# Poker Video: Misc/Other by bellatrix (Micro/Small Stakes)

## Math Attacks: Episode Eleven

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### Math Attacks: Episode Eleven by bellatrix

Bellatrix in onto chapter 11 with a topic of toy games and half street games. No homework this week.

Bellatrix takes you on a journey through The Mathematics of Poker by Bill Chen and Jerrod Ankenman, breaking down each chapter one at at time. Warning - if you haven't figured it out by now, there will be math!

### Video Details

• Game:
• Stakes: Micro/Small Stakes
• 39 minutes long
• Posted almost 2 years ago

## Comments for Math Attacks: Episode Eleven

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#### random_99

161 posts
Joined 12/2007

How often do you use game theory at the tables?

#### Diddy

23 posts
Joined 07/2011

I do not understand the second quotation.

Why is c=1+(alpha)?

From my mathimatical standpoint c=(alpha)*P.

#### bellatrix

826 posts
Joined 12/2007

I do not understand the second quotation.

Why is c=1+(alpha)?

From my mathimatical standpoint c=(alpha)*P.

Do you agree that c=P/(P+1)?
If you do and define alpha = 1/(P+1)

1-alpha = 1 - 1/(1+P) = (P+1)/(P+1) - 1/(P+1) = ((P+1) - 1)/(P+1) = P/(P+1) = c

c = 1-alpha

#### paulethomson

53 posts
Joined 01/2008

What I don't understand about the definition of alpha is that isn't it based on the assumption that the better has 50% value and 50% bluffs. Why should we make this assumption? What if the better has 25% value hands and 75% in his range?

#### paulethomson

53 posts
Joined 01/2008

I also have a question regarding the idea that the bottom of our bluffing range gains equity by betting it. I am unable to verify this point. Shouldn't our EV increase with our equity?

For example, let's look at a specific hand. Let's say we have hand .9. Let's assume that it's in our bluffing range. If we check it behind, it's not going to win very often, but it will win all the times the Villain is holding hands between .9 and 1. So the hand has some equity when we check back, therefore won't the EV of that hand be the equity * pot and be +EV. However, if we bluff with it, the villain is going to call with a balanced range, so the EV of the same hand will be 0?

Can you help me with the inconsistency in my thought?

#### bellatrix

826 posts
Joined 12/2007

What I don't understand about the definition of alpha is that isn't it based on the assumption that the better has 50% value and 50% bluffs. Why should we make this assumption? What if the better has 25% value hands and 75% in his range?

both are playing optimally. As soon as the bettor moves away from that equilibrium strategy, you can exploit it and are in Section II of the book. It is assumed that both players will try to play optimally and it is up to us to determine those lines for the two players.

#### bellatrix

826 posts
Joined 12/2007

I also have a question regarding the idea that the bottom of our bluffing range gains equity by betting it. I am unable to verify this point. Shouldn't our EV increase with our equity?

For example, let's look at a specific hand. Let's say we have hand .9. Let's assume that it's in our bluffing range. If we check it behind, it's not going to win very often, but it will win all the times the Villain is holding hands between .9 and 1. So the hand has some equity when we check back, therefore won't the EV of that hand be the equity * pot and be +EV. However, if we bluff with it, the villain is going to call with a balanced range, so the EV of the same hand will be 0?

Can you help me with the inconsistency in my thought?

Yes, this is a bit counterintuitive, but basically you don't want to get caught "bluffing with the best hand". If you bluff with the worst hand, you can be sure that ANYTHING that you fold out will be better, so you have accomplished the ultimate goal of bluffing, which is to fold out a better hand. As soon as you have "some" vague hope of winning at SD you can start bluffing less and less the better you move up in your range until you reach that magical other point where you actually start betting not to fold out better but to be called by worse.

I would rather get caught bluffing with the nut low than some Ahigh type of hand on the river, because the Ahigh hand still had a chance to be winning at SD, the nut low doesn't. As Mike Sexton would say "the only way he has to win is to bet (bluff)". And by that you conserve a tiny bit of equity in the pot - fold equity.

#### paulethomson

53 posts
Joined 01/2008

both are playing optimally. As soon as the bettor moves away from that equilibrium strategy, you can exploit it and are in Section II of the book. It is assumed that both players will try to play optimally and it is up to us to determine those lines for the two players.

Thank you sooo much for your quick reply... you're AWESOME! Unfortunately, I don't think I was clear with my question. I'm going to restate it and try and be clear... I really appreciate your time and don't want to waste it.

