I regret it now) was to illustrate that we aren't using the KC for it's intended purpose and would probably be better off using something else. I didn't do it very well. The drive behind the KC was to attach value to information. The more accurate the info. (signal/noise) the more valuable it is. We're assuming that the info. is 100% and back solving for a bank.

I flat out disagree with the bold part. I explained before why the environment of our poker scenario and the original Kelly situation are similar except for not having complete information as it relates to the inputs. Do you want to elaborate on the "attaching value to information" you bring up and how it applies to the KC. I don't understand it.

I just don't like it. I think anything we come up with will be too small because of other variables (risk) that we haven't put into the equation. You're trying to convince me that we'll be okay because we'll use conservative estimates (some random number between x and y percentage of our WR).

First off, because this is important, be aware that we are largely excluding psychological factors. On a non KC note, DO NOT play any limit you are uncomfortable playing. If we played scared money, we are in essence lowering our win rate but it is not immediately apparent that we are changing the inputs (decreasing both win rate and standard deviation) so this does not conflict with how the KC works and a motivation for using a conservative win rate. My second point of impracticality also applies here which stated that the KC requires that we move down as soon as we fall below the minimum required, something much harder to do than one would think due to things like apathy, ego, etc.

Everything else is characterized by our inputs so in essence, the unknown variables that you refer to affect our return distribution. If you want to question the validity of a normal distribution being inaccurate in modeling an actual distribution due to factors like tilt or rushes, then I won't debate to the contrary because I personally think that there is some skew and kurtosis to an actual return distribution even over a large sample size. The counter to this is that the Central Limit Theorem applies over a large enough sample size that washes this away. I have a problem with this because the Central Limit Theorem requires the observations to be independent which I don't think is the case here. However, apparently this normal distribution is what is readily accepted and what was used to construct the more standard bankroll management rules so I had nothing I could argue against this. This is pretty much where the last discussion I had about the KC ended.

I'll just ask you to keep punching numbers in until we come up with something that equals 40 BI's at 100 6max. Which is what I think I'd need to take a shot at that level (yes, I'm apparently that bad [at 6max]). Then you could use the KC to justify the math behind some random number that I came up with to make me feel comfortable. A 1/8 Kelly wager or whatever...

You're right. Not a good example. I was ignoring utility... Rogue nailed me on this earlier. Or maybe I'm not... I dunno. I'm coming from the perspective where money has no utility (to me) except for having enough of it to buy into the highest limit game that I can beat, tomorrow.

Oh, We can use past data to predict the future. (Although, I think current data is more useful. I don't bury my head in weather history to see if I think it will rain today.) I just don't want to only use past data when deciding how much of our current capital we should EXPOSE to variance.

How do you play hands in poker? Do you not use past data to determine how much we bet (expose to variance)? Obviously there is an expected value issue in poker hands because we want to maximize our equity. However, this is exactly the same as wanting to maximize our bankroll as outlined by the KC. We can use tells and generalizations but the actions of a player on previous hands should be king in your decision making. Note that there is high correspondence between expected value and variance.

I think that is sorta the point of KC. If by "advance faster" you mean "bet more". But that kinda implies that we're betting "some thing" that we can beat a given limit. I don't think that we are... maybe we are. I guess we're betting a certain percentage of our BR that we can be successful when we move up. And, we'll lose that bet when we have to move down.

I'm not sure if you are aware of this or not, but we have to be able to beat a given limit for the KC to apply. If we can't, the KC will basically tell us not to play the limit (or not to "bet").

But, our risk of ruin won't necessarily be smaller than someone with a lower win rate. They could have a bigger bank... now I'm being a nit. Sorry.

I'm going to "no comment" this... I hope you understand the point of the original statement I made...

I see your point here. I guess we could use conservative inputs for the KC to quickly find a level that we couldn't beat (assuming we could beat the first level or infinitely small levels). Then we could move down to a level that we could beat, input even more conservative values, study/improve our win rate, and build a bigger bankroll (than we previously moved up with) to take a shot.

Do note that taking shots at a higher limit is out of the realm of the KC for the reasons of needing to have inputs which we don't have.

