Bankroll Statistics

[gtrunner57]

I noticed that on page 88, Malmuth writes "Even though it is true that the standard deviation is larger in no-limit hold'em, the real expert will also have a much larger win rate, meaning that he probably won't need as much money to ensure survival."

However, Malmuth's formula for a bankroll requirement as a function of standard deviation and win rate on page 59 has the standard deviation being squared over the win rate. This would imply that the standard deviation has a larger effect on the bankroll requirement, assuming similar changes in standard deviation and win rate.

I have been very impressed by the analysis and reasoning that Malmuth uses throughout his arguments in Poker Essays. I am just trying to figure out his reasoning on this issue. Is it because the standard deviation does not increase very much compared with a large increase in win rate when comparing limit to no-limit hold'em? Even if it does not increase proportionally as much, any increase would seem to have a large effect since the standard deviation itself is much larger than that of win rate.

I am hoping for someone to clear this up for me. Thanks for any replies.

[pzhon]

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Malmuth's formula for a bankroll requirement as a function of standard deviation and win rate on page 59 has the standard deviation being squared over the win rate.

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Yes.

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This would imply that the standard deviation has a larger effect on the bankroll requirement, assuming similar changes in standard deviation and win rate.

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The standard deviation may be about 3 times as great. The win rate depends on the game and skill level of the winning player, but it might be 5 times as great. So, the bankroll needed to play NL would be 9/5 as great as playing limit with the same blinds, for the same level of safety.

However, comparing limit and NL with the same blinds is not natural. It is more natural to ask what the swings will look like for a given monetary win rate, or which game is more profitable at a given level of bankroll. In low stakes games, NL is significantly safer. NL $100, with a $1 big blind, may give an expert the profit of $5-$10 limit with the swings of $3-$6 limit. Your comparisons with $1-$2 limit, showing that you need a larger bankroll to play NL $100 than $1-$2 limit, is irrelevant, as NL $100 should be compared with a much larger limit game.

[gtrunner57]

Ah, this makes a lot of sense. Thank you pzhon.