Rational Bankroll Management – Part 1

1. Introduction
This is the first part of an article series about bankroll management (BRM) in poker. Using some form of bankroll management is vital for poker success. A serious poker player wants to maximize his profits in the long run. To achieve this, he has to protect his bankroll from ruin, and this principle of bankroll management is well known. What’s the best way to go about it is less well known.

In this article series we’ll study rational bankroll management. We define this as BRM based on the way we play, combined with rules derived from statistics. We want to use BRM adapted to our situation (the games we play, our win rate, our standard deviation, and our risk tolerance) instead of simple rules of thumb like “50 buy-ins for no-limit”. We’ll use mathematics to define a model for BRM that we can use to calculate the best BRM scheme, given our situation.

Our goal is make good decisions about which games to play and how high to play, given our bankroll, our skills, and our risk tolerance. Good BRM decisions will maximize our profit over the long run, which is what we strive for as serious poker players.

It’s been said that “all the best gamblers go bust sometimes”. Amateurs with gamble in their blood are perhaps attracted to the “romantic” notion of risking everything. But from a money making perspective, bankroll ruin is completely irrational and unnecessary.

In this series we’ll assume that we are rational gamblers who want to make the most money in the long run. Our only concern is to maximize profit and not to satisfy emotional urges. For us, variance and swings are obstacles that we want protection against. Our protection is our understanding of what variance is, mathematically speaking. Armed with this knowledge we can design plans for moving up and moving down in such a way that we both maximize our income and protect our bankroll.

1.1 Bankroll management in its simplest form
Before we begin our discussion of rational bankroll management, let’s talk about how most players practice BRM today. The most commonly used method is to choose some bankroll size that seems big enough to play our chosen game and limit, and then we play and hope we don’t bust.

BRM in its simplest form then means following certain rules that give you an acceptably low risk of ruin, based on your subjective assessment. Many rules of thumb exist for bankroll requirements, some better than others. For example:

  • 300 BB for fixed limit
  • 50 BI for NLE
  • 100 BI for PLO

And so on. All such rules have in common that they use some assumption about what is “enough” to play, and that you are “safe” if you stick with the scheme. This can work, especially when combined with rules for when to move down. For example:

  • 300 BB for fixed limit, but move down if the bankroll drops to 300 BB for the limit below
  • 50 BI for NLHE, but move down if the bankroll drops to 50 BI for the limit below
  • 100 BI for PLO, but move down if the bankroll drops to 100 BI for the limit below

To make this type of simple, rule-based BRM scheme complete, we can add a rule for when to move up:

  • 300 BB for fixed limit
    • Move down if the bankroll drops to 300 BB for the limit below
    • Take a 100 BB shot at the next limit if you have 300 BB for the current limit plus 100 BB for the next limit
  • 50 BI for NLHE
    • Move down if the bankroll drops to 50 BI for the limit below
    • Take a 10 BI shot at the next limit if you have 50 BI for the current limit plus 10 BI for the next limit
  • 100 BI for PLO
    • Move down if the bankroll drops to 100 BI for the limit below
    • Take a 10 BI shot at the next limit if you have 100 BI for the current limit plus 10 BI for the next limit

Most players who use BRM to guide their moving up and moving down between limits use some variation of the rule-based scheme outlined above. Not necessarily as strict, but it’s based on some idea about what is “enough” to play a certain limit. Then they add a lower threshold that tells them when to move down, and an upper threshold that allows them to start shotting at the next limit.

Here is an example of this kind of planning in practice:

Example 1.1: A rule-based BRM scheme for no-limit Hold’em
Bob begins his NLHE career at $25NL. He has decided to start with a 50 BI ($1250) bankroll, using the following BRM scheme:

  • Start with 50 BI ($1250) at $25NL
  • Move down to the limit below ($10NL) when the bankroll drops to 50 BI ($500) for $10NL.
  • Take a 10 BI shot at the next limit ($50NL) when the bankroll grows to 50 BI ($1250) for $25NL plus 10 BI ($500) for $50NL
  • Keep climbing in limits using the same system (50/50+10)

Bob uses the more compact notation 50/50+10 to describe his system. The first number, 50/50+10, is the number of buy-ins he wants in the bankroll before he considers himself established at the limit he is currently grinding. The second number, 50/50+10 is the threshold for moving down to the limit below. The third number, 50/50+10 is the minimum capital he needs, beyond his minimum 50 BI for the current limit, in order to take a shot at the next limit.

