The question that has bugged me since then, however, is where did this number come from? The cash game requirements (300xBB for limit, and 20xBuyIns for NL) come straight from Matt Matros' math, where he assumes you're an average/solid player and that there is a 5% chance at any given time of losing 150 big bets due to standard variance and deviation.
In contrast, the 40x buy-in rule for SnGs came from advice found in a variety of poker books I own, as well as conventional wisdom found on poker forums. In other words, everyone seems to parrot these numbers, but no one explains where it comes from. In other words, what math is it actually based on? And is it the same for MTT and STTs? And doesn’t the size of your buy-in depend on other factors besides just the size of your 'roll? For instance, how good of a player are you? I mean, if you're crushing the $2 games (e.g., in the money, or ITM, of 45% or greater), you probably need proportionally less money in your account to handle the swings than if you were playing in $20 games (where, presumably, your win rate is reduced due to the fact that you're up against better players; e.g., your ITM may only be 35-40% in these tougher games). To help answer the question (or muddy the waters, depending on your point of view), Mr. Multi brought me something called the Kelly Criterion.
In 1956, J. Kelly Jr., of Bell Labs, developed a formula that could be used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the so-called Kelly Criterion strategy will theoretically do better than any different strategy in the long run. The betting and investing schemes outlined in the books "Beat The Dealer" and "Beat The Market," respectively, for instance, are both based on the Kelly formula. It's also said that investors like Warren Buffett and Bill Gross routinely use a variant of the Kelly system when deciding how much to invest in a particular stock.
Now, the Kelly Criterion is designed to be used in any gambling situation in which you have a positive expectation, and it essentially tells you how much of your bankroll can you bet without risk of going broke. Note the words "positive expectation"; i.e., if you play in poker games where your outcome is negative (i.e., you are a losing player once we remove the variance and rake) then you will eventually lose your entire bankroll-- regardless of the buy-in size. No formula can help you overcome poor play.
Okay, back to the Kelly Criterion. What does it actually do for us SnG players? In a nutshell, it answers the question: What is the highest level of SnG that one can play in, given a specific bankroll size, and after how many losses should one move down?
Okay, you're thinking, so what is the formula and how do I use it? Let's start with the original Kelly equation: f* = (bp-q)/b, where f* is the fraction of the current bankroll to bet, p is the probability of winning, q is the probability of losing. Working through the math, and substituting ITM (In The Money) and ROI (Return on Investment) terms into the equation, the formula can be reduced to: f* = ROI * ITM / (ROI + 1 - ITM). Next, inverting the formula gives us an equation that calculates the recommended number of buy-ins for a given skill level:
Recommended # of Buy-Ins = (ROI+1-ITM)/(ROI x ITM).
Finally, I took the formula one step further and plotted a set of parametric curves for typical ITM values:
Now, if we go back and look at the conventional wisdom advice of 40x buy-ins, we see that it's valid ONLY if you're a pretty decent player. I consider myself a solid SNG player (my OPR numbers currently are 45% ITM and 19% ROI. I believe I can sustain the 45% ITM number if I play tight/cautious/aggressive poker, but the ROI is running pretty high and will probably be more like 10% in the long(er) run. If I look at the curve for 45% ITM and 10% ROI, I see that I can get away with playing at levels 1/15 of the size of my bankroll. With a $150 bankroll, for instance, I could theoretically play at $10 tables. Unfortunately, the level of competition at those levels is higher than what I'm used to play at (and what generated my current good numbers), so I need to temper the numbers a bit. If I assume, for instance, that my ITM would drop to 40% at the higher buy-in level, and my ROI would also drop to around 5%, the Kelly Criterion says I need something like 33x buy-ins to play at the $10 level. This is more commensurate with the conventional wisdom of 40x buy-ins, so it's probably pretty close to the bankroll I would need.
Unfortunately, this means I'm currently in a kind of in a funny no-mans land between the $2 and $5 games that I seem to be able to crush, and the tougher $10 games that Kelly says I can play at given my current win rates, but not if I assume a lower win rate. Hmmmm. Perhaps the answer is somewhere in between 15 and 33…. call it 25 buy-ins to allow me move up to $10 SnGs. And because I'm building my roll up from basically nothing in my new account, I can't yet move up. In the meantime, I'll stick around at the $5 levels and just keep working on building the 'roll. Per the charts, when I hit $250 in the account, I can start playing $10 games.
Clear as mud, eh?
All-in for now…
-Bug
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