A Problem with Calculating Variance

[brianmarc]

The variance calculation (Malmuth, "Computing Your Standard Deviation"; 2+2 Publishing, originally published in "Gambling Theory and Othre Yopics")implies that if you have a series of winning sessions with few corresponding losing sessions, your variance will increase. This is because standard deviation is unconcerned about the DIRECTION of the variance - ie., a large gain is treated equally to a large loss. I have a problem with this as an operational tool to assist you decide how large your bankroll should be. Also, with the implication that lower variance is better than higher variance. To illustrate, let's say you never have a losing session. As your winnings increase this formula, as I understand it, says you should keep on increasing your bankroll! I suspect the problem with this concept is that it assumes a normal, or at least a symmetrical, distribution about the mean return. In fact, an expert player's session returns are going to be skewed to the right, while the poor player's distribution will be skewed to the leaft. (This is not unlike the type of analysis done on investment returns, where high variance that is positivley skewed is not used to penalize the risk-adjusted returns of that set of results. In this world distributions such as the three-parameter lognormal are used, in conjunction with alternative risk measures, such as semi-deviation to eliminate the bias agaisnt high-return, high-variance time series). Comments?

[Cosimo]

If you play long enough, you WILL have losing sessions. Over the course of a few hours, your cards dominate your outcome. I lack the tools to assess my own results, but someone else here could probably give you a good answer on whether their results are close to Gaussian.

Also, EV and SD should be calculated for each level and game. Stats for a live game should be treated separately from online, and even site to site. Further, as you play and learn more, it makes sense to recalculate your EV and SD for your improved skills.

I'm not quite sure what your point is. If your current play implies that old samples are no longer relevant, then why maintain stats based upon them?

[Bozeman]

Actually, you are incorrect. Variance is figured with respect to the mean, so adding more winning sessions will not increase your variance. Adding a few loosing ones will, though,
Craig

[Nottom]

I think you are interpreting the facts a bit wrong here.

Lets say for example that we have a winning player that never loses, but he has a high varience, because he either wins a tiny amount or wins a huge amount.

Wins (for the sake of simplicity assume equal lengthed sessions):
$20
$10
$500
$10
$20
$600
$700
$10
$40
$1000

Now his Std Dev for these sessions is about $373 and thats pretty big, but his average win is $291/session. Since his varience is small compared to his win rate (assuming someone could sustain such numbers) his bankroll requirements are small dispite a high varience.

This is the sort of thing you will see if you hit a run of better than average wins. Your varience (Std Dev) may go up but you winrate will go up even more and result in a decrease in BR needs.

[Cosimo]

I think he's trying to say that (1) as your bankroll grows and you play higher limits and/or (2) as you become a better player, your EV increases, hence skewing your distribution. This is especially true if you start out as a losing player (win one, lose one) and then get better (win win win).

This is the reason why I mentioned that winning players don't have 100% winning sessions (though, strictly, the percentage depends on the session length). I think that percentage is around 55%, but I don't have enough samples to justify that.