 

[WurlyQ]

We might go too far too fast and cost ourselves. Consider two players w/ same abilities, BR's, and opponents. Both use KC to determine which limits they will play. They crush every limit up to 10/20. Then one decides to move up and the other ditches the KC to stay where he's at. While player one goes off to slowly lose a percentage of his BR at 15/30, player two stays home and builds his at 10/20. By the time player one's KC tells him to move down, player two could have 1.5-2 times the bankroll he needs to move up.

Depending on how slowly player one lost (or more ominously, how long he played before he hit that unpredictable, worse than anything he has ever experienced, losing streak) player two could have enough money to comfortably stay where he's at with much less of a chance (than player one) of having to move down.

Or he could move up. He'd have much less of a chance of having to move back down than player one had. He might even have enough to withstand player ones losing streak and stay at 15/30.

Your example lacks statistical significance. You are laying out one possible scenario independent of probabilities. The two players need to have the same win rates and standard deviations as the other player at both 10/20 and 15/30 for this example to make sense. Also, the absolute amount that they expect to make at 15/30 needs to be higher than at 10/20. Following these requirements, at the point one moves up and one stays, the player who moved up could just as easily run good while the person who stayed run bad. If you want to apply scenario analysis like this, you would need to analyze the possible scenarios and their likelihood in a Monte Carlo Simulation type testing for this point to hold weight.

 

[Double-A]

Your example lacks statistical significance.

Maybe so, but that doesn't necessarily mean we should dismiss it. It holds some weight outside of statistics. It gets us thinking about our concept (subjective) of optimal. As well as, the vehicle we use to get there. Maybe moving up faster won't serve our intended purpose? Maybe we won't have a higher chance of maximizing our money by moving up sooner? Besides, it's fun.

You are laying out one possible scenario independent of probabilities.

Sure, but it's a daisy. We could add a lot more. Only being limited by our imagination.

The two players need to have the same win rates and standard deviations as the other player at both 10/20 and 15/30 for this example to make sense.

They have the same abilities. By that I meant that Player1 and Player2 would have identical WR's and SD's at both limits. But their win rates at 10/20 and 15/30 would be different. Maybe 15/30 is a level they can't beat? If Player2 moved up he would suffer the same fate as Player1. Note that, depending on which disaster we decide to throw at him, Player1 could actually be a winner (long term) at 15/30 and still suffer this horrible fate.

Also, the absolute amount that they expect to make at 15/30 needs to be higher than at 10/20.

Why? I don't understand this point.

Following these requirements, at the point one moves up and one stays, the player who moved up could just as easily run good while the person who stayed run bad.

Sure. Player1 could go on the run of his life. I hope he does. All the while, he could attribute his amazing success to using the KC! Unfortunately, I foresee some really bad things happening to Player1...

If you want to apply scenario analysis like this, you would need to analyze the possible scenarios and their likelihood in a Monte Carlo Simulation type testing for this point to hold weight.

I'm not sure we'll have to go that far.

 

[Double-A]

Real quickly, I know I come off as a condescending crackhead sometimes during some of the longer threads I get involved in but...

Not at all.

We are faced with a scenario where we have a bankroll, win rates and standard deviations for all the available limits, and we're trying to find the limit we can play. Therefore we have an equation where bankroll, win rates and standard deviations are constant and we're trying to figure out what limit to play (or bet size to make). Stated in another way, at any given point in time where we're trying to decide what limit we should play, our bankroll as well as our past win rates and standard deviations are known constants.

I'm not sure what you're saying with the bolded parts. If we don't have a WR and SD for a given limit, we can not apply the KC.

Uh, I'm confused. Do we know what our WR and SD will be for "all available limits" or just the limits we have played? Are our win rates identical for all levels? The lowest through the highest? Of just the ones we have played?

Do you want to elaborate on the "attaching value to information" you bring up and how it applies to the KC. I don't understand it.

I tried to... not very well I might add. My horse player with the phone line to the future? Maybe we should let Kelly do that? "A New Interpretation Of Information Rate"

First off, because this is important, be aware that we are largely excluding psychological factors...If you want to question the validity of a normal distribution being inaccurate in modeling an actual distribution due to factors like tilt or rushes, then I won't debate to the contrary because I personally think that there is some skew and kurtosis to an actual return distribution even over a large sample size...This is pretty much where the last discussion I had about the KC ended.

Sorry for the chop... I agree with all of your points of impracticality. I'm just trying to add a few more dangers (to optimal). Like finding the level we can't beat, catastrophic runs that our stats can't predict, or SLOWLY finding out that we made more money at last level.

