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.
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Radeh's picture

Nice article! Good job at

Nice article! Good job at putting a rather complex concept into words :)


Goats!!! MORE GOATS!!!

xSCWx's picture


If there was no wait between games would the 40 edge/1500 risk situation become optimal over the 30 edge/100 risk situation? This has crossed my mind a lot in the really marginal "wait for a better spot" situations because I have always advocated finding your better spot in the next game (assuming that taking the slight edge now will be +$/hour if we just join a new game immediately afterwards) but my reasoning neglects the wait between games.

In other words, my understanding of the "never-ending stream of positive expectation plays" is because the game will end eventually, but if you have a never-ending stream of games then don't you essentially have a never-ending stream of positive expectation plays?

r1tony's picture

very nice read skates...

very nice read skates...

Skates's picture

Yes and No... In order for

Yes and No...

In order for the 40/1500 situation to be better than the 30/100 situation, you would need the bankrolls are infinite (outside of the games), the game will continue after the hand regardless of outcome (so you start another game immediately), and that rake is non-existant or on a per-hand basis (not a per-game basis).  None of these conditions are ever met in HUSNGs.  At microstakes, you could say that virtually everyone is a "weak" player and you don't have to spend time waiting for new games, so the game continuity part is solved, as well as the infinite bankroll thing (because there will always be people ready to play for $2).  However, the rake bit kills it.  You would need a way to account for the amount you've paid in rake and covering that edge.  You might be able to come up with something for this, as well as make a point for pushing edges requiring higher chip risk in microstakes matches, but what I've written above is pretty much accurate in all cases.

Of course at higher stakes, you cannot handle any of the three conditions at all, so you're stuck with optimizing the growth of your one buy-in.

The biggest issue is, some people take this whole "wait for a better spot" ideology out of control.  You could completely ignore it and be a highly winning player moving up through the stakes; you just won't be playing optimally vs. the weakest of your opposition.  People tend to convince themselves they have edges they don't, or decide not to call 800 into a pot of 2200 on the flop with a flush draw because they're set on finding a better spot.  This is obviously not what we're talking about.  We're talking about finding better exploitive strategies against players that we have significant edges on.


cymonguk's picture


Good article..


I have been playing a lot of people at low stakes who call 90%, check-call/raise 100%, this caused me many issues early on


In this case limping or low raising then trying to outplay on the flop and turn worked best for me, many of these players are calling cbets because they believe you will try to cbet any two cards, and because they will hold any ace/king/queen, or any pair they have at the flop to see the turn



PrimordialAA's picture

out of curiosity, is a 40

out of curiosity, is a 40 chip edge that risks 1500 chips equivalent to a 8 chip edge that costs risks 300 chips (also equivalent to a 2.67 chip edge that risks 100 chips) ?

and if so, then a 5 chip edge that risks is SIGNIFICANTLY better than a 40 chip edge that risks 1500

is this all on the right track?

incog13's picture

Nice article

This has been something I have struggled with for a while. My variance has skyrocketed since I have implemented shoving over raises that commit >8% of starting stacks with some not insignificant range and with big stacks in terms of chip amounts. I have gone as far as limping all my buttons and trying weird things like that. Now I think I have found a decent balance between raising a number of my buttons and also keeping pots small postflop (checking back more on certain flops and balancing checking back range against aggros). Thanks for putting these things in a more quantitative form Skates!





Skates's picture

Thanks for all of the

Thanks for all of the positive replies everyone!

In regards to Primo's question:  The technical answer is no, they aren't equivalent.  We have some growth curve determining the value of different plays, and it is almost certainly non-linear despite the Kelly function because our game state changes after each hand (stack depths and position changes, so our edge does as well).  

However, what you're talking about is fixing a win probability for a win/loss ratio of one.  A 100-chip edge risking 1500 chips is the equivalent of winning a wager for 1500 chips 53.33% of the time.  A 20-chip edge risking 300 chips also has win probability of .5333.  So it might make sense to compare them.  But again, we have the Kelly criterion coming into play, which says we should wager 6.6667% of our bankroll with this edge to optimize our growth.  So, the Kelly criterion suggests the best amount to wager with the same edge/risk ratio would be a 6.667-chip edge risking 100 chips.  Of course, this has the problems I outlined above.

To respond to your last claim... If your edge is small (51.33% in the case of a 40-chip edge risking 1500), then yep, a 5-chip edge risking 187.5 chips is SIGNIFICANTLY better than a 40-chip edge risking 1500.  On the other hand, if your edge is large (75% in the case of a 750-chip edge risking 1500), you'd much rather take a 750-chip edge risking 1500 than a 50-chip edge risking 100 since you don't really have that endless stream of positive expectation wagers.  So the amount that you actually want to wager on each fixed edge/risk amount is gonna be a function of how significant the edge is.