Instead of looking at it in context of a constant distribution. Can we look at it in terms of the following scenario. (if you want you could probably skip to the bottom of the post for my analysis. But if it doesn't make sense, you could look at the scenario I created)
- Pot is \$2
- Bet Size - \$1
- X must check-call or check-fold
- Y either bets \$1 or checks.
- X hand distribution is made up of the following range: 100% bluff catchers
- Ys total hand distribution BEFORE BETTING is made up of the following range: 70% bluffs 30% value

Below is how I solve for Y to have a optimally balanced strategy.

Y is going to bet \$1 into a pot of \$2. Y is risking \$1 to \$2, so he needs his bluffs to work 33% of the time to be able to bet any two cards profitably. Therefore, X needs to call 67% of the time to make the Hero indifferent to bluffing.

Y is going to be t\$1 into a pot of \$2. Therefore, X will be getting 3-1 on a call. Therefore, he will want to have a ratio of 3 value hands for every 1 bluff combo. So the correct bluffing frequency of his betting range is 25%.

OPTIMAL BALANCED BETTING RANGE:
He will bet all his value hands: 30%
He will balance it with the correct bluffs: 10%
-----------------------------------------------------
TOTAL BETTING RANGE = 40% of all hands

OPTIMALLY BALANCE CHECKING RANGE = 1- 40% = 60% of all hands

So he's going to bet 40% of his hands for value, and he's going to check back the other 30%.

EV for Y on Bluffs:
Y bluffs, X calls, Y loses = .10* .67 * -1 = -.067
Y bluffs, X folds, Y wins = .10 * .33 * 2 = .066
-------------------------------------------------------------
Y'S EV of BLUFFING = ~0

EV for X to Call:
Y bets, X calls, Y wins: .75 * .67 * -1 = -.5025
Y bets, X calls, Y loses: .25 *.67 * 3 = .5025
----------------------------------------------------
X'S EV on a Call = 0

So we know that we found balanced range because Y is indifferent to betting or bluffing. And X is indifferent to calling or folding. And we know it's optimal because Y is betting all of his value hands, right?

And if that's all true. We also know the following:
Y is "bluffing with this frequency of his dead hands" is equal to: .10 / .70 = 14.2%
The ratio of bluffs to value bets = .10 / .30 = 25%

Now in the book isn't alpha defined the same way? Isn't Alpha the frequency of dead hands which is optimal to bluff? So if we use the definition in the book.

Alpha = 1 / (P + 1)
Alpha = 1 / (2 +1)
Alpha = 1/3 = 33%

So where am I going wrong? In the book at the bottom of pg 113, it says, "Recall that he holds half nut hands, and half dead hands...." And if this is true, then I do get the same answer for Alpha as in the book. But isn't this a very limiting case? Aren't their plenty of times when we will have a large frequency of bluffs in our range on the river such as 70% bluffs and 30% value... when a bunch of draws miss on the turn?

#### paulethomson

53 posts
Joined 01/2008

I would rather get caught bluffing with the nut low than some Ahigh type of hand on the river, because the Ahigh hand still had a chance to be winning at SD, the nut low doesn't. As Mike Sexton would say "the only way he has to win is to bet (bluff)". And by that you conserve a tiny bit of equity in the pot - fold equity.

Yeah, I'm pretty sure I understand what you're saying. Aren't you simply saying that we should bet all of our value hands, check-back our showdown able hands, and balance our value hands with the worse hands in our range (excluding obviously blockers and things like that which would manipulate the villain's calling range in our favor)?

Yes, this is a bit counterintuitive, but basically you don't want to get caught "bluffing with the best hand". If you bluff with the worst hand, you can be sure that ANYTHING that you fold out will be better, so you have accomplished the ultimate goal of bluffing, which is to fold out a better hand.

Well if our hand on the river has 0% equity isn't the EV of that hand 0 EV if we check-back. Now if we bet that same hand, and the Villain folds better hands, but he will also call with a balanced range so that the EV of our bluff is still 0 EV. How did we really gain equity with this hand if in the end the hand is still 0 EV?

I understand that by including bluffs from the bottom of our range that the Villain then has to defend against them which allows us to get paid off with our value hands. But how does it increase the EV of the bluff? Am I confusing something between the EV and equity?

As soon as you have "some" vague hope of winning at SD you can start bluffing less and less the better you move up in your range until you reach that magical other point where you actually start betting not to fold out better but to be called by worse.

I'm a little confused by this. If we're in position on the river aren't we always just betting any hand with 50% equity or more against the Villain's calling range? And then don't we simply balance with the correct ratio of bluffs to make the Villain indifferent to calling? I'm interpreting your post as saying that we should bet a hand which has less than 50% equity against the Villain's calling range.