How do you play hands in poker? Do you not use past data to determine how much we bet (expose to variance)?

I flip a coin. Heads, I bet. Tails, I bet. I do use past data when playing, if that's the only thing I have or my exposure to risk will be small (small bet/bluff, small pot, etc...). But I do give more weight to current information and might throw out past data entirely if the pot/bets get big enough.

I'm not sure if you are aware of this or not, but we have to be able to beat a given limit for the KC to apply. If we can't, the KC will basically tell us not to play the limit (or not to "bet").

Do note that taking shots at a higher limit is out of the realm of the KC for the reasons of needing to have inputs which we don't have.

So, how do we move up? It seems like you are saying (previously): That our goal is to optimize our bankroll (get it to a point we can move up, fast, so that we can make more money). That the KC is the vehicle we will use to get there. But, that we can't move up because we don't have the inputs for the KC.

 

[WurlyQ]

I don't have too much time today so I'm going to address the points I think are most important.

Maybe so, but that doesn't necessarily mean we should dismiss it. It holds some weight outside of statistics. It gets us thinking about our concept (subjective) of optimal. As well as, the vehicle we use to get there. Maybe moving up faster won't serve our intended purpose? Maybe we won't have a higher chance of maximizing our money by moving up sooner? Besides, it's fun.

Sure, but it's a daisy. We could add a lot more. Only being limited by our imagination.

They have the same abilities. By that I meant that Player1 and Player2 would have identical WR's and SD's at both limits. But their win rates at 10/20 and 15/30 would be different. Maybe 15/30 is a level they can't beat? If Player2 moved up he would suffer the same fate as Player1. Note that, depending on which disaster we decide to throw at him, Player1 could actually be a winner (long term) at 15/30 and still suffer this horrible fate.

Why? I don't understand this point.

Sure. Player1 could go on the run of his life. I hope he does. All the while, he could attribute his amazing success to using the KC! Unfortunately, I foresee some really bad things happening to Player1...

I'm not sure we'll have to go that far.

Regarding why a player needs to make more money on an absolute basis at 15/30 than 10/20: Bankroll maximization means we choose the limit that grows our money the fastest on a risk adjusted basis. Therefore if we do not make more money on an absolute basis at 15/30 than 10/20, then in order to maximize our expectation, we would play 10/20 until our expectation of winning at 15/30 on an absolute basis exceeds our expectation of winning at 10/20.

When I say absolute basis, I mean nominal amount of money earned per hand (I'm excluding hands/hour as a variable). Winning 15/30 at 3bb/100 ($90/100) is greater than winning 10/20 at 4bb/100 ($80/100) on an absolute basis.

Optimal here for me is very defined. It is what limit we should be playing on any given hand to maximize our expectation on a risk adjusted basis.

Not at all.

Uh, I'm confused. Do we know what our WR and SD will be for "all available limits" or just the limits we have played? Are our win rates identical for all levels? The lowest through the highest? Of just the ones we have played?

I tried to... not very well I might add. My horse player with the phone line to the future? Maybe we should let Kelly do that? "A New Interpretation Of Information Rate"

Sorry for the chop... I agree with all of your points of impracticality. I'm just trying to add a few more dangers (to optimal). Like finding the level we can't beat, catastrophic runs that our stats can't predict, or SLOWLY finding out that we made more money at last level.

I flip a coin. Heads, I bet. Tails, I bet. I do use past data when playing, if that's the only thing I have or my exposure to risk will be small (small bet/bluff, small pot, etc...). But I do give more weight to current information and might throw out past data entirely if the pot/bets get big enough.

So, how do we move up? It seems like you are saying (previously): That our goal is to optimize our bankroll (get it to a point we can move up, fast, so that we can make more money). That the KC is the vehicle we will use to get there. But, that we can't move up because we don't have the inputs for the KC.

I don't have time to read the whole Kelly paper now but I will when I get the chance.

What is current information as you define it? Almost every decision I make in poker puts heavy emphasis on past data. On any given hand in poker, on any given street, my decisions are based on the strength of my hand and how I expect my opponents to react to my action. I determine how I expect my opponents to react almost entirely on past data.

