“What does poker have to do with mathematics? I hate mathematics anyway!”
Whether we like it or not, and whether we are aware of it or not, mathematics governs every decision we make while at the poker table. David Sklansky wrote:
“Every time you play a hand differently from the way you would have played it if you could see all your opponents' cards, they gain; and every time you play your hand the same way you would have played it if you could see all their cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.”
This quotation may be written in plain English, but it is based on mathematical principles. The idea upon which it is based is that if we could see an opponent’s cards, we would be able to determine an action which maximizes our expected value (EV) i.e. make a decision which could be mathematically proven to be optimal. For those of you who don’t know what EV is, it is the amount of money we expect to make on average for a certain scenario e.g. if we flip a coin and every time it lands on heads you give me $3 and every time that it lands on tails, I give you $1, the EV for me of this proposition is=0.5x3-0.5x1=$1. Note that your EV is -$1 due to the fact that money is neither created nor destroyed in this example (this is not the case in poker though due to rake). What this means is that if one or more players have a +EV proposition, one or more players have a –EV proposition. This is important because every time your opponent makes a –EV decision, you make a +EV decision (assuming only two players).
It is impossible, of course, to know for sure what your opponents’ hole cards are (unless you play on Ultimate Bet ). For this reason, we need to adopt the concept of a range. Almost all of you should already know what a range is, but for those of you who don’t, a range is simply the possible holdings an opponent (or you yourself) could have. For example, if a regular with 14/12 stats raises from UTG, he might have AA, but that’s not all he could have, he more realistically will raise with the following hands AA-99, AK, AQ and occasionally suited connectors and smaller pocket pairs for deception and balance. This means that this player’s range for this hand in particular (ranges are not static) is AA-99, AK and AQ (plus the other hands they might use for balance). Now that we have an idea of what a range is, the fun can really begin.
The prerequisites for this Concept of the Week are as follows:
Knowledge of how to use StoxCombo (See COTW 1 : Equity Exploration)
Basic understanding of algebra (I’m sure everyone already has this)
Understanding that of what EV actually is (Value x probability of occurrence)
Patience (this is key – the maths is not very hard, but it is important to understand how we derive the formulae – also, due to the number of variables we are considering, the formulae derived are quite long)
Whether we notice it or not, there is actually a large amount of mathematics in our pre-flop strategy, it is just that with the very limited information that we have available; the mathematics that is used is imprecise at best. This is not to say that the mathematics is not important, it is exceptionally so, but it is to say that there is a larger chance for error when playing pre-flop due to the width of most peoples ranges
2.1 Opening the pot with only one player left to act
So picture this, it just folds around to you when you are in the small blind. What do you do? The answer is that, like all things in poker, it depends. While the solution may not be obvious, the objective is: Make the play that will on average, maximise our EV (usually anyway, sometimes we will make suboptimal decisions for meta-game purposes). So what factors are important in working out what play is the best one available? Answer: Stack sizes (a deeper stack gives more implied odds which are just really chances for your opponent to make mistakes which increase your EV), your opponent’s range (which in turn depends on your range, his perception of you, your relevant history etc.) and your opponent’s post-flop tendencies (these are especially important as these allow you to make plays which will maximise your EV by causing your opponent to make mistakes). Note that your opponent’s range is important because it determines the EV of hands which we may choose to play. I know these concepts are so simple and common sense for regulars here at 2+2, but it is important to understand the mathematical reasoning as to why these concepts are important.
Note: In the following maths examples, it is assumed that there is no play after the flop i.e. there are no implied odds etc. In reality this is not the case clearly, however these examples given can be adjusted (and will be later on) in recognition of these implied odds.
Theory:
Say hypothetically that in the SB we raise to 3BB (Effective stacks of 100BB). This means we are investing 2.5BB (in addition to the 0.5BB that we have already posted) to win 1.5BB (because while it may be our small blind it belongs to the eventual winner of the pot and not us).
Assumptions:
Our opponent in the small blind will call with c% hands
Our opponent will 3-bet (raise) to 10BB with r% of hands
This means he will fold (100-c-r)% of the time.
When we are called by the BB we have an equity of e (as a decimal)
We cannot profitably call a 3-bet out of position i.e. we must raise or fold
We will 4-bet to 35BB with b% of hands
Our opponent will either fold or 5-bet shove with s% of hands in response to a 4-bet.
