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Never Go Broke Playing Poker

Bankroll Management and Poker Variance

Stu UngarStu Ungar is considered by many to be the greatest Texas Hold’em player to have ever lived. In the final hand of a 1992 WSOP $50,000 no limit hold’em heads-up match, Mansour Matloubi pushed all-in on a board of 3-3-7-K-Q. Ungar thought for a few seconds and said, “You have 4-5 or 5-6 so I’m gonna call you with this” and turned up T-9, winning the match with ten-high!

However, despite winning the WSOP Main Event in 1980 and then again in 1981, by 1997 Stu Ungar was deeply in debt and had to be staked the $10,000 buy-in in order to play the WSOP Main Event that year, which he won, splitting the $1,000,000 prize pool with his staker, Billy Baxter. He then proceeded to spend all of his winnings over the next few months.

By 1998, the three-time WSOP Main Event champion was broke again. In October of that year, poker player and former casino owner Bob Stupak agreed to stake him and gave him an advance of $25,000. Stu Ungar was found dead in a $48/night hotel room with only $800 left from his $25,000 advance of just a month before.

While Ungar’s story is probably the worst example of a player with terrible bankroll management (fuelled by an addiction to drugs), he is certainly not the only poker player to quickly lose all of his winnings.

Learning to Manage Your Poker Bankroll

Since there is a LOT of information to cover, we have separated our “Poker Bankroll Management” article series into 6 topics:

The following factors contribute to variance in poker:

Tournaments Cash Games
Win rate Win rate
ROI ROI
Number of Entries  
Prize distribution  
  1. Never Go Broke Playing Poker - Learn about poker variance and how to use your bankroll to protect yourself from it!
  2. Can Your Poker Bankroll Handle MTTs? - The number of entries has a huge impact on poker variance. Is your bankroll big enough to handle it?
  3. 4 Essential Tips to Protect Your Bankroll - Learn several techniques to reduce your variance, allowing you to invest more of your bankroll each tournament.
  4. Tournament Poker Bankroll Calculator - Use our tables to determine your ideal buy-in based on your ROI and the size of tournaments you prefer to play.
  5. Cash Game Bankroll (not yet published)
  6. Bankroll Management for Losing Players (not yet published)
 

Even if you are only interested in one of the topics, we strongly recommend that you read through the articles in order as each subsequent article builds on the information presented in the previous topics.

Skill Alone is Not Enough!

Most poker players would much rather learn how to figure out exactly what their opponent is holding, or how to bluff like a champion than learn such a boring task as managing their poker bankroll. Perhaps it’s because we all like to focus on winning, whereas successful bankroll management requires that we admit that not only are we going to lose, but at times we are going to lose big and often!

poker bankroll management

Simply put, poker bankroll management means ensuring you have enough money to weather the bad runs that WILL happen, no matter how good you are. Learning how to manage your poker bankroll is the single most important skill for every poker player whether you play professionally or are a fish in your local home game. In fact, winning players with no bankroll management frequently end up far worse off than losing players who have mastered this skill!

You need to use your bankroll to protect you from short-term poker variance.

Always remember that no matter how good you are, poker is still a form of gambling - there is always an element of luck that will play havoc on your poker bankroll in the short term. While skill will always win out in the long run, the short-term is much longer than most players realize. You will need good bankroll management to avoid going broke from the short-term downswings thereby allowing time for your skill to ensure you are profitable in the long run.

For a more detailed look at the mental aspects of managing your bankroll, we highly recommend Alec Torelli's 4 Steps to Never Going broke. Beyond the mathematical analysis we conduct in this article series, Alec has some great tips to help identify whether or not you are playing withing your means.

Poker Variance

“Variance” is a statistical measurement that indicates how spread out a series of numbers is. "Poker variance" typically refers to the swings every player will experience. Here is a great definition of poker variance:

The difference between how much money you expect to win on average over the long run and the results you have in the short term.

Let’s use the simplest of examples: flipping a coin, the odds of which are 50% for heads and 50% for tails. If you flip a coin 10 times, the average outcome is 5 heads and 5 tails. However, here are the actual probabilities for each possible outcome for 10 flips:

Result

Expected Occurences
(in 1,024 Attempts)

% Chance
10 heads & 0 tails 1 0.1%
9 heads & 1 tails 10 1.0%
8 heads & 2 tails 45 4.4%
7 heads & 3 tails 120 11.7%
6 heads & 4 tails 210 20.5%
5 heads & 5 tails 252 24.6%
4 heads & 6 tails 210 20.5%
3 heads & 7 tails 120 11.7%
2 heads & 8 tails 45 4.4%
1 heads & 9 tails 10 1.0%
0 heads & 10 tails 1 0.1%

The first thing to note is that over 10 flips, we will only get the “expected” result of 5 heads and 5 tails 24.6% of the time (roughly one quarter). If we flip a coin 1,033 times and are betting on heads, then we should expect that we are going to lose at least 10 times in a row at least once! Of course, we can also expect to win 10 times in a row at least once. However, if we don’t have enough money to keep playing through the bad runs go, we could go broke before we get the opportunity to recover our losses with a 10-flip winning streak!