As an example, in a value betting scenario, against a calling station (characterized from past data), I'm value betting thinner because my risk reward is greater than against a tough player (also characterized from past data), whom I'm less likely to value bet thinner because my expectation is lower and my variance most likely higher.

Regarding any given limit we are trying to move up to, we have to expect to make more money on an absolute basis than the current limit for reasons outlined earlier.

 

[Double-A]

WurlyQ, I'm starting to get confused...

Optimal here for me is very defined. It is what limit we should be playing on any given hand to maximize our expectation on a risk adjusted basis.

Do you think a poker player should use the KC to find that limit?

If they did, should they use the KC to determine their BR size for that limit?

Optimal bankroll size for me is the amount at which we can move up to the higher limit because it allows us to increase our bankroll the fastest. We have a higher chance of maximizing money if we move up earlier do we not?

Do you think a poker player should use the KC (with conservative inputs) to determine how large his bankroll needs to be to move up in limits?

Regarding why a player needs to make more money on an absolute basis at 15/30 than 10/20: Bankroll maximization means we choose the limit that grows our money the fastest on a risk adjusted basis. Therefore if we do not make more money on an absolute basis at 15/30 than 10/20, then in order to maximize our expectation, we would play 10/20 until our expectation of winning at 15/30 on an absolute basis exceeds our expectation of winning at 10/20.

When I say absolute basis, I mean nominal amount of money earned per hand (I'm excluding hands/hour as a variable). Winning 15/30 at 3bb/100 ($90/100) is greater than winning 10/20 at 4bb/100 ($80/100) on an absolute basis.

I get it. I still want to eliminate it as a point of contention for my Player1/2 scenario. Where, one player using the KC (with conservative inputs) to decide when to move up (with the goal of maximizing money) gets trumped by another player who doesn't. To eliminate it, let's assume that they both expect to make more money (on an absolute basis). Player2 wants to move up with Player1, but he can't. His Mom told him not to... she doesn't trust the KC.

What is current information as you define it? Almost every decision I make in poker puts heavy emphasis on past data.

Sorry, but I'm going to put this on a back burner for now. I'd like to try and narrow down the other points we are debating. If we're going to relate my response to this question to our debate then let me know and I'll get back to it.

Regarding any given limit we are trying to move up to, we have to expect to make more money on an absolute basis than the current limit for reasons outlined earlier.

Can we strike out "expect" and put "hope" in there? Or how about "predict"? And again, will we look to the KC to give us this hope? Or use it to predict how we will perform?

Sorry, for the profundity. In the background my wife is watching a patriotic documentary on the history of American National Parks.

 

[RogueRivered]

Fortune's Formula talks about all this from Kelly's paper. One aspect I don't understand, though, with regards to horse racing information, is that the gambler could bet his entire bankroll on the inside information and maximize its growth. However, as someone who is also interest in horses, I thought race betting was a parimutuel pursuit (betting against the other bettors), which determines the odds. The odds aren't in until all the bets are placed. If the gambler used his inside information to make a sure-thing bet, and he bet a large enough amount, his odds would be severely reduced. Assuming his bankroll was large enough, his bet could effect the odds so much that his return would be practically nothing (in comparison to his bet). And people would notice. When they saw a last minute bet that completely changed the odds, they would become suspicious that someone had inside information. My question is, how does the gambler make any money? Where is this bet being placed on a race that has already happened? The book talks about bookies, so there must be this extra layer of middlemen that are actually taking the bets and paying out. In effect, the gambler is stealing from these middlemen. He isn't gambling, he is just taking advantage of the fact that he knows the results before they do. I hope they don't know where he lives, because he may not much longer if he keeps this up.

At any rate, sorry for the digression. When reading Kelly's paper, I think of the information rate on a noisy channel to be similar to our calculated win rate at poker. We are using information from our database to compute our likelihood of winning money, but since we are unsure of the accuracy of the inputs, it is noisy information. How much of our bankroll should we invest in each bet given the amount of noise and the flow of information? If we have no information, like moving to a new limit, then Kelly says we shouldn't bet. Moving up isn't a bankroll maximizing event, it is an information gathering event. We only know how to maximize our bankroll using inputs with which we have experience. As we invest more money and effort into learning our inputs at the next level, we can slowly start to see whether our bankroll will be maximized by playing at the higher or lower level. It's an ongoing process.

