300BB "Safe" Bankroll.

[RocketManJames]

I've heard the 300BB bankroll number tossed around quite a bit. I read that for a winning player, 300BB is generally safe, based on typical win rate and typical variance of winning players. It follows that if you lose 300BB, then chances are you're a losing player.

How does the flip side work? If you win 300BB are you just as unlikely to be a losing player? I was thinking about this a bit, and I don't think it's symmetric, because if you win a pot, you win a good chunk, whereas if you lose one, you lose a much smaller amount. To further clarify, my current belief is that it is probably easier for a losing player to win 300BB than a winning player to lose 300BB.

This, however, is countered a bit by the fact that there is indeed a rake and extra money is being taken away from you when you win.

So, that said, how many BB would you have to win (irrespective of sample size) such that you could conclude with high statistical confidence that you were a winning player?

-RMJ

[Copernicus]

I believe, without going back to ancient dog-eared text books, that you are correct in your impression that the hypothetical distribution of wins and losses must be skewed toward the winning side if you are a winning player. If it were symmetrical the implication would be that you are equally likely to be at any point in the distribution as you are at its mirror image, which looks more like a break-even player than a long term winner.

Your last question is handled well by Mason in "Gambling Theory and Other Topics" in a chapter called "What about the losers".

Rather than using statistical tests of significance of a sample mean vs the true mean, he takes the more practical approach that any players results are highly likely to be withing 3 standard deviations of his expected results. He then provides tables of the losers "upper limit" (ie the point 3 standard deviations from the mean on the positive side) for various means/standard deviations and lengths of time. The implication is that results beyond that upper limit are strong evidence that the player is in fact a winner.

The main conclusion is that standard deviation dominates the mean for "short periods of time" and that "short" can be pretty long. Just one example, a 75 bet per hour LOSER playing for a 1000 hours (so he has a mean loss of 75000) has a 3 standard deviations positive result of +20,000 if his standard deviation is 1000.

That player might realistically be a 20/40 player losing less than 2 big bets per hour. You probably have a feel for how easy it is for a winner to lose 1000 in an hour of 20/40, much less a loser, yet this losing player still has a reasonable chance of winning after 1000 hours.

[BruceZ]

After sufficient time, the distribution of a player's winnings becomes a symmetrical normal distribution centered at his average, whether that be positive or negative. In theory, it is just as difficult for a 1 bb/hr winning player to lose 300 bb as it is for a 1 bb/hr losing player to win 300 bb provided both players play with the same standard deviation. An exception to the theory is that in the short term a 1 bb/hr losing player may find a table of idiots having a party, and it is more likely for him to win a large amount than it is for the winning player to lose an equivalent amount. These conditions should average out after a reasonable amount of time. If they don't, then the players are not really playing at the same standard deviation.

A significant winner may be unlikely to lose 300 bb, but a winning player who is only slightly better than break-even can easily lose that much and more. Similarly, you can win 300 bb and only be a small winner or even a loser. All that tells you is that you are probably not a big loser. To determine the statistical confidence that you are a winner, you have to take into account the number of hours played and compute your standard deviation. If you are 1 SD above 0, then you are 84% confident that you are a winner. If you are 1.6 SD above 0, then you are 95% confident that you are a winner. If you want to know if you are a 1 bb/hr winner, then you must be the same number of SDs above 1 bb/hr. Whether you are in fact a 1 bb/hr winner or a break-even player, it can take an extremely large number of hours to find that out to a desired degree of confidence. In fact, if you are exactly a 1 bb/hr winner or exactly a 0 bb/hr winner, you can never prove that even to 84% confidence. The best you can do is to be 84% confident that you are within some range around those numbers, and the range gets smaller with time. However, if you are a 1 bb/hr winner, you can at least become confident relatively quickly that you are not a loser.

[BruceZ]

In fact, if you are exactly a 1 bb/hr winner or exactly a 0 bb/hr winner, you can never prove that even to 84% confidence.

Actually you can, but you have to get lucky. You have to at some point be 1 SD above 1 bb/hr, and then you can say with 84% confidence that you are at least a 1 bb/hr winner. Most people will never achieve this. Only 16% will be 1 SD above their true EV at any given time. An interesting question is what percentage of people can ever achieve 1 SD above their true EV in their lifetimes? That number would be somewhat higher than 16%. It's a purely mathematical question. The people who achieve this won't know who they are.

[Wake up CALL]

"The people who achieve this won't know who they are."

Why won't they?

[BruceZ]

Why won't they?

It's a slippery slope.

You can't know that you are 1 SD above your true EV unless you know what your true EV is. If you've played so long that you know precisely what your true EV is to 1 SD of confidence, that means you've placed your EV in a range within +/- 1 SD where 1 SD has become a very small amount. If your actual win rate then moves 1 SD above this, at that point you cannot say that you are 1 SD above your true EV because the whole confidence interval for your EV shifted along with your results.

Another way to say it is that if you are 1 SD above something, that something is not an EV that you have any confidence in, by definition.

You can know that you are 1 SD above 1 bb/hr, but then are you a lucky 1 bb/hr winner, or are you actually a 1.5 bb/hr winner? If you are an X bb/hr winner, you can know that you are at least a Y bb/hr winner as long as Y < X. But you can't know that you are 1 SD above your true win rate of X bb/hr. You may be, but you can't know that because what you know is based on your confidence interval which is no longer centered on X.

[Ulysses]

[ QUOTE ]
You can know that you are 1 SD above 1 bb/hr, but then are you a lucky 1 bb/hr winner, or are you actually a 1.5 bb/hr winner?

[/ QUOTE ]

Finally, a concise answer for people who ask me why I've never bothered to calculate this stuff. I know how many hours I've played and how much I've won. Don't know what my confidence levels are, I'm just pretty confident that I play good. [img]/images/graemlins/grin.gif[/img]

[BruceZ]

That doesn't mean it doesn't have value. You can always say that your EV lies within certain ranges with some degree of confidence. It's just that it takes a long time for the range to become narrow and the confidence to become high simultaneously. Until that happens, you can say something with high confidence, but the range will be wide. Or you can say something about a tight range, but the confidence will be low. You might in fact be a 1 bb/hr player, but only able to say with decent confidence that you are "not a loser". To distinguish between 0.5 bb/hr and 1 bb/hr with good confidence requires a lot of hours. In time this improves, and the ranges become tight. Also, you can always say that to some confidence your EV is strictly above a certain value (1-sided confidence interval). However, as I pointed out in my last posts, you can never say that you are performing some number of SDs above your true EV, even when that happens to be true.

[Ulysses]

[ QUOTE ]
That doesn't mean it doesn't have value.

[/ QUOTE ]
No disagreement. I just don't think it is of all that much importance to most recreational players.

[BruceZ]

I just don't think it is of all that much importance to most recreational players.

It helps you answer 3 types of questions: 1) How much am I likely to win? 2) Am I likely to lose my whole bankroll? 3) Am I good/bad or just lucky/unlucky? People ask these questions all the time, and I assume most of them are recreational players. If they are interested in objective answers to these questions, then the analysis is important for them. Without it, the average recreational player has no real feel for swings and bankroll requirements. He is likely to assess his abilities based on subjective opinions which are frequently way off, and on meaningless chance fluctuations of results.

Besides, it takes next to no effort to track these statistics. Once you have a spreadsheet set up, all you have to do is take a few seconds to enter how much you won for a session and how long you played, something you probably do anyway, and it calculates everything else. There is no reason not to track them, and there are many benefits. I figure if you're going to play thousands of hours of poker, you might as well record what happened.