If we are betting on coin flips, then it should be obvious that we will need to maintain a bankroll of at least 10 times our bet in order to cover the expected 10 losses in a row we will incur. Note that I say “expected”; this is not bad luck, but rather an expected result because probability tells us this should happen on average once for every 1,033 coin flips.

Oh wait…I just said “on average”. That means it could happen more than once during a series of 1,033 coin flips. There’s also no guarantee that immediately following a run of 10 losses that the next run of 10 flips will have more heads than tails. In fact, there’s a 25% chance that the next 10 coin flips will only break even, and a 38% chance that the next run of 10 will be a loser as well!

For the rest of this article, and the other articles in this series, we will place the mathematical proofs in a yellow section. That way you can skip the analysis if you just want to know the results. For more information on variance in poker (specifically in cash games), check out this article on poker swings and variance.

Calculating Maximum Poker Variance

A coin flip is not an ideal example because break even bets can have a ridiculously long stretch of winning or losing.

This happens because once the results deviate from the expected (i.e. there are more heads than tails or vice versa), there is no inclination to recover. That is, if after 1,000 coin flips you have flipped heads 50 times more than tails, then there is no reason to expect a recovery (since the expected result of each additional flip is 50% heads and 50% tails). In other words, once you have flipped heads 50 times more than tails, the expected result is that you will always have 50 more heads than tails.

Luckily, it’s almost impossible for a poker player to be a truly break-even player, so this is not an issue for our analysis.

Instead of continuing with our initial example of coin flips, let’s look at playing heads-up sit and go tournaments. To calculate your required poker bankroll, we need to know the following information:

  • Win rate (we will assume you will win 55% of the time)
  • Buy-in plus fee (we will analyze a $100 sit & go on PokerStars)
  • Prize distribution (a heads-up sit & go awards all of the prize pool to the winner, in this case $196.24)

With these assumptions, your expected return on investment (ROI) is 7.9%.

55% x $196.24 = $107.93Carbon Poker
- $100 entry
= $7.93 (7.93% ROI)

If you are willing to accept a 5% chance of going broke during each stretch of 1,000 tournaments, you would need a poker bankroll of $3,330 (33.3 buy-ins). To reduce this to a 0.3% chance of going broke, you need a bankroll of $4,020 (40.2 buy-ins).

Using the randbetween function in excel, we simulated 1,000 matches with a 55% chance of winning $196.24, 200 times. For each group of 1,000 matches, we recorded the largest downswing (in terms of buy-ins). A downswing is the difference between the previous peak and the lowest low valley until your winnings exceed the previous peak. Taking the standard deviation of the maximum downswings, we can calculate poker variance. 

poker varianceThis chart of winnings is from the 45-man sit & go example we will discuss in our next article.

The results of our heads-up sit & go analysis downswings are presented below:

Range of Largest Downswing 8.5 – 51.9 buy-ins
Avg. Max Downswing 19.6 buy-ins
Std. Dev. of Max Downswing 6.85

Assuming a normal distribution, we expect 95% of the results to lie within 2 standard deviations of the average. Therefore, to have a 95% chance to avoid going broke over a stretch of 1,000 sit & go’s, we need a bankroll of 33.3 buy-ins (19.6 + 2 x 6.85 = 33.3). For a 99.7% chance, it’s the average plus 3 standard deviations.

Note that heads-up is the only example that will result in a normal distribution. However, the empirical rule still gives a fairly accurate assessment for poker bankroll requirements. However, if you want to be truly safe, add an additional 20% to the bankroll requirements for all non-heads-up calculations.

Even though we have assumed you are a winning heads-up sit & go player (with a 55% win rate and 7.9% ROI), you still need to have a significant poker bankroll because over a stretch of 1,000 tournaments there is a 5% chance that you could suffer a downward swing of 33.3 buy-ins. If you have a higher win rate or a better ROI (both of which reduce the likelihood of long stretches of losses), you can avoid going broke with a smaller bankroll.

However, as we add more variables (more entries, smaller chance of reaching the money, and variable prizes), the amount of poker variance increases. More poker variance means you need an even bigger bankroll. Continue reading about bankroll management to learn how much larger your poker bankroll would need to be for 10-player and 45-player sit & go’s, and learn how you can mitigate some of the poker variance mentioned above.

This post was shared on the Alec Torelli’s Best of Poker Series

Do you have a bankroll management story you would like to share, or perhaps some advice to add? We would love to hear your stories and thoughts on this article in the comments below!

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