Double-A
Visionary
Silver Level
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.