An interesting quote from the end of Kelly's paper:

"Although the model adopted here is drawn from the real-life situation of gambling it is possible that it could apply to certain other economic situations. The essential requirements for the validity of the theory are the possibility of reinvestment of profits and the ability to control or vary the amount of money invested or bet in different categories. The "channel" of the theory might correspond to a real communication channel or simply to the totality of inside information available to the investor."

Sounds kind of like poker to me.

P.S. I'm really enjoying the Taleb books, but I'm reading The Black Swan first.

 

[Double-A]

Fortune's Formula talks about all this from Kelly's paper. One aspect I don't understand, though, with regards to horse racing information......My question is, how does the gambler make any money?......He isn't gambling, he is just taking advantage of the fact that he knows the results before they do.

Kelly might be side stepping this with, "the gambler still gets to make his bet at fair odds". So, it's not EXACTLY the parimutuel world that you and I know. Just switch it to baseball games...

Also, the gambler isn't "gambling" when his information of race results (from the future) is 100% correct. But since NO form of communication is full proof he is gambling. He's gambling on how accurate his information is.

At any rate, sorry for the digression. When reading Kelly's paper, I think of the information rate on a noisy channel to be similar to our calculated win rate at poker. We are using information from our database to compute our likelihood of winning money, but since we are unsure of the accuracy of the inputs, it is noisy information. How much of our bankroll should we invest in each bet given the amount of noise and the flow of information?

Yeah, they are similar. But HOW they are different might matter more. Our database tells us what has happened to us in the past while Kelly's channel tells us what is going to happen in the future.

Also, when you say "bet" you're meaning "buy in", correct?

If we have no information, like moving to a new limit, then Kelly says we shouldn't bet. Moving up isn't a bankroll maximizing event, it is an information gathering event.

I like that. This is also where some of the debate is coming from. In order to get inputs for the KC we have to get out there and play some poker (gather information). Where should we start and how much money do we need? I dunno, let's use the Kelly Criterion. Oops!

An interesting quote from the end of Kelly's paper:

"Although the model adopted here is drawn from the real-life situation of gambling it is possible that it could apply to certain other economic situations. The essential requirements for the validity of the theory are the possibility of reinvestment of profits and the ability to control or vary the amount of money invested or bet in different categories. The "channel" of the theory might correspond to a real communication channel or simply to the totality of inside information available to the investor."

Sounds kind of like poker to me.

We (poker players) can reinvest profits and control our investment. But, there are substantial differences between using past data to make predictions and having a "phone line to the future" or insider information.

Taleb, is great. Worth multiple reads. Funny as well.

 

[Emrald Onyxx]

I still don't get it............ But i wanted to place a RELATIVE quote too.

 

[RogueRivered]

Kelly might be side stepping this with, "the gambler still gets to make his bet at fair odds". So, it's not EXACTLY the parimutuel world that you and I know. Just switch it to baseball games...

Yeah, I guess things are handled differently for sports betting. I've never tried that (but I saw your blog). Once, when I was a kid, a tried to con the neighbor boy into accepting a bet on a football game that was going to be shown later that day on TV. He didn't realize that it was tape-delayed for the West Coast and I already knew the results. I won that bet, but I think I came clean and told him my trick (that was more fun than winning the money).

Also, the gambler isn't "gambling" when his information of race results (from the future) is 100% correct. But since NO form of communication is full proof he is gambling. He's gambling on how accurate his information is.

True, that is where the gamble part comes in, but isn't it still information from the past? We just don't know how accurate it is.

Yeah, they are similar. But HOW they are different might matter more. Our database tells us what has happened to us in the past while Kelly's channel tells us what is going to happen in the future.

I don't see that -- it's still all information about the past as far as I can tell (things that have already happened, and how accurately they might reflect the future. The noisier the channel, the less sure you can be of the future results.)

Also, when you say "bet" you're meaning "buy in", correct?

Yeah, that's the only way I can think of to match up that part of the theory.

I like that. This is also where some of the debate is coming from. In order to get inputs for the KC we have to get out there and play some poker (gather information). Where should we start and how much money do we need? I dunno, let's use the Kelly Criterion. Oops!