Is my premise wrong that if we're in position on the river aren't we always just betting any hand with 50% equity or more against the Villain's calling range? And then don't we simply balance with the correct ratio of bluffs to make the Villain indifferent to calling? (I'm just going to assume that this is correct...)

Now let's imagine a scenario which I think exists with the constant distribution hand. So let's say that we can bet X% of hands for value, and we need to balance it with Y% of bluffs. The Y% of bluffs is going to come from the very bottom of our hand distribution. Now even though they are from the bottom of our hand distribution, the Y% of bluffs still has a tiny bit of showdown equity only against the very very bottom of the Villain's range (which would be check-folding), but they have 0% value against the Villain's check-calilng range.

Now when we bet our bluffs, they're going to be called by a balanced range so the EV of that bluff will be 0 EV. But if we check-back the hand, the EV of the hand will be +EV because it's going to beat the very very bottom of the Villain's checking range which was planning on check-folding. So instead of being able to realize the tiny bit of equity by checking back, we turned the hand into a bluff which ended up being 0 EV.

Now I understand that this might be the best way to maximize the EV of our entire range because we need to bluff some hands, so the Villain will need to call. But I still can't quite wrap my head around the idea that this somehow raises the equity of our bluffs.

Not sure if I explained this any better. Thx again.

#### bellatrix

826 posts
Joined 12/2007

I'm gonna need a bit more time to respond to these. Please give me a day or two, ok?

#### paulethomson

53 posts
Joined 01/2008

I'm gonna need a bit more time to respond to these. Please give me a day or two, ok?

Sure. I appreciate whatever time you have.

I guess another way to state my contention that our EV is higher in the 2nd hand is that even at the number 1 which is the worse hand in our distribution, it will even have a tiny amount of equity when we check-it-back, since it will split the pot the smidgen of the time that X also has 1. So even the worse hand in our distribution range has some equity, and therefore is +EV when we check. Whereas, as soon as we bet it, it has 0 EV because the Villain will call with a balanced range.

#### paulethomson

53 posts
Joined 01/2008

just a friendly reminder post in case you forgot about my questions

#### paulethomson

53 posts
Joined 01/2008

just checking in again.

#### paulethomson

53 posts
Joined 01/2008

another post as a reminder. if you're having troubles answering the questions, can you please just let me know. Or if you're having trouble finding the time to respond. I'd just appreciate it, so I'm not waiting indefinitely.

#### Amygdala

2 posts
Joined 07/2010

could someone please show me why c = p(1-c) = p / (p+1) ?

#### Amygdala

2 posts
Joined 07/2010

I seriously have trouble with the math of it all, but isn't the whole point of alpha and optimal play to force our opponent to play optimal too.

When both villain and hero play maximally exploitative strategies EV is always 0.

When villain stops playing optimally and hero keeps on playing optimally, villains EV will decrease and Hero's EV wil increase.

Just as long hero is playing optimally hero is indifferent about villains calling of folding frequencies just as long his/her frequencies aren't optimal.

I don't get why being indifferent about our opponents frequencies is mentioned in the book and these videos, because we do want them to differ from being optimal...... uhmmm.. right??

#### bellatrix

826 posts
Joined 12/2007

I do think there are some definitional misunderstandigs here. For example, this quote:

When villain stops playing optimally and hero keeps on playing optimally, villains EV will decrease and Hero's EV wil increase.

is true, but not really the point. It doesn't help us find the optimal strategy as the best strategy against an opponent that deviates from optimal is an exploitative one. We do want to our opponent's strategies to differ from optimal, of course. Every time you deviate from optimal strategy, you give your opponent ammo to take an exploitative line.

The thing is that for finding the optimal line we need to act as if the opponent is also playing exploitative. You need to find that balanced strategy that can't be exploited that maximizes EV. But this will not be the best strategy overall if our opponent can be exploited.

I do feel it is a problem with the wording as in "optimal" is the "best of the best", hence it should make us the most money, right? Wrong. While optimal strategies in general are quite good and won't lose us money, exploitative strategies are the one that really make us the most money.

So in summary:
- Try to find the best exploitative strategies against your opponents. This will make you the most money and is described mostly in Part II of the book. This is also what you try to do with reads, Pokerstove, etc. and when hands are discussed in forums. It includes knowledge of typical lines, etc.
- However, at the same time you want to be the person that can't be exploited, where the opponent cannot do anything to counter you, where no matter what line he takes the best he can do is the optimal strategy. To find that, you need to assume opponent is playing optimally.

Hope that helped a little...

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