We will call a 5-bet shove with a frequency of f%
When we call a 5-bet shove, we will have an equity of v%
Now when we raise there are three options:
1. Our opponent folds: This gives us an EV=1.5(1-c-r) (NB: I’m converting percentages into decimals now)
2. Our opponent calls. There are two outcomes:
a. We win 3.5BB, EV=3.5ce
b. We lose 2.5BB, EV=-2.5c(1-e) which is the same as EV=2.5c(e-1). The sum of these two are EV=c(6e-2.5)
3. Our opponent raises to 10BB. Now we have two options:
a. Folding: EV=-2.5(r-b) which is equivalent to EV=2.5(b-r)
b. Raising to 35BB. Our opponent now has an option of:
i. Folding: EV=10(r-s)
ii. Raising All-in. This gives us two options:
1. Calling: There are two outcomes:
a. We win 100.5BB, EV=100.5fv
b. We lose 99.5BB, EV=-99.5f(1-v) which is the same as EV=100f(v-1). The sum of these are EV=f(200v-99.5)
2. Folding: EV=-35(b-f) which is equivalent to EV=35(f-b)
This means that the overall EV of raising is given by:
EV=1.5(1-c-r)+ c(6e-2.5)+2.5(b-r)+ 10(r-s)+ f(200v-99.5)+35(f-b) – Equation 1
I know this equation is long and took a while to get to, but it is actually quite useful in measuring overall equities against a villain and hence determining how to best exploit them.
Example:
Let’s assume that our opponent is a player with the following ranges:
They will call with: TT-22,AJs-A2s,K9s+,Q9s+,J9s+,T8s+,97s+,87s,76s,65s,AJo-A2o,K9o+,Q9o+,J9o+,T9o i.e. c=0.300
They will raise with: JJ+,AQs+,AQo+ i.e. r=0.042
They will shove with: QQ+, AK i.e. s=0.026
The remaining unknowns are: e, b, f and v
By looking at calling a shove first, we can determine f and v.
When our opponent shoves, the pot will be 135BB and we will have 65 BB behind. This means we will be getting 135:65 odds. Let’s say 2:1 after rake (5BB). This means we need 33.3% equity to break even.
Against the range of QQ+ and AK, the following range has enough equity to call a shove: 22+, AK
NB: This assumes that we actually 4-bet with 22 and represents the widest range possible. This will be checked later.
This means that f=0.071 and that on average, our v=0.395 on the assumption of this wide range.
At this point, some of you will object: “What?! You can’t possibly call a 5bet shove there with 22!” But what we must realise here is that 22=JJ assuming that we 4-bet with 22-JJ. Do you see why?
The remaining two unknowns are b and e.
In order to determine b, the calculation is quite simple. If the hand is in our ‘call a shove’ range, we can profitably 4-bet with it obviously right? Think again!
JJ and 22 have around 35% equity against our opponent’s shoving range meaning that if we elect to 4-bet, we can call a shove. Let’s work out what equity we need against our opponent’s shoving range assuming we are going to call a shove.
1. We either win 13BB when our opponent folds and this occurs (r-s)/r times therefore EV=13(r-s)/r
2. Our opponent 5-bet shoves and we call s/r times.
a. We win 113BB v times
b. We lose 87BB (1-v) times
Our overall EV=13(r-s)/r+s(200v-87)/r– Equation 2
We need an EV=0 to make it a break-even 4-bet/call. r=0.042, s=0.026
This means we cannot 4-bet and call with JJ-22 but we can with QQ (which has 40.2%) equity against our opponent’s range. This means our true 4-bet and call range is QQ+ and AK which gives a true v=0.500 and f=0.026
The question is: can we 4-bet a hand which we intend to fold to a shove? The answer to that depends on one thing and one thing only – before I tell you what it is, can anyone guess?
Spoiler:
It depends only on whether we can get enough folds out of our opponent’s 3-bet range i.e. if our opponent is 3-betting with enough hands to which he will fold to a 4-bet. Notice that this depends solely on the value of s/r and not hand strength. This proves the concept of 4-betting light – you can’t do it profitably unless your opponent is 3-betting light.
Some of you may say: “Because we are not 4-bet bluffing ever, aren’t we exploitable?” Yes we are, technically speaking, but in practice, we are actually exploiting our opponent’s tendency of never 3-betting light by never 4-betting light. By exploiting our opponent’s tendencies, we ourselves deviate from a mathematically balanced strategy and thus become exploitable. See Sklansky’s No Limit Hold’em Theory and Practice for more about winning the battle of mistakes.