If everything goes smoothly, he will quickly grind in $500 and take a $50NL shot with this money. If the $50NL shot fails, he still has 50 BI ($1250) for $25NL, so he returns to this limit and tries again. If the $50NL shot succeeds, he keeps playing $50NL until he has 50 BI ($2500) for that limit, and then he considers himself to be established there. His next task is to grind in 10 BI ($1000) more for $100NL, so that he can take a shot at the next limit, rinse and repeat.

How will this bankroll management scheme work for Bob? When we try to answer this question, we run into the fundamental problem of this type of BRM planning:

1.2 Why simple, rule-based BRM schemes are problematic to use
The biggest weakness of a simple, rule-based BRM scheme is that it does not contain any information about the player’s win rate or the variance he expects to meet in the game.

Why is this a problem? Let’s think about what will happen to various player types, where we for now only consider their win rates. Let’s first assume that Bob is a steady low limit grinder who wins 10 bb/100. So he wins 1 BI per 1000 hands on average.

The probability that this player will face a -50 BI downswing is negligible, and even if that happens, he will be protected by his move-down rule. The 50/50+10 scheme will work well for this player, and he will quickly climb in limits to begin with. Still, his win rate might drop significantly when he reaches higher limits (for example, when he reaches $200NL, where he will play in games with many professional grinders).

If his win rate drops, but is still positive, Bob will still be protected by his BRM scheme. As long as he wins and has the discipline to move down when necessary, he will never experience bankroll ruin. But 50 BI might not be enough to properly deal with the swings he now experiences. He will survive them as long as he moves down when he should, but he might waste a lot of time bouncing up and down between limits.

For example, let’s say that the swings at $200NL are so big that Bob has to try 5 times before he establishes himself there. His move from $100NL to $200NL might have gone quicker if he had used a stricter scheme, say 100/100+20. It will take more effort at $100NL to get ready for a shot, but the chance of success has gone up. So the total effort to move up might be less.

The same problem would have occurred already at $25NL if Bob had started out as a small winner there, say 2 bb/100. A low win rate equals big swings, and he might have to pay a visit or two to $10NL before climbing up to $50NL for good. Having to make several attempts to move up takes time and effort, and can be bad for morale. So a stricter BRM scheme might be better overall.

But for both the 10 bb/100 winner and the 2 bb/100 winner, we can say this:

As long as we have a positive win rate and a reasonable scheme for moving between limits, we will not experience bankroll ruin. If we have the discipline to stick with the plan and move down when we should, we will eventually move up.

That’s a good start. During this article series we’ll see that we can do better than the simple rule-based systems by applying a bit of mathematics/statistics. Our goal is to design a bankroll management scheme tailor-made for us, based on our win rate and variance in the games we play. A unique BRM scheme for every player, not crude rules of thumb that tries to apply to everyone.

But the biggest problem associated with trusting the simple rule-based schemes is this:

Bankroll management does not protect losing players against ruin!

The simple schemes outlined previously circulate on poker forums and new players quickly get an idea about how big a bankroll should be to be “safe”. But all these rules are based on two things:

  • We have a positive win rate
  • We have the discipline to stick with the scheme during adversity

In practice, new players will often run into problems on one or both of the following areas:

  • They could be losing players, even when playing their best. Then no bankroll will be big enough.
  • They win when playing their A-game, but they struggle hard with tilt. The effect of tilt is to make them losing players overall.

This naturally causes some frustration if you believe bankroll management to be a “magical” recipe for success. BRM is for winners, not losers. BRM alone does not give you the right to survive, you need poker skills for that.

If you’re a losing player, you don’t need a system for BRM, you need a budget (how much you allow yourself to lose). Splashing around for fun as a losing recreational player is fine, if you can afford it and think the fun is worth it. But we will not consider this player type in our discussion of bankroll management.

2. Two components of bankroll management: Win rate and variance
To get to a rational system for bankroll management, we need some building blocks, first and foremost an understanding of win rate (also called “expected value”, “EV” or “expectation”) and variance.

Simply put:

Win rate
Our win rate in a game is what we expect to win (expressed as profit per game, per 100 games, or some other convenient unit). We distinguish between our true win rate (our hypothetical, fully converged win rate that we would have if we played infinitely long) and our observed win rate (the win rate we have now, given the amount we have played).

Win rate is a function of how well we play and how well the opposition plays. It’s the difference between our skills and their skills that determines our win rate. Good poker players actively seek out opponent that play worse than them, and then they play them. This is the skill of game selection, and it’s critical for your win rate.

Variance
Variance is a statistical property than tells us something about how far our results will deviate from the expected results. For example, let’s say our true win rate in a poker game is 10 bb/100. Does this mean we’ll have 10,000 x 10 bb/100 =1000 bb after playing 10,000 hands?

