Basic Bankroll Management Math: An Instructive Example
I've gotten asked about this twice in the past week, which is usually when I decide to make a post about something. Here I'm going to break down the basics of bankroll management from a basic math standpoint using a simple dice game as an example.
Introduction to Risk of Ruin and Expected Value
Suppose you have $100,000,000 to live off of for the rest of your life, but have no way to gain an income of more than $1/hour ever for the rest of your life. You have no reason to think you'll die any time in the next 50 years.
Someone offers to roll two 6-sided dice with you on the following conditions. They will pay you $100,000,000 if the sum of the dice is anything besides 2, but if it comes up a total of 2, then you pay them $100,000,000.
Risk of ruin is your chance of going broke. In this scenario, the risk of ruin is 1/36 since that's your chance of rolling a 2.
Expected value is how much you will profit on average from an action. In this scenario, the expected value is (35/36)(100000000) + (1/36)(-100000000) = $94444444, so about $94.4 million.
Despite the fact that your expected value is extremely high, there's no way you could take this bet with the dice roll because your risk of ruin is also extremely high.
An Improved Scenario
Instead of betting your $100,000,000 all on one roll, what if you broke your bankroll in half, betting $50 million on the first roll, and $50 million on the second roll. In poker, this would be like playing a game with 2 buy-ins instead of 1.
Since we're betting half of the money and everything else stays the same, our expected value is cut in half for each individual roll and becomes $47,222,222, about $47.2 million.
So what happens to our risk of ruin? A lot of people would think it's cut in half, but that's not true. The only way we can be broke after two rolls is to lose them both, and the chance of that happening is (1/36)² = 1/1296. Instead of cutting our risk of ruin in half by moving from 1 "buy-in" to 2 "buy-ins", we've cut it down by a factor of 36!
By doubling the number of buy-ins we use for our bankroll management in this dice game, we've cut our expected value in half for each roll which has a negative effect, but we've cut our risk of ruin down by a much higher factor.
The Relationship to Poker
Because of the variance in poker, you always have some risk of losing no matter what your expected value is. However, you have to achieve a balance between your risk of ruin and your expected value. Here's an example of dealing with a $5000 bankroll, assuming you can win at all games involved:
Playing with 500 buy-ins at 10nl is making a mistake on the side of being too conservative. Playing with 5 buy-ins at 1000nl is making a mistake on the side of being too aggressive. Playing with 50 buy-ins at 100nl is about right.
So yeah there you go. Now when people ask me to explain how and why bankroll management works and/or is necessary, I can just link them to this. Good luck.