In this instance, our opponent doesn’t appear to be 3-betting light. This means that my gut feeling is telling me that f=b.
To work out whether we should 4-bet light, we need to work out how many folds we will get. Once we get 3-bet, the pot is 13BB and we are considering investing another 32BB to win these 13 BB. So every time our opponent folds (denote this o) we win 13BB and the rest of the time we lose 32BB.
13o-32(1-o)>0
13o-32+32o>0
45o>32
o>32/45 which is equal to 0.711 or 71.1%
This means our opponent needs to fold 71.2% of the time for this to be profitable.
Remember, o=(r-s)/r where o>0.711 – Equation 3
In this case o=(0.042-0.026)/0.042=0.381<0.711
Therefore b=f=0.026
Now the only unknown is e.
Now this is easy, because the EV of raising and folding to a 3-bet is: EV=1.5(1-c-r)+c(6e-2.5)-2.5r
In this case EV=1.5(1-0.300-0.042)+0.300x(6e-2.5)-2.5x0.042
0=0.987+1.8e-0.75-0.105
0=1.8e+0.132
e=-0.07
This means that we can raise/fold with any two cards profitably! Against our opponents overall range when called we have 41.5% equity meaning e=0.415. NB: This took forever for PokerStove to grind out.
Note that this still isn’t quite the 1.5BB that is out there but anyway it is a pretty close.
Notice however, that ranges are dynamic, not static i.e. constantly changing. This means that the optimal play in this situation is for a given hand in time. The next time it is folded around to you in the SB, the BB’s calling and raising range may have adjusted to your opening range.
MAKE SURE YOU DO THIS BEFORE READING THE NEXT CHAPTER!!! This is because this chapter is a prerequisite to the one that follows.
It is folded around to you in the SB. Using the same assumptions that I did, find the optimal raising, 4-betting and calling a shove ranges against a villain with the following range:
By looking at calling a shove first, we can determine f and v.
When our opponent shoves, the pot will be 135BB and we will have 65 BB behind. This means we will be getting 135:65 odds. Let’s say 2:1 after rake (5BB). This means we need 33.3% equity to break even.
Against the range of TT+ and AQ+, the following range has enough equity to call a shove: 22+, AQ+
NB: This assumes that we actually 4-bet with 22 and represents the widest range possible. This will be checked later.
This means we can profitably 4-bet and call a shove with 55+, AK and AQs (Note that 99-55 are only very slightly +EV and so one could make a case for folding these to reduce variance if on a limited bankroll)
Notice how much wider we can 4-bet/call a shove because this opponent 3-bets with a polarised range?
Can we 4-bet/fold profitably?
Equation 3: o=(r-s)/r where o>0.711
o=(0.094-0.047)/0.094
o=0.500>0.711
This condition is not met, and so we cannot profitably 4-bet bluff against this opponent. Therefore b=f.
b=f=0.060 and v=0.464
Now the only unknown is e.
Now this is easy, because the EV of raising and folding to a 3-bet is: EV=1.5(1-c-r)+c(6e-2.5)-2.5r
In this case EV=1.5(1-0.462-0.094)+0.462x(6e-2.5)-2.5x0.094
0=0.666+2.772e-1.155-0.235
0=2.772e-0.724
e=26.1%
Which means we can still open any two cards! Our average e=0.440 NB: This took a long time on PokerStove to grind out!
Notice how our profit is much less against this opponent because he plays a far less exploitable strategy than the previous opponent described in the example. It is also important to note that playing against this opponent will have a much larger variance than playing against the villain described in the example. Note that our e value will be lower than our theoretical calculated one because we are out of position post-flop.
Ok then that's good for you. Hopefully you feeeeeeel the math when you make the decisions. Seriously though, the math is soooo important, if I knew how to exploit every opponent perfectly by intuition, I'd be playing the nose bleeds. Once you know how to exploit your opponent, he has no chance.
lol I was joking btw, even included the barry greenstein "math is idiotic" quote hoping you'd catch that
btw, I like that you are doing all the math behind the stuff, very very useful for people who haven't done all this stuff before, but you chose kinda awful 4b sizing? It's skewing all of your calculations to needing way more FE than normal, and having to call with way looser ranges than optimal
anyways, no more interrupting until you finish writing
I know you were. With regards to the idiotic 4-bet sizing, I know, but the point is here, to not give everyone all the answers, but to teach them how to think about the answers. Also, if you 4-bet to say 25BB which is more optimal and get shoved on, you need way more equity to call. Play around with different sizings if you want. I am here to help people help themselves, not to spoonfeed everyone.
