## Optimizing Growth: Not A Bankroll Article

The following is a notion that you can find all over the 2+2 Heads Up NL forum.  It is never really questioned by anyone except low stakes players who say "I want to lower my variance" or "I'm gonna wait for a better spot." The forum regulars just shoot them down and say "cEV = \$EV", leaving the newbie poster scrambling to find out what that means and to adapt the 2+2 style of thinking.  I am planning to dethrone it.  The concept...

2+2 HUSNG Theory (in complicated words): The probability of you winning a HUSNG is equal to the number of chips you have divided by the total number of chips in play.  Then each chip has a fixed value in cash, equal to a fraction of the prize pool.  In a \$100+\$5 game, your 1500 chips are worth \$100.  When you have 150 chips left, they are worth \$10.  When you have all 3000 chips, they are worth \$200.  We write that cEV = \$EV (chip expected-value = cash expected-value).  Then in all plays we are trying to maximize our \$EV, as we would in cash games.  So, by the transitive property, we are always trying to maximize our cEV.

2+2 HUSNG Theory (in easy words):  Make the play that wins you the most chips on average.

Sounds so good so far, right?  Actually, the whole idea is correct up until the really obvious part that we glance over, "Then in all plays we are trying to maximize our \$EV."  What it should read, is: "Then in all plays we are trying to maximize our \$EG (Expected Growth)."  People who are familiar with using the Kelly Criterion for bankroll management or wager sizing may already understand this concept, but I will try to break it down in simple terms anyway.

Suppose you have \$100 and are given the option to take a 60/40 coin flip in your favor.  Suppose also that if we are not broke after any coinflip, there will always be another weighted coinflip.  It might not always be a 60/40, maybe a 55/45, maybe a 51/49, but every time, you have a slight edge.   The question is, how much should you wager on each coinflip in order to make your bankroll grow the fastest with your risk of ruin going to zero?  This is the concept of maximizing your growth as opposed to your expected value.  To maximize \$EV, we see that we should wager all \$100 for an \$EV of \$120.  However, to maximize \$EG, we should actually only wager \$20 (20% of our bankroll).  For those mathematically inclined, this is determined by a function f = ((b - 1)p - 1) / b where b is the odds received on the wager (in this case b = 1) and p is the probability of winning.

We know how this applies to bankrolls, and that's why we have some basic guidelines to measure up to, but how does it apply to HUSNGs?  Clearly you can't pass up a 60/40 for your whole stack, or even a 55/45 vs. most players, so why do we care?  The truth is, the reason that we don't pass up those edges is because as the blinds raise, our edges decrease, and the number of opportunities to take those "coinflips" decreases.  The optimizing growth strategy depends on there being a never-ending stream of positive expectation plays, which we don't have in HUSNGs.  So, at some point, we just gotta pick our spot and get it in.  But wait!  That doesn't mean this concept is useless.  We have another rule that we can now apply.  Lets look at chip amounts instead of percentages.

Skates' Law (since I might as well name it after myself if I'm going to reference it):  A 40-chip edge that risks 1500 chips is less valuable than a 30-chip edge that risks 100 chips.

Do you get it?  We're trying to optimize our growth, not our expected value.  In fact, there is some curve for growth in HUSNGs, but it would change from player to player an opponent to opponent, and since our edges are hard to quantify sometimes already, there is almost no hope of ever being able to solve for it.  However, the basic idea should stick.  You want to be able to pick up the largest chip edges you can while risking the smallest number of chips.  This is also why if you feel as though you are being outclassed by a better player, you might consider making pots bigger and playing more aggressively. In that case, since you are a losing player, you'd rather take a 40-chip loss risking 1500 chips than a 30-chip loss risking 100 chips.

Think about Skates' Law the next time you're playing someone terrible.  Do you really want to raise 90% of your buttons and cbet close to 100% of the time if they're calling everything preflop and check-raising the flop half the time?  What if instead, you limped every button and just picked up small pots playing intelligently in and out of position.  Which strategy would win more often in a HUSNG?  Which strategy would win more money in a cash game?  I'll give you a hint... they don't have the same answer.