Gambler's Ruin - The Importance of Bankroll Management
Introduction
In this article
- The Gambler's Ruin problem
- The probability of going broke playing on different limits
- Interpretation of the results
Our model
Consider a poker player with $50 starting capital and with the following strategy: We sits down at a poker table with a full buy-in (100 Big Blinds), and keep playing No Limit Hold’em (BSS) until either we lose all our money or double up. We keep using this strategy until our bankroll goes up to $150 or we go broke.
Note: This model is perfect for DoN SNGs, and also reasonable for BSS cash games. It can also be easily adjusted for FL games.
The probability of going broke (Gambler's Ruin problem)
In this part we introduce you to the mathematical foundations behind our model. If you are interested in calculating it yourself, you can do so using the formulas below. If on the other hand you don't care for the mathematical background at all, you can skip right to the results for some interesting information to be found there.
Consider a game that offers a probability p of winning 1 unit and a probability (1 – p) of losing 1 unit. If a player begins with h units, and intends to play the game repeatedly until he either goes broke or increases his holdings to N units, what is the probability of going broke before reaching the desired goal? This is commonly known as the Gambler's Ruin problem, an interesting idea from the field of probability theory.
If a player is currently holding h units (and p ≠ 0.5), the probability of going broke before reaching N units is
P = 1– (1–rh)/ (1–rN),
where r = (1–p)/p.
We can also consider the number of rounds that the player can expect to play before either going broke or reaching his goal. For this we get
Eh = (r + 1)/(r – 1)*[h – N*(1 – rh)/ (1 – rN)]
Note: If p = 0.5 (a fair game), we get
P = 1– h/N,
E = (N/2)2 – (h – N/2)2
Playing No Limit Hold'em cash games
In the case of BSS cash games the "unit" we play for is a full stack, and:
- p is the probability of successfully doubling up
- P is the probability that we go broke before our bankroll reaches $150
- h gives the number of full buy-ins we have at the given limit
- E is the expected number of sessions we play before we either go broke or reach $150
The model does not consider us stepping down a limit in case our bankroll falls below a certain treshold, or stepping up in case we win. We stay on our chosen limit until we either go broke or triple our money and reach $150. The model is of course a simplified version of reality, however the results still provide us with a lot of interesting information and food for thought.
The results
Summary
NL50:
Doesn't really matter if you're a losing or winning player - on this limit you will most probably part from your starting capital in a very quick fashion. However it is also true that if you are a losing player, this is where you have the highest chances of "getting rich quickly". One can say this is ideal for a gambler, however if you are thinking of a poker career, there are much better ways...
NL10:
This is already a bit less of a lottery, but still a very risky business. Losing players will have some time to enjoy playing with their starting capital, although it's also true that they will have less chance to reach the coveted $150 playing on this limit than on NL50. The probability of going broke is still quite high even for the profitable players - 13% chance of bankruptcy in case of a 60% winning player.
NL2:
Bankroll Management in action. The winning players practically cannot go broke. Those however who consider poker more of a pastime than a profession, can still enjoy the thrill of poker action for a long time with their $50 starting capital.
And after everything has been said and done - which limit should you play then? This of course is up to you. We have provided you with all the necessary information to make an educated decision on how to invest your money. The choice is yours.