You are definitely right. And that is not what I originally proposed, by any means. I already know my inputs at the level at which I intend to play. I'm just trying to grow another bankroll at a second poker site with my info from the first poker site. What I wonder is how similar the play will be between the two sites. I mean, my play will be fairly similar, but what about all the new opponents? I assume that if you take a cross-section of player abilities at the low levels, and assume they will choose poker sites in roughly a similar way, then I think the level of play will be similar. Judging by Vpip alone, which you can see from the lobby, Full Tilt players look looser at the same stakes than pokerstars players (not sure why), but I like to play against looser opponents.

We (poker players) can reinvest profits and control our investment. But, there are substantial differences between using past data to make predictions and having a "phone line to the future" or insider information.

I do have a little bit of insider information versus most of my opponents. I have a history of nearly 200K hands against similar competition -- I doubt most of them do, too. That's part of my advantage when I choose to apply Kelly to play against them.

Taleb, is great. Worth multiple reads. Funny as well.

Yeah, he gives you so many interesting things to think about and research. I tried to look up the Yevgenia Krasnova writer he mentioned, though, and it turns out she is fictitious. She's more of an autobiographical figure to him, but she serves as a good way to make his point.

I had another thought about my friend moving to LV to play poker. If he turns $700 into $100,000, think of all the pride and joy he will feel. I bet it will be more than someone turning $250,000 into $500,000. Bottom line, I think, is that he has more upside than downside, even if the upside is very unlikely.

Another thing that always fascinated me was the way people are bored with the lottery until the prize gets to be huge. Betting $1 to win $2 million doesn't generate much interest, but when the jackpot grows to $300 million, people start to notice and line up for blocks to play. What does that say about the value of money? When the amounts get high, the difference of utility in money isn't that great.

 

[Double-A]

True, that is where the gamble part comes in, but isn't it still information from the past? We just don't know how accurate it is.

I don't see that -- it's still all information about the past as far as I can tell (things that have already happened, and how accurately they might reflect the future. The noisier the channel, the less sure you can be of the future results.)

The result of tonight's baseball game is independent of any past data. Our win rate IS past data. By itself, our win rate doesn't predict the future at all. It's YESTERDAYS lottery result.

Imagine a game where we are drawing colored balls out of a bag. We're going to wager on what color we will draw next out of 1000. The casino decides it will let us do one of two things before we wager: 1) Draw one ball at a time (without looking in the bag), record our result, and replace the ball. We can do this 1000 times. 2) Dump all the balls out on the floor and count how many of each color there are.

One is your estimation of your possible chances of winning based on the results of your draws. The other is your actual chances of winning.

 

[Emrald Onyxx]

The result of tonight's baseball game is independent of any past data. Our win rate IS past data. By itself, our win rate doesn't predict the future at all. It's YESTERDAYS lottery result.

Imagine a game where we are drawing colored balls out of a bag. We're going to wager on what color we will draw next out of 1000. The casino decides it will let us do one of two things before we wager: 1) Draw one ball at a time (without looking in the bag), record our result, and replace the ball. We can do this 1000 times. 2) Dump all the balls out on the floor and count how many of each color there are.

One is your estimation of your possible chances of winning based on the results of your draws. The other is your actual chances of winning.

I finally get it!

Thanks for the analogy!

 

[WurlyQ]

A little lacking on time again because I do want to get some hands in so let me know if I miss any critical points. For the most part, I agree with Rogue about the utility of information in regards to how accurately it predicts the future and how this is pretty much the "noise" in the channel according to Kelly in his paper. You can pretty much assume I agree with his viewpoints unless otherwise stated including the following:

I will state this again as a vital part of the optimality of the KC: the accuracy of the input data. Therefore if our inputs are inaccurate or the data is not available, the KC becomes largely flawed. If we have not played a limit yet, the KC is inapplicable in most shapes and forms.

The result of tonight's baseball game is independent of any past data. Our win rate IS past data. By itself, our win rate doesn't predict the future at all. It's YESTERDAYS lottery result.

I flat out disagree with this statement. Am I correct in assuming that you are questioning the ability to use a sample of results from the past as a means to predict the future?

If you have two baseball teams that have played 100 times in the past and team A beat team B 99 times, who would you bet on to win the next game? Would you simply ignore the fact that team A has a 99% win rate (in a 100 game sample) when calculating your expectation and risk? Sure, it is past data and is not perfect information as it relates to what is going to happen in the future, but it IS a predictive piece of information. All other factors equal, if we were getting a 1:1 bet on the game and we were trying to determine how confident we were that team A was going to win, we can perform hypothesis testing to see how confident we can be that we will win. I guarantee that hypothesis testing would throw out an overwhelmingly large certainty in which we win this bet.