The math should be a thought process in addition to (no pun intended) the actual numbers. Go ahead, modify the equation to suit your purposes, in fact I encourage it.
I know you were. With regards to the idiotic 4-bet sizing, I know, but the point is here, to not give everyone all the answers, but to teach them how to think about the answers. Also, if you 4-bet to say 25BB which is more optimal and get shoved on, you need way more equity to call. Play around with different sizings if you want. I am here to help people help themselves, not to spoonfeed everyone.
The math should be a thought process in addition to (no pun intended) the actual numbers. Go ahead, modify the equation to suit your purposes, in fact I encourage it.
understood, it's just the one thing that's stood out to me so far as and just don't want the newbies out there to assume that 35bb = optimal 4b-sizing because they read it in a COTW, that's all
2.2 Opening the pot with multiple opponents left to act
CAVEAT: I was exceptionally tempted to leave this section out altogether because of its computationally exhaustive nature. This has some optional homework at the end for some very hardcore players, but it is more confusing.
It should be very apparent to anyone who read the previous section (and especially those who did their homework – have you done it??? – DO IT!!!) that solving this problem even with assumptions and with only one player left to act can be very computationally heavy. Imagine this, we are UTG (and so have 8 players left behind us who each have their own ranges and tendencies), how do we decide whether raising a hand is profitable? Obviously, it is a hell of a lot more complex!!!
Due to the computationally exhaustive nature of solving this problem, I will show a simplified (you will soon find that word exceptionally ironic) version assuming we are on the button and the SB (Player A) and BB (Player B) are the only two players left in the hand. There are many different combinations of what could happen in this scenario:
Player A can either: Call, Raise or Fold
Player B can either: Call, Raise or Fold
This means that there are 9 combinations of what could happen when we raise (Note with 8 players to act, there are potentially 6561 combinations of actions). Remember though, Player B’s ranges for each action depend on what Player A does. This gives rise to conditional probability. This makes playing against even two players very tricky, yet alone against 8 more!
Assumptions:
1. Everyone has exactly 100BB
2. We open to 3BB – we will either raise or fold, not call
3. Player A will call with Ca% of hands
4. Player A will raise to 12BB with Ra% of hands
5. This means Player A will fold (100-Ca-Ra)% of the time.
6. If Player A calls, Player B will:
a. Call Cb|Ca% of the time - Overall time it occurs=Ca* Cb|Ca
b. Raise to 15BB Rb|Ca% of the time - Overall time it occurs=Ca* Rb|Ca
c. Fold 1- Cb|Ca%- Rb|Ca% - Overall time it occurs=Ca* (1- Cb|Ca- Rb|Ca)
7. If Player A raises, Player B will:
a. Call Cb|Ra% of the time - Overall time it occurs=Ra* Cb|Ra
b. Raise to 40BB Rb|Ra% of the time - Overall time it occurs=Ra* Rb|Ra
c. Fold 1- Cb|Ra%- Rb|Ra% - Overall time it occurs=Ra* (1- Cb|Ra- Rb|Ra)
8. If Player A folds, Player B will:
a. Call Cb|Fa% of the time - Overall time it occurs=(1- Ca-Ra )* Cb|Fa
b. Raise to 12BB Rb|Fa% of the time - Overall time it occurs=(1- Ca-Ra )* Rb|Fa
c. Fold 1- Cb|Fa%- Rb|Fa% - Overall time it occurs=(1- Ca-Ra )* (1- Cb|Fa- Rb|Fa)
9. We can profitably flat call a 3-bet in position
10. We must raise all-in or fold to a 4-bet
11. Our equity when we call, or when called will be given by the pre-flop equity
12. If Player A calls, and Player B raises, Player A will always fold i.e. Player A never slow-plays a big hand out of position
Potential Events:
1. We fold pre-flop: EV=0
2. We raise pre-flop:
a. Player A calls, Player B calls. Our equity in this scenario is given by ecc. There are two potential outcomes:
i. We win 6BB: EV= 6*ecc*Ca* Cb|Ca
ii. We lose 3BB: EV=3(ecc-1) *Ca* Cb|Ca. In this scenario, our EV=3(3ecc-1)*Ca* Cb|Ca
b. Player A calls, Player B raises to 15BB. We now have three options:
i. Call 12BB (recall Player A always folds): Our equity in this scenario is given by ecr. There are two potential outcomes:
1. We win 33BB: EV=
2. We lose 12BB: EV=
ii. Fold: EV=
iii. Raise to 30BB: Player B now has two options:
1. Fold: EV=
2. Raise All-in. We now have two options:
