A Blended Approach to Bankroll Management
Neither article details strategies for increasing risk, in exchange for greater profitability. There are a few theories you should consider whenever making adjustments to your bankroll standards based on the risk/reward ratio you are willing to accept.
Kelly Betting
Most modern day betting systems are either a version of or in some way related to the Kelly Strategy, published by J. L. Kelly in 1956. Edward Thorp famously utilized Kelly’s work in the blackjack book “Beat the Dealer.” The Kelly Strategy provides the percentage of an optimal bet amount, given a particular bankroll with known odds.
Modified versions of Kelly Strategy are still being used today, from Wall Street portfolio management theory to sophisticated patrons at the local horse track. Understanding the Kelly bet and its factors will help you adjust whatever bankroll system you do choose with the added bonus of providing a guideline for taking shots outside of your bankroll. The original formula is a bit complicated for my taste so I prefer to use the version below.
Bet Size = Bankroll * (ROI * ITM/ (ROI + 1-ITM))
The formula in its simplest form is expectation divided by the likelihood (odds) of a positive result, for a single bet. For that reason, the Kelly Strategy is highly volatile, dangerous to a poker player’s bankroll, and emotional health. Modern-day systems have been adjusted using a utility-based approach. Understanding utility and how it should be factored when adjusting your approach to bankroll management is crucial. Prospect Theory is my favorite explanation of utility and how it affects decision making in games of chance.
Prospect Theory
Psychologists say humans feel pain from losses about twice as strongly as joy from gains. This is a major component of Prospect Theory known as loss aversion. The fundamental principle of Prospect Theory states as your net worth increases, you value gains (as a %) less and fear losses more. In other words, finding $10 means less to a millionaire than it would your average college student. On the other hand, a first year student would fear losing half of his net worth much less than a millionaire would. The relationship is shown in the Indifference Curve model below.
The above intersect represents the point of value at which winning or losing has no effect, also known as Risk Indifference.
A common poker misconception is “the money doesn’t effect me.” As poker players, this is a rarified ideal; it does have some affect on almost everyone. Some people deal with the adrenaline rush and emotional affect of big-money play better than others, but those players truly indifferent to the money are a lot less common than you are led to believe. Identifying players who are most likely to be affected will be massively profitable.
Our theory shows there is an inverse relationship; the larger your bankroll, the less each additional dollar affects you. Take a few moments as you approach your final table to run a search on your opponents. Note who has large cashes, and who is likely playing way out of their bankroll. Some people don’t even realize their game has changed; adrenaline doesn’t exactly facilitate rational self-assessment. It can be as simple as an unconscious range adjustment or maybe a slight change in bet sizing. If you’re thinking, “I’m good,” have someone else watch and confirm that assessment…just to be sure.
Endowment Effect
As a player become uncomfortable from playing out of their bankroll, the Endowment Effect begins to take over. According to behavioral economics we place a much higher value on things we own than on things we do not. When you see a player begin to tighten as a tournament approaches the money, the bubble is altering his perceived value of the chip stack he “owns.”
In the beginning, each stack was worth roughly the tournament buy-in. As cashing becomes a likely result, the tightening player is unintentionally using an ICM-like thought process. In his mind, cashing has a much higher value than the additional equity he could gain from risking his tournament life. Good poker players understand the value of a big stack and the importance of not just cashing, but going really deep. Your bubble-scared opponent does not – the fear stops him from playing back at you. Players taught to “push the bubble” are using the Endowment Effect to its full strength.
Taking Shots
At some point you are going to get the “gotta take a shot” itch, and want to play out of traditional bankroll restrictions. If you’re using a volume-based bankroll system, go ahead and scratch…but be sure you’re not taking too much of a shot. Standard bankroll management calls for a maximum of 2.5% of your bankroll to be at risk for any one buy-in.
To be certain I’m not getting out of control, I never play tournaments with a buy-in higher than a full Kelly bet. I make an adjustment I call my “fuzzy factor” – the amount that I feel warm-and-fuzzy risking and playing for. Determining your fuzzy factor can be done mathematically, but it should be simpler than that.
Find a tournament at your Kelly buy-in and look at the final table payouts. Would the money make you less willing to gamble on a coin flip, or bluff off your stack if it’s the right play? If the answer is no, or you’re not sure, rethink your maximum buy-in accordingly. Don’t be afraid to make constant adjustments or tweaks. Under no circumstance should you adjust your buy in up without a significant change in your bankroll or personal wealth. Doing so would defeat the entire process.
