Bankroll article

[Wm. Jockusch]

Having seen the reaction to this article, I just want to state my reason for writing it. Prior bankroll articles were answering the wrong question! For bankroll purposes, the correct question to answer is this one:

If I have $B, and some known win rate, what is the chance that I will be able to play at my limit forever, without going broke?

No prior poker article answered this question. In his Poker Essays book, Mason acknowledges that he answered a different question, but he never answers the correct question. That is what my article does. To my knowledge, no prior poker bankroll article answers this question. (Some math papers do, but most poker players don't read those.)

If I were writing the article over, I would make this more clear.

[jfk]

The article is fine. Look forward to seeing parts II and III. Just because the topic is frequently discussed on the forum, it is still very useful to hear a comprehensive explanation from a single, well considered voice.

[BruceZ]

Here are links to posts on this forum dealing with bankroll requirements and the risk of ruin formula. The derivation based on the coin flip game is the same as the one in this article.

Bankroll formulas

Derivation of bankroll formulas

Online calculators for bankroll and risk of ruin


Here is a derivation of the risk of ruin formula for coin flip games:

Risk of ruin for coin flip freezeout

[BruceZ]

I just noticed one issue with this article. Since we are taking the probability of winning a coin flip to be h > 0.5, it is not exactly true that the standard deviation is 1 chip, as would be the case if h = 0.5. Instead the standard deviation in chips is sqrt[1 - (2h-1)^2]. This means that if the standard deviation is S, then S = sqrt(b^2 - W^2) where b is the value of 1 chip (bet size), so the value of 1 chip is not S as stated in the article, but instead b = sqrt(S^2 + W^2). The result of this is that the final formula for risk of ruin

R($nS) = ((S-W)/(S+W))^n

should be replaced with

R($nS) = ((b-W)/(b+W))^n

where b = sqrt(W^2 + S^2), and b is the value of 1 chip (bet size). This is the form of the equation that appears in my derivation that I linked to above. The difference is small for h fairly close to 0.5, so the formula in the article is usually a sufficient approximation for the games we are interested in modeling. In fact, you can see that this same error is made in the paper by Patrick Sileo which I reference.

[Wm. Jockusch]

Bruce is correct. Fortunately for the situations we are interested in, the error is usually very small. For instance, if your win rate is 10% of your standard deviation, the error comes results in a base of roughly 0.818, instead of the correct value of roughly 0.819. That said, it's still an error, and thanks to Bruce for pointing it out.

[Izverg04]

[ QUOTE ]
Prior bankroll articles were answering the wrong question! For bankroll purposes, the correct question to answer is this one:

If I have $B, and some known win rate, what is the chance that I will be able to play at my limit forever, without going broke?

[/ QUOTE ]

Your statement is strange, given that your question has been answered long ago, and given that this is not the "correct" question either.

The correct question should be: given the full range of +EV gambling setups (different games, different limits, multitabling choices), each one with a different hourly rate and variance, which setup is optimal for you at a given bankroll?

I hope you will address this question in the future articles, in terms of the certainty equivalent or a similar concept, or they will also be incomplete.

[pzhon]

[ QUOTE ]

The correct question should be: given the full range of +EV gambling setups (different games, different limits, multitabling choices), each one with a different hourly rate and variance, which setup is optimal for you at a given bankroll?

[/ QUOTE ]
Indeed, one part of the answer is that advantage gamblers should tend to play in games for which they are overbankrolled according to the Kelly Criterion or their preferred fractional Kelly system. This is a consequence of the way games get tougher as you move up, rather than being scalable as is assumed by proportional betting systems. Polls have indicated that most 2+2-ers are overbankrolled according to the usual heuristics, but it is unclear whether this is rational.

I'd like to see analyses of the expected amount of time it takes for good players to build a bankroll from level X to level Y in each variant of poker, particularly in the variants whose statistics are unfamiliar to me.

[Izverg04]

[ QUOTE ]
[ QUOTE ]

The correct question should be: given the full range of +EV gambling setups (different games, different limits, multitabling choices), each one with a different hourly rate and variance, which setup is optimal for you at a given bankroll?

[/ QUOTE ]
Indeed, one part of the answer is that advantage gamblers should tend to play in games for which they are overbankrolled according to the Kelly Criterion or their preferred fractional Kelly system. This is a consequence of the way games get tougher as you move up, rather than being scalable as is assumed by proportional betting systems.

[/ QUOTE ]
Yep, that was the main thing I was thinking about, and not something that many poker players are aware of. Although I wouldn't generalize for all advantage gamblers, overbankrolling effect only applies to people who play skill games like poker and backgammon. This effect applies to a much smaller extent to blackjack players, sports bettors and the like, who can scale their bets without running into diminishing returns, unless they bet really big.

Another related result is that blackjack players should trade away exactly half of their winrate if they are offered a flat wage in return, while poker players should always trade away less than 50%, typically around 30% when playing at optimal stakes -- because they should be overbankrolled, compared to blackjack players.

Remember a poll in a high-stakes forum maybe a year ago that asked this question?

Ok, found it: Poll . Too bad, the answers are missing. But I remember that 50% said they'd give up less than 10% which struck me as "wrong" for being a little too low.

[BillC]

In our paper on bjmath.com, there is a formula for the expected exit times for fractional Kelly betting. It is the "obvious" function of the win rate r.