No, since poker is a game with a random component (the cards). In the long run our observed results will converge towards the expected results, but in the short run (say, some tens of thousands of hands, or some weeks) it’s very likely our observed results will deviate significantly from our expected results.

Variance is partly a function of the random component in the game (the randomness of the deck), partly a function of the specific rules of the game (some poker games have more variance than others), and partly a function of our playing style and our opponents’ playing styles. If you play tight and passive, you’re variance will be lower, while loose-aggressive play increases variance.

Since variance causes swings and emotional stress, it’s generally a good idea to take the low variance route if we have the choice between alternatives that are otherwise equivalent. But this doesn’t mean we should try to reduce variance wherever we can. If increased variance is accompanied by increased win rate, we should embrace the variance and make sure we’re properly bankrolled for it.

A good loose-aggressive player plays this way because it maximizes his win rate. A consequence is that he experiences more variance than a tighter player. More variance means bigger swings, so he probably needs a bigger bankroll (but not necessarily, more about this an a bit). This is a trade off he is willing to make in order to win as much as possible. And he protects himself by using a bankroll management scheme suited for his style.

A tight player has lower variance, which reduces his swings. Does this guarantee he can get away with a smaller bankroll? Maybe, but not necessarily, if his tight play keeps significantly reduces his win rate.

The mathematics behind the last statement will be discussed later, but for now let’s establish two important principles:

– Increased variance increases our bankroll requirements
– Decreased win rate increases our bankroll requirements

It follows that we want to maximize our win rate and minimize our variance. If these two considerations pull in opposite directions (and they usually do in poker) some compromise must be made. The obvious choice for a serious player is to maximize win rate, accept the variance, and use a BRM scheme that provides adequate protection against it.

There is a mathematical formula (the risk-of-ruin formula) that tells us how win rate, variance and bankroll requirement are related to each other. We’ll get there eventually, but before we do, we’ll build a better understanding of win rate and variance by playing around with a couple of “toy games”.

We start by defining win rate in a strict way, suited for our purpose:

2.1 Definition of win rate/EV
Assume we’re playing a game with n different outcomes. For example we can flip a coin, which gives us two outcomes (heads or tails). Furthermore, assume that each outcome is associated with some probability, expressed as a number between 0 (never happens) and 1 (always happens). For example, in a coin flip both heads and tails are associated with the probability 1/2 (50%). Finally, assume that each outcome is associated with some value. For example, we might wager $1 on a coinflip, win $1 on heads and lose $1 on tails.

The general definition of expected value (EV) which for our purpose is the same as win rate, can now be written as:

where the probabilities for the n outcomes are noted p, while the values are noted x.

In other words, for each outcome we multiply the probability and the value, then we sum over all outcomes. To illustrate the process, we’ll calculate the EV for a simple toy game:

EV for Dice Game 1
We wager $1 per throw in the following dice game:

  • Our wager is returned on 1, 2, 3 and 4
  • We lose our wager on 5
  • We win twice our wager on 6

This game has six outcomes (the die can land on 1, 2, 3, 4, 5, and 6). A fair die has equal probability for all six outcomes, so the probability for each outcome is 1/6. The value for the different outcomes are $0 for 1, 2, 3, 4, -$1 for 5, and +$2 for 6. Note that the payout of +$2 is the net payout (we get our $1 back, and then win $2 in net profit). We plug the numbers into the EV formula and get:

EV (Dice Game 1)
=(1/6)(0) + (1/6)(0) + (1/6)(0) + (1/6)(0)
+ (1/6)(-1) + (1/6)(2)
=(1/6)(1)
=0.1667

We profit from this game, and we should play it. Our EV is 1/6 times our wager, which becomes $1/6 =$0.1667 per throw.

One important property of EV is that it’s additive:

In other words, if we throw several times, the total EV equals the sum of the EVs for each throw. So if we play Dice Game 1 one hundred times, our total EV is $16.67:

EV (Dice Game 1, 100 throws)
=100(Dice Game 1, 1 throw)
=100(0.1667)
=16.67

When we know the EV for the game, we can calculate the variance:

2.2 Definition of variance
Win rate/EV tells us what we can expect to win (or lose) on average when playing a game. The variance tells us something about how far our result can deviate from the expected result. For us poker players the most interesting part of this is the size of the swings our bankroll will go through.