If team A won only 60 times out of the previous 100, the information would not be nearly as powerful. However, even when our win% is only 60%, we could be much more confident in our 60% if team A won 60,000 out of the previous 100,000. This is essentially the power of sample size and what is happening with the KC and what Kelly refers to as the value of information. Kelly uses "insider information" and how "reliable" it is, but the reliability of information is pretty much the same thing as the "certainty" of our hypothesis testing as it relates to the KC. The more certain we are with the information, the more powerful it becomes.

Imagine a game where we are drawing colored balls out of a bag. We're going to wager on what color we will draw next out of 1000. The casino decides it will let us do one of two things before we wager: 1) Draw one ball at a time (without looking in the bag), record our result, and replace the ball. We can do this 1000 times. 2) Dump all the balls out on the floor and count how many of each color there are.

One is your estimation of your possible chances of winning based on the results of your draws. The other is your actual chances of winning.

Scenario 1) is used to find scenario 2) and sample size determines the accuracy of our findings (or certainty or whatever you want to call it). You are entirely correct in stating that we can never be 100% sure what the color distribution of the balls inside the bag are no matter how many times we draw a ball from the bag. However, lets say we pull a ball from the bag 1,000,000 times and get a red ball 430,300 times, a blue ball 260,500 times, and a green ball 309,200 times. We can be fairly confident that ~43% are red balls, about ~26% blue balls, and about 31% green balls. If there were only 100 balls in the bag, I would bet with very high confidence that there were 43 red balls, 31 green balls, and 26 blue balls. This is the power of sample size.

This example is essentially what sampling to find the expectation and standard deviation of the return distribution in our situation in this thread is. This is also why I state that our inputs need to be accurate and as a result, why we need a large sample size to be confident in our inputs.

 

[aliengenius]

Fortune's Formula.


a most excellent book

 

[RogueRivered]

Kelly's channel isn't about the future, it is simply communicating things that happened in the past. But since it's noisy, how confident are we to bet on it? We may have misheard.

Thanks, AG, I think you were the one who put me onto that book in the first place. I agree, a most excellent book.

 

[Double-A]

A little lacking on time again because I do want to get some hands in so let me know if I miss any critical points. For the most part, I agree with Rogue about the utility of information in regards to how accurately it predicts the future and how this is pretty much the "noise" in the channel according to Kelly in his paper. You can pretty much assume I agree with his viewpoints unless otherwise stated including the following:

WurlyQ, there's no time limit. Take all the time that you need.

However, I do think that you missed some critical points. Namely, that in regards to our debate, I have become confused. To me, you seem to be switching the focus of you argument. That's probably not the case but...

To help me narrow down the points of our debate, could you answer the following questions? In regards to our goal being, to play at a limit where on any given hand we have maximum expectation (risk adjusted):

Do you think a poker player should use the KC to find that limit?

Should they use the KC to determine their BR size for that limit?

And in regards to moving up sooner rather than later being optimum:

Do you think a poker player should use the KC (with conservative inputs) to determine how large his bankroll needs to be to move up in limits?

 

[RI_ER_SA]

The reason why this shouldn't be used for poker is that the accuracy of the model is highly dependent on the accuracy of your inputs where your win rate is much more important than standard deviation. The accuracy of your inputs doesn't converge until you have a huge sample size and by then, any winning player should have moved up. The math is there.

true.

-You must be very stringent in moving down when you hit your breakpoint to move down, something harder than most people think due to common human psychology.

true it is an ego war inside

 

[Double-A]

I flat out disagree with this statement. Am I correct in assuming that you are questioning the ability to use a sample of results from the past as a means to predict the future?

Yes.

If you have two baseball teams that have played 100 times in the past and team A beat team B 99 times, who would you bet on to win the next game?
Would you simply ignore the fact that team A has a 99% win rate (in a 100 game sample) when calculating your expectation and risk? Sure, it is past data and is not perfect information as it relates to what is going to happen in the future, but it IS a predictive piece of information.