a. Fold: EV=
b. Call. There are two potential outcomes:
i. We win BB: EV=
ii. We lose 100BB: EV=
c. Player A calls, Player B folds. Our equity in this scenario is given by ecf. There are two potential outcomes:
i. We win 4BB: EV=
ii. We lose 3BB: EV=
d. Player A raises to 12BB, Player B calls. We now have three options:
i. Call: There are two potential outcomes:
1. We win BB: EV=
2. We lose BB: EV=
ii. Fold: EV=
iii. Raise All-in: Assume Player B always folds. There are two potential outcomes:
1. Player A folds: EV=
e. Player A calls. There are two potential outcomes:
1. We win BB: EV=
2. We lose BB: EV=
f. Player A raises to 12 BB, Player B raises to 40BB. We now have two options:
i. Fold: EV=
ii. Raise All-in: There are now four potential outcomes:
1. Player A calls and Player B folds:
2. Player A folds and Player B folds:
3. Player A calls and Player B calls:
4. Player A folds and Player B calls:
g. Player A raises to 12BB, Player B folds. We now have three options:
i. Call: EV=
ii. Fold: EV=
iii. Raise to 30BB:
h. Player A folds, Player B calls. Our equity in this scenario is given by efc. There are two potential outcomes:
i. We win 3.5BB: EV=
ii. We lose 3BB: EV=
i. Player A folds, Player B raises. We now have three options:
i. Call: There are now two potential outcomes:
1. We win BB: EV=
2. We lose BB: EV=
ii. Fold: EV=
iii. Raise to 40BB: Player B now has two options:
1. Fold. EV=
2. Raise All-in: We now have two options:
a. Fold:
b. Call: There are two potential outcomes:
i. We win BB: EV=
ii. We lose BB: EV=
j. Player A folds, Player B folds. EV=
It is evident, that from this that there are around 30 potential outcomes raising could have. I am not going to determine the EV equation for this scenario. I know how to, and I have given everyone else who has read (and understood) the previous section the necessary skills to do this also. I am not determining it because to be blunt, in a practical environment, the equation is impractical to use. There are so many variables, which make this equation very tricky to solve. Namely, these variables are our opponents’ ranges. Note, some of these scenarios will never even occur at a table: How many times have you raised from the BTN, had the SB and BB both raise, you move all-in and both call? In a cash game, I have never-ever seen it – and I have seen some f**ked up stuff happen at the tables. Anyway, there is a simpler way to solve these problems. This is additional homework which I will give answers to at a much later date (I am going to be really busy for the next week and a bit, but will endeavour to give answers asap). First of all, determine the chance of you being 3-bet. Then determine which hands you can profitably raise and fold to a 3-bet with. Then work out 4-bet and stack off ranges. It shouldn’t be too hard. Use the opponents which I gave in the above examples. Villain 1 is SB and villain 2 is BB. If villain 1 3-bets, assume villain 2 has the same range as his 4-bet range and the same range as his stack-off range.
Just as an aside, working out which hands to steal with is probably the most important and so I will present a simplified example. Let’s assume that if we are 3-bet we will always fold. I will come back to this though, I promise.
understood, it's just the one thing that's stood out to me so far as and just don't want the newbies out there to assume that 35bb = optimal 4b-sizing because they read it in a COTW, that's all
Yeah for sure, you are correct sir. It is a valid point to make. So everyone 4-bet to 35BB=spew.
Two Plus Two
). For this reason, we need to adopt the concept of a range. Almost all of you should already know what a range is, but for those of you who don’t, a range is simply the possible holdings an opponent (or you yourself) could have. For example, if a regular with 14/12 stats raises from UTG, he might have AA, but that’s not all he could have, he more realistically will raise with the following hands AA-99, AK, AQ and occasionally suited connectors and smaller pocket pairs for deception and balance. This means that this player’s range for this hand in particular (ranges are not static) is AA-99, AK and AQ (plus the other hands they might use for balance). Now that we have an idea of what a range is, the fun can really begin.
and just don't want the newbies out there to assume that 35bb = optimal 4b-sizing because they read it in a COTW, that's all
Linear Mode