Let win rate/EV be defined like previously for a game with n different outcomes, each associated with a probability p and a value x. The variance (V) of the game can then be written as:

In other words, for each outcome, take the difference between its value and the EV. Then square the difference and multiply it with the outcome’s probability. Then sum over all outcomes. We can now calculate the variance for Dice Game 1:

V (Dice Game 1)
=(1/6)(0 - 0.1677)^2 + (1/6)(0 - 0.1677)^2
+ (1/6)(0 - 0.1677)^2 + (1/6)(0 - 0.1677)^2
+ (1/6)(-1 - 0.1677)^2 + (1/6)(2 - 0.1677)^2
=0.8056

Like EV, variance is additive. The total variance for a series of throws is found by summing the variance for each throw:

So the variance for 100 throws in Dice Game 1 is equal to 100 times the variance for one single throw:

V (Dice Game 1, 100 throws)
=100(0.8056)
=80.56

The last mathematical definition we need is standard deviation (SD):

2.3 Definition of standard deviation
The standard deviation is simply the square root of variance:

Standard deviation and variance are two ways of measuring the same, namely how wide the spread between observed results and expected results can be. Statistical formulas typically use the standard deviation, so we need this definition in our work.

The standard deviation for Dice Game 1 is then:

SD (Dice Game 1)
=sqrt(0.8056)
=0.8975

Where “sqrt” is the square root. Unlike EV and variance, the standard deviation is not additive. This follows immediately from the definition:

So if we throw 100 times in Dice Game 1, the total standard deviation is sqrt(100) =10 times the standard deviation for a single throw:

SD (Dice Game 1, 100 throws)
=sqrt(100)(0.8975)
=10(0.8975)
=8.975

We have now defined all the statistical concepts we’ll use throughout this article series when we discuss bankroll management for various games. How we use EV and standard deviation to calculate bankroll requirements will be discussed in detail in future articles. But before we do this, let’s study a variation of our dice game to illustrate the effect of variance. We define a new dice game with the same win rate as Dice Game 1, but different variance:

3. Illustrating the effect of variance
We define a new game, Dice Game 2, where we lose our wager on 1, 2, 3, 4, and 5, and win 6 times the wager on 6. EV, variance and standard deviation for the new game are easily calculated:

EV (Dice Game 2)
=(1/6)(-1) + (1/6)(-1) + (1/6)(-1) + (1/6)(-1)
+ (1/6)(-1) + (1/6)(6)
=(1/6)(1)
=0.1667

EV for Dice Game 2 is the same as for Dice game 1, namely +$0.1667. But the variance and standard deviation are much higher:

V (Dice Game 2)
=(1/6)(-1 - 0.1677)^2 + (1/6)(-1 - 0.1677)^2
+ (1/6)(-1 - 0.1677)^2 + (1/6)(-1 - 0.1677)^2
+ (1/6)(-1 - 0.1677)^2 + (1/6)(6 - 0.1677)^2
=6.8056
SD (Dice Game 2)
=sqrt(6.8056)
=2.6087

Let’s summarize and compare our two toy games:

Dice Game 1
We wager $1 per throw in the following dice game:

  • Our wager is returned on 1, 2, 3 and 4
  • We lose our wager on 5
  • We win twice our wager on 6
  • The expected value of the game is EV =0.1667
  • The variance of the game is V =0.8056
  • the standard deviation of the game is SD =0.8975

Dice Game 2
We wager $1 per throw in the following dice game:

  • We lose our wager on 1, 2, 3, 4, and 5
  • We win 6 times our wager on 6
  • The expected value of the game is EV =0.1667
  • The variance of the game is V =6.8056
  • the standard deviation of the game is SD =2.6087

The variance for Dice Game 2 is more than 8 times higher than for Dice Game 1, with a corresponding difference in standard deviation. What are the consequences when we play these two games?

The answer to this question pierces the core of rational bankroll management. Most readers will intuitively realize that if we have the option of choosing which game to play, we should choose Dice Game 1. Dice Game 2 exposes our bankroll to bigger swings without any additional reward. Bigger swings can cause problems for us, particularly if we’re playing with limited funding.

Le’s say we only have $5 in our pocket to play for. In Dice Game 1 there is only one outcome (rolling 5) that loses. 5 out of 6 times we’ll either get our wager back or win twice the amount. The risk of going broke before we have built our roll big enough to survive the expected swings in the game is fairly small.

But in Dice Game 2 we’ll lose our wager 5 times out of 6. We expect to make the same profit as in Dice Game 1 in the long run, but the most likely outcome is that we won’t reach the long run since we’ll go broke early. Another way of phrasing this is that both games are profitable, but it’s much less likely for us to realize the profit potential in the high-variance game (when we start out with a small bankroll).

Of course, if we’re loaded with cash the variance becomes irrelevant. We still choose Dice Game 1, since there is no reason to choose the higher variance game when the win rate is the same. But now our choice is more dictated by principle than by a need to protect our bankroll (since we are over-bankrolled for the game).