Prospect Theory can help you understand how your management approach should change as your bankroll blossoms. In the beginning, your bankroll is a fairly small amount of your net worth (at least, it should be). Your primary focus is learning the game, moving up after you have a good feel and some success. When you start, there is little difference between doubling your bankroll and losing it. Doubling it may even be worth more to you than losing it would hurt. As a result, you should be comfortable playing with riskier guidelines. Once you’ve made it, rethink bankroll management.
Adjusting your buy-in requirements, and ultimately your risk level, is dependent on your game, net worth, and disposition. As a result, some pros recommend 200, or even 300 buy-ins, ensuring they stay in action – their livelihoods depend on it. Find a balance between moving up and maintaining your roll, and adjust it often as you climb.
The Blended Bankroll
In modern portfolio theory, investment managers try to combine high volatility with lower risk investments to maximize returns and minimize risk. Portfolio Theory offers a good medium to express one of the weaknesses of current bankroll practices. Your 30% ROI would be the envy of Warren Buffet, however, your Return on Asset (ROA) is abysmal.
Let’s say you use a 100 buy-in system. Getting a 30% return on 1% of or your bankroll, in reality, you’re only getting a .3% return on your bankroll. At 150 buy-ins, your ROA goes down to .2%, and we miss out on even greater value.
As an investor, you want to safely get the most out of your bankroll. If I have to be satisfied with .3%, I want to sure that I comfortably maintain it. You could use a volume-based system and invest your cash outs elsewhere. But if you want to keep moving up in poker, use a blended system. In this sort of system, you’re playing similar games at different buy-ins. You safely maximize your ROA while keeping your 100 buy-in average (or whatever your comfort level is).
Let’s start with a small deposit, say $50 on Full Tilt. The best micro game, for its structure and rake, is the $1.10 45-man SNG. I am perfectly happy playing this game with 50 buy-ins. So my floor would be $1.10 * 50= $55. Then I decide to mix in the $3.30 90-mans because they have a decent structure. I want at least 75 buy-ins (75 * $3.30 = $248) to play the 90-mans comfortably. Combine the two: (1/6 * $248) + (5/6 * $55) = $86. Below $55, I play only $1.10s. From $55-$86, I play one $3.30 and five $1.10s. Continue on in the same way as shown in the chart; remember the ratio of games is critical.
Generally speaking, larger field MTT’s mean higher ROIs, but since the 90-mans are turbo structures, let us assume a 20% average ROI for both. If you were to play $1.10s @ 20% ROI you earn about $.22 per game. It would take roughly 875 games to get to $248, the minimum bankroll for $3.30s. With the blended system it would take roughly 381 games less making the blended system about 40% faster. Mixing games isn’t always easy, and perhaps it lowers your average ROI. But even if you increased the $1.10 ROI to 30% it would take 643 games to reach the next level with a 100 buy in system, 110 games more than using a blended system. The chart below shows your theoretical return according to the blended system, using the limits set out in the previous chart.
I would love to give you a simple formula, but a bankroll management system has to be more than that. Your system must be personalized. When you’re building, take calculated risks, always be aware of the consequences, and make certain you are able to handle the swings. Learn what you can about yourself and your game before determining what you’ll play.
I asked Jennifear if she would be kind enough to give me her thoughts on my work before I put it out into the world. Her response reinforces the need for discipline so well; I felt, with her permission, I should include it below.
“While I'm all for a blended system, I have a few issues with it that should be taken into consideration.
In practice people tend to stick with the higher end games or not give the lower games as much attention or care, causing their results to suffer. If you are going to use a blended system, then don't just get up for the high-end games and neglect the lower-end games.
Using a blended system increases variance, and allows players to run hot over 500 games and end up playing levels that they are simply not ready for at times. With a standard system, it would require a statistical outlier to run up into games that you are not ready for too quickly. If you are going to use a blended system, every time you move up, you should be able to clearly and succinctly list the weaknesses your typical opponents have, and reasons why you are able to beat the new level. If you cannot do this, don't move up. There's no reason to get involved in games where your ROI is going to be negative just because you are "properly rolled" because you ran hot.
Using an average buyin system, for most people, is nothing more than an excuse to play tournaments that they are not properly rolled for. When using a blended system, be certain that you are doing it for the right reasons, namely to increase your bankroll at the appropriate rate, not to play as high as possible. Playing at the highest level you show a profit at doesn't always yield the best hourly rate. Example: 1% ROI in $22 SNGs will make you less money than 5% ROI in $11 SNGs.
While I throw these caveats of caution in here, the system, when applied properly and with the right intentions, will indeed yield better results for most people. Please take care and caution when using this system and you should flourish!”
