Whenever I say "bet" I am essentially implying limit.
You originally stated this:
Secondly, you can't reverse the Kelly to find an optimal bankroll. Or, I should say that you don't need to. The optimal bankroll is infinite. But, it doesn't work anyway because the Kelly assumes that you'll be able to continue making bets at a fraction of a penny. You can't. If you use the Kelly to determine your bankroll size, and lose your first bet, then you'll have to move down in stakes. That move will only require half of your current bankroll (or less {$25NLHE to $10NLHE}) and you won't be playing "optimally" any more.
Which I responded to with this:
We are not finding an optimal bankroll. We are finding the minimum amount necessary to be able to play a given limit with minimal risk of ruin.
... and then came this:
You can't have your cake and eat it too... The lower you want your risk of ruin the larger bankroll you'll need (assuming we don't get better at poker or find worse opponents). You minimize one and the other gets larger.
Or are we just gonna pick whatever number we feel is a "minimal risk of ruin" out of thin air? We don't need a formula to do that...
I'll say 10%.
...and this:
I'm not sure what you're trying to say because you just make an obvious statement that is not going against anything I said... the part you quoted is the end goal we are trying to achieve (how large our bankroll needs to be for us to play a given limit)...
... and finally this:
I'm saying that we shouldn't be using the KC to figure out how large our bankroll needs to be to play a given 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).
We don't know anything except what we have collected from past experience and can't be too sure about that anyway. You seem to have made similar points previously but STILL want to use that data to calculate our risk exposure. I don't want to do that.
I'm saying by the time we have statistically significant inputs (win rate and standard deviation) we will most likely not be playing the current limit because of the number of hands required (sample size) meaning we will most likely have moved up. 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.
If you want to discuss whether or not using a normal distribution to characterize our return distribution for any given hand, then we have a very interesting debate.
You seem to think that by using a "conservative" version of our past win rate and possible standard deviation we can over come any dangers that our miscalculations (craziness) may expose us to.
I'm asking you, "How do we come up with that conservative version?" Do we cut our (actual past) win rate by 10%? Do we double our standard deviation? If we do then aren't we just pulling numbers out of thin air?
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.
The KC won't get us anywhere "safely". It's designed to be maximally aggressive. So much so, that Kelly himself would consider being any more aggressive to be "insane".
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... It is only maximally aggressive if the inputs are the actual values. Let's say we have a game where we win our wager 80% of the time and lose our wager 20% of the time (this basically defines our win rate and standard deviation). If instead, we use the Kelly to calculate our necessary wager using conservative values of winning our wager 55% of the time and lose our wager 45% of the time (this is in essence, a conservative win rate and standard deviation approximation), then the chances we go broke are extremely small even if we only have a finite amount of "bet" sizes available to us.