Serj's picture

so, which strategy would win

so, which strategy would win more money then? :) I can't figure that out because I didnt played HU cash too much.

Cards are war, in disguise of a sport.

AQs's picture

wow this article open my

wow this article open my eyes,i always played thinking about "2+2 husng theory",but now I really want to adjust my game according to this (or switch to cash game :) ) but I got some problem to figure out how does it apply in the gameplay (I maen not obvious spots like folding 44 facing a deep 3bet-shove,I guess evry win player already does)


riskoriented's picture

Since the thread got bumped,

Since the thread got bumped, I have some questions now:

How do you reconcile your strategy that you prefer a 30chip edge over a 40chip edge when it lowers our risk with the assumption that we do not have to beat the rake every hand, but over time our job is to accumulate small edges that allow us to beat the rake?

Second, how are we expected to differentiate between a 30chip edge and a 40chip edge while playing? Surely if we see an edge, we should take it, and we should make an educated guess about which edge is higher, rather than making an educated guess about the balance between the edge we're taking and the amount of variance it implies?

Finally, isn't the point of bankroll management to enhance our growth by taking on _more_ risk? Do you think that players would have faster growth if they approached a true-kelly bankroll (10-20 buyins depending on ROI) rather than trying to lower their variance or increase their roi in each individual game? For the uninitiated, the kelly criterion says that we maximize growth by wagering an amount roughly equivalent to our ROI. This means, if your ROI is 10%, we should always wager 10% of our bankroll (move up and down in order to maintain 10 buyins) for the maximum long-term growth. I think if people are concerned with faster bankroll growth, then aggressively bankrolling will do a lot more for them than trying to lower their variance while sitting on 50 buyins.

::: Sage


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Skates's picture

Rake is paid per game not

Rake is paid per game not per hand.  Accumulating small edges per hand will eventually overtake rake per game.  This is along the same lines of saying the probability of a winning player making money over 5000 games is close to 1, but the probability of a winning player making money over 1 game might only be .55.

We can't necessarily differentiate that strongly between the two edge types when they arise in play.  The idea is to craft your basic strategy accordingly.  If your villain is super lag preflop, even though playing super lag back is probably the most effective way to make money in a cash game, assuming you're good at it, tightening up preflop is a way to attain close-to-comparable edges while risking less chips.  One example of something that could come up postflop in a specific hand is what to do when facing a river bet with a weak made hand where villain's range is merged.  You might think that villain bet/calls some percentage and bet/folds some percentage, and that your hand is good as it stands some percentage.  In a cash game, you just find out which has higher equity and make that play.  In a HUSNG, you might consider just clicking call if the difference in chip equity isn't that high.  This spot comes up alllll the time, at least in my games :).  Other things to consider would be b/3bet flush draws OOP vs. c/c vs. c/r against an aggressive villain, or "pot controlling" the turn with midpair or top pair against someone who can c/c flop and c/rai turn vs. just betting it and picking off bluffs.  Basically, you're just thinking about the lines that you take with a variety of hand-types and then coming up with a cohesive strategy to play against your villain so that you're increasing your edge as much as possible without risking your stack over and over again.

This article wasn't about bankroll management, however, I can still respond to your last question.  Yes.  Play based on the Kelly Criterion with the greatest Kelly value you can while being able to handle it emotionally.  I don't recommend going greater than 1/2 though because most people overestimate their edges.  Now, I'm well-enough rolled for whatever and Kelly-wagering doesn't make as much sense anymore.  During most times of day, you can make more money playing $230s than $570s just based on traffic.  However, when traffic isn't an issue, like at lower stakes, yea, follow the Kelly criterion.

bonaparte's picture

Cool article.  This cements

Cool article.  This cements some gut feelings I have been having about the game.

Jmonderson's picture

What exactly is a chip edge

What exactly is a chip edge and how is it calculated?

Ph33roX's picture

Calculating Edge

Hey Jmonderson,

when we play a heads up tournament vs someone we have an edge against, our equity in that tournament is higher compared to when we play an opponent who we have no edge against. For example, let's imagine we register to a 1,000$ husng and we start with 1000 chips. each chip we have is worth 1$ right? well, not in every case. Let's also assume we play a pretty terrible opponent, one which we have a 60% winrate against. In this case when we start the tournament each chip is worth more than 1$. If we're going to win this tournament 60% of the time, our stack is worth 0.60 (our winrate) times 2000$ (the prize pool), which is 1200$. So each chip is worth 1.2$ instead of 1$.