We end Part 1 with a graphical illustration of the effect of variance for our two dice games. We use a online variance simulator to generate graphs.

The simulator uses the units “per 100 hands” in the calculations, so we plug in EV and standard deviation expressed as “per 100 throws”. EV is additive, so the EV for 100 throws is 100 x EV for one throw, namely 100 x 0.1667 =16.67 per 100 throws. This is the EV for both games.

The standard deviation grows as the square root of the number of throws (see the previously defined formulas), so the standard deviation for 100 throws is sqrt(100) =10 times the standard deviation for one throw. We get SD =10(0.8975) =8.975 for Dice Game 1 and 10(2.6087) =26.087 for Dice Game 2.

Now we let 10 players (let “Num. of trials to run” =10) throw first 1000 times each (let “Num. of hands” =1000) and then 10,000 times each. We do this for both games and plot the profit graphs for all 10 players:

3.1 Variance simulation for Dice Game 1

The expected result after 1000 throws is 1000 x 0.1667 =+$166.70 . This is the dotted straight line on the graph. We see that the results of our 10 players are distributed around the expected result, with actual results ranging from about +$115 to +$200. The player with the largest deviation is about $52 below expected profit.

Even if all players expect to make +$166.70 on 1000 throws, there is a significant spread around the expected result. 6 players make more than they should and 4 make less (this is of course a random distribution, and a new simulation with the same input would distribute the players differently).

If we increase the number of throws to 10,000, we get:

All 10 players have solid profit. The observed profits range from about +$1550 to +$1950, where 10,000 x 0.1667 =+$1667 is the expected result. We see that the spread around the expected result is less, relatively speaking, than for the first simulation.

The player with the largest deviation is about $283 above expected profit, while the largest deviation for the 1000 throws simulation was $52 below expectation. But $283 relative to $1667 (283/1667 =17% deviation) is a smaller relative deviation than $52 relative to $166.70 (52/166.7 =31% deviation). This illustrates that the relative deviation from the expected result decreases as the number of throws increases.

This is the effect of the long run. The more we play, the more similar our observed results will be to the expected results. Mathematically, this stems from the fact that our profit grows linearly with the number of throws (EV is additive), while the standard deviation is non-additive and grows more slowly, as the square root of the number of throws (see the previously defined formulas).

Then we do the same simulations for Dice Game 2, with the same EV but far higher variance and standard deviation.

3.2 Variance simulation for Dice Game 2
First the simulation for 1000 throws:

The spread of the 10 players after 1000 throws is significantly larger than for Dice Game 1, from about $35 to $210. The largest deviation is then about $132 below the expected profit of $166.70. We can see clearly from the graph that it’s much more randomness in the final results than what we observed for the corresponding simulation for Dice Game 1.

Then we increase to 10,000 hands and get:

The spread is now from about +$1400 to +$2000, so the largest deviation is about $333 above the expected result. The corresponding spread for Dice Game 1 was from about $1550 to $1950, with the largest deviation about $283 over the expected result. The effect of the larger variance in Dice Game 1 is still significant, even after 10,000 throws.

3.3 Conclusion from variance simulations for dice games
The simulations we have done here with exactly known EVs and standard deviations for two toy games have given us a taste of things to come in future articles. We observe that games where we have the same win rate can have very different profit curves, and it’s the difference in variance that causes this.

In future articles we’ll look at win rate/EV and variance/standard deviation for poker. These quantities are far more complicated to calculate in real poker games than in our simple toy games, and in practice they can’t be solved for exactly. But we can estimate them from tracker software (such as HoldemManager and PokerTracker) and the estimates can be used for bankroll estimates.

Our next task is to learn how to use win rate and standard deviation for poker games to estimate bankroll requirements. We’ll use the risk-of-ruin formula for this, and that will be the topic for Part 3. But before we do that, we’ll spend Part 2 investigating how our definitions of EV, variance and standard deviation work in a simple poker toy game (the AKQ game).

4. Summary
We have started our discussion of principles for rational bankroll management. In this article we have defined the problem of bankroll management and gotten a basic understanding of win rate/EV and variance/standard deviation from exactly solvable toy games.

In Part 2 we’ll discuss how our definitions of win rate, variance and standard deviation can be applied to poker games. We’ll use a simple poker toy game (the AKQ game over 1/2 street with fixed-limit betting) to illustrate this.

Good luck!
Bugs – See more at: http://en.donkr.com/Articles/rational-bankroll-management—part-1-406#sthash.Go52WHkR.dpuf

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