I don't know which team I would bet on. I probably would ignore the fact that team A has a 99% win rate. If someone where using that piece of information to predict the outcome of tonight's game then I'd ask them if they were willing to lay me 99-1 on team B.

If team A won only 60 times out of the previous 100, the information would not be nearly as powerful. However, even when our win% is only 60%, we could be much more confident in our 60% if team A won 60,000 out of the previous 100,000. This is essentially the power of sample size and what is happening with the KC and what Kelly refers to as the value of information. Kelly uses "insider information" and how "reliable" it is, but the reliability of information is pretty much the same thing as the "certainty" of our hypothesis testing as it relates to the KC. The more certain we are with the information, the more powerful it becomes.

I understand the power of sample size. Kelly's gambler is blessed with something different. A better knowledge of true odds than the odds maker.

To adapt his edge to a poker scenario: He doesn't only have the same database/stats as we do on another player, he knows something more. Kelly was trying to attach value to THAT information. Like, maybe the other players girlfriend will be using his account today (we'd be better off throwing our database out the window) or the other player wrote a bot to play exactly the way he has in the past (we should ignore everything BUT our database).

Interesting to note: From what we've said, you'd be more likely to over expose yourself to risk against the other players girlfriend (over valuing past data) and I'd be more likely to miss out on some profit against the bot (under valuing past data).

This example is essentially what sampling to find the expectation and standard deviation of the return distribution in our situation in this thread is. This is also why I state that our inputs need to be accurate and as a result, why we need a large sample size to be confident in our inputs.

To keep talking about my balls: If half of the players chose to "sample" the balls in the bag and the other half chose to dump the balls out and actually look at them. Which group would have a better chance of making +EV bets? To borrow heavily from Taleb here: The sampling group has a better knowledge of the random color generator (balls in the bag) than anyone who hasn't done any sampling. But the dump the balls out gang knows EXACTLY how the random color generator works.

The "dump the balls out" gang will still be exposed to risk but not to the risk of overvaluing past data.

 

[Double-A]

Yeah, that's the only way I can think of to match up that part of the theory.

You are definitely right. And that is not what I originally proposed, by any means. I already know my inputs at the level at which I intend to play. I'm just trying to grow another bankroll at a second poker site with my info from the first poker site. What I wonder is how similar the play will be between the two sites. I mean, my play will be fairly similar, but what about all the new opponents? I assume that if you take a cross-section of player abilities at the low levels, and assume they will choose poker sites in roughly a similar way, then I think the level of play will be similar. Judging by Vpip alone, which you can see from the lobby, Full Tilt players look looser at the same stakes than PokerStars players (not sure why), but I like to play against looser opponents.

I understand most of what you originally proposed. We've sorta picked up that ball and ran with it... in directions that weren't intended.

Some of that comes from my assumptions of what we mean by bankroll. And also, my assumptions of the why and how we would want to expose that bankroll to risk (playing poker).

I wish you the best of luck with your experiment.

 

[Double-A]

Kelly's channel isn't about the future, it is simply communicating things that happened in the past. But since it's noisy, how confident are we to bet on it? We may have misheard.

I'm going to reshuffle the variables a little...

Assume a parimutuel horse racing game where we can only bet on one horse, to win. There are ten horses that have raced each other a million times. Every player has a complete data base of the results of each race.

All the squares are betting on horse because they like their names, or what ever.

All the sharpies have their heads buried in their databases, creating their own odds lines by assigning win percentages to all of the horses. From the past data, the sharpies conclude that it's an even race. Each horse has a ten percent chance to win. The sharpies look to the tote board for horses going off at better than 9-1.

We get a phone call from the future on a perfect line (no noise). The message? The winner will be either the 1,2, or 3 horse. WE know that horses 4-10 have a ZERO percent chance to win. So, we assign a 33% chance to horses 1, 2, and 3 and look to the tote. We're going to bet on whichever horse (1-3) is going off at better than 2 to 1 odds. And, we should use the Kelly Criterion to size our bet...

For the sharpies, to use the KC spells almost certain doom (failing to maximize bankroll growth). Even if they get lucky, and one of our horses goes off at better than 9 to 1, they won't be betting enough.

Fun to note: If none of our horses went off at better than 2 to 1 then the squares (picking horses at random) would have the same percentage chance to win their bets as the sharpies (using past data)!