Lets continue with our story here. In the first hand of the match we get dealt A7o and the blinds are 10/20. our opponent openshoves for 1000 chips and turns his hand face up - he has KQs. We open pokerstove and see we have 54.2% equity. The max EV decision is clearly calling. What happens to our stack when we fold? we lose 1 big blind, and it shrinks to 980. What happens when we call?

54.2% of the time we win and our stack doubles to 2k. 45.8% of the time we lose our stack is gone. The average result is that our stack increases from 1000 to 1084, so we win 84 chips on average by calling. Pretty great compared to losing 20 chips right?

No. And let's talk about why. The average result (a stack of 1084 chips) is made up from the average of two scenarios, one where we have none of the chips and one where all of the chips. In both those situations, an edge no longer exists. 0 chips are worth 0 dollars, and 2000 chips are worth 2000$, so each chip is worth 1$ again. What does this tell us? That going into the first hand of this match, the 1000 chips we can win from our opponent, are worth less in dollar ev than the 1000 chips we have right now. Hence the skates quote:


A 40-chip edge that risks 1500 chips is less valuable than a 30-chip edge that risks 100 chips.


What we learn from this is that while calling A7o against the openshove is the max ev play, it's not the winrate optimizing play. The winrate optimizing play is to fold. We have to sacrifice expectation in this hand to increase our expectation in the match. That was a pretty shocking discovery for me when I first found out about this concept. If we can use math to prove that sometimes the max ev play is not the best play, why do great players advocate to always play each hand in the max ev way, on matter what? Why are people are often laughed at when they argue for "waiting for a better spot?".

It's because winrate optimizing math is a pretty delicate thing, it's because it takes time to understand it's implications and how it might change our decisions in real scenarios. So I would really advocate to anyone hearing about this concept for the first time, do not try to implement it into your thought process. It can only hurt your expectation. Play each hand the max ev way and don't wait for better spots.

If you want to ask more specific questions, feel free to do so.

javir's picture




I understand all the logic on this article, but I disagree with the approach.... I understand I am a beginner and maybe after you read this you might think "poor guy, he will never make it" or "OMG what a fish"....  But I read this article like 4 times and I couldn't even sleep last night thinking about what I am just about to say... I am not a math nerd but logic seems different to me on this one... I KNOW I am VERY likely to be wrong... I would apreciatte guidance in order to desestimate whatever bs I am talking.

I will continue with the example of a $100 HUSNG to make it simple... I disagree with the value of the starting chips for a reason..... you can't never cash out... So, in  this case is plain simple: 3,000 chips are worth $190 (because of the rake) and 2,990 or 10 chips are worth $100.. since you haven't been able to capitalize your net winnings yet, in this case the value per chip due reflect a change... for the person who has 10 chips each represents a $10.00 value, and for the person who has 2,990 a value of $0.03. If we think that hero's chips are worth $100 and Villain's chips $90... we shouldn't be taking exact coin flips because long term you will lose... For example: You are both dealt AA every hand long term you will be 50/50 but when you lose vs flush you will lose $100 Vs $90 when you win... in a 1,000 hands sample where there is always a flush on board thats a lose of $500, I know you wont fold your hand if you have the knowledge of the villiain's holding, but maybe we should at the beginning of the game....

When we are talking about +EV and +EG I totally agree we should exploit to the maximum those situations where we definitely have an edge... but we should consider that post flop play take the chances of winning higher or lower, so I assume we are speaking about the end game here.... For instance Would you call your A 7 Vs QK on the very first hand of a regular HU?? I know I wouldn't I could win the hand but according to logic, I am not getting the odds to win: 0.9:1 (risk $100 to win $90) in a 54%vs47% situation... I wouldn't take those odds.

However, what would happen if that situation is effective when you have 2,250 chips? each chip is worth $0.04 and you are risking 750 or $33 meaning you are getting 2.7:1 odds ($33 to win $90) so I would even take the QK against the A7!!!!!!!!!!!!!

This is my approach in logic to the HU betting, thats why on situations we can call with J10s, Q9s etc, even considering the fact that the oponnet MIGHT have our hand dominated. When we bust him out, we capitalize our money... 

There is something I also take in consideration: The value of your hand!!!!

If you are down to 10bb (wether blinds increased or you got stacked) each hand costs $10 to be played... meaning folding might be to expensive to our purposes and we need to widen our shoving/calling range, simply cuz we can't afford folding anymore, same applies to 5 or 7bb... As in the hyper turbos, your cost per hand is considerably higher than in a regular speed.

As I said I am not by any means Math geek... I would kindly appreciatte a review of this saying and will be awaiting for a more in depth insight.. Thanks.