I prefer books with the academic thoroughness to employ somebody qualified to actually proof read the manuscript... but that's just me.

A maths book about poker. Its unlikley to improve your poker much, but its gives an intersting insight into the theory of poker games.

Example 20.5 might have an erratum in middle of p. 257 at:
"X should fold immediately if the value of folding is less than value of calling"
Seems like it should be more instead of less.
Correct me if I'm wrong.

There seems to be an erratum at p. 261 Case 1 where the inequality sign in the equation should be "<" not "_>".

The turn bet example on p260-262 is interesting. As I understand it, it is a Holdem Limit-like situation where player Y has 99+,AQ+ and player X checks As4c on the turn OOP with a 5 bet pot. The game theory optimal (GTO) solution given by MOP is for Y to bet the turn with the top 60% of his hands and X to call 80% of the time. The interesting thing here, that is not discussed in the book, is that it implies that Player Y should optimally bet either TT or 99 (each have outs to straight) here. We assume all rational players would bet AA,AK,AQ,JJ here for value. Hands KK, QQ, and either TT or 99 should be a checked. Betting more hands is spewy and betting fewer hands loses value. Someone could do the math to verify this.

In real life situations things are more fluid. Player Y might do better by exploiting flaws in Player X. If X deviates from 80% calling he can be exploited. If Y knew player X was a 100% calling station, X could be exploited better by checking both TT and 99. If player X was a little more risk-averse at 60% calling, then Y could possibly bet both TT and 99 to exploit X. In either case Y does better than by playing GTO style.

This can be applied to a NL situation but the various bet sizes and raise options make it very complex.

Pokerlogist, and potentially others...

I will preface my statements by saying that I have not yet finished the book, but am well on my way.

I agree with your statement, in principle, but I think there are a few things we need to keep in mind here when we look at this book.

This is not just a math book. It is better described as a statistics book, which by definition deals with uncertainty. As is often the case, I believe the authors, in order to demonstrate ideas and give the reader a starting point are setting up models to be interpreted (toy games, later examples, etc.) By doing this, it allows the reader who has a good understanding of the models to have the ability to make adjustments based on the information gleaned to match the real life situation they may be facing. I think most "good" poker players are able to do this by feel and likely don't even know specifically why they are doing it, or do know why they are doing it (this guy "always" likes to Cbet....approaches 90-100% in math terms) but don't know the math behind the decisions. If we as somewhat average players (I am WAY giving myself the benefit of the doubt, lol) can get a handle on some basic concept of the math, we may be able to elevate our game somewhat to what others are able to do by feel.

Because there are figuratively infinite variables and combinations of variables (in terms of math... bluffing percentage, calling percentage, tightness, looseness, percentage the opponent is multi-barreling...the list goes on and on) the equations become prohibitive to "solve" for real world situations. Also, the math used in this book is not adequate to describe the multitude of possibilities for these models as they approach real world situations...correct me if I am wrong Cangurino, but I believe that we would need to use fairly advanced Calculus to even begin to make this approach which if far beyond the math most of us have at our disposal. The only way I have been able to make heads of tails of it is that I actually had a statistics class as part of my undergrad.

So far I have found this book to be very interesting, though to say the least a tough read. In my mind I can see where it may be helpful but I have yet to get to the point where I am anywhere near ready to employ the concepts.

I am hoping I will get there soon.

I'm looking forward to what others have to say.

Fammy

I need some help if one of you math gurus has a spare second.

On page 112, which is the introduction to half-street games, towards the bottom of the page it says:

(pot size)(frequency X folds) = (bluff bet)(frequency X calls)
P(1 - c) = c
c = P/(P + 1)

I don't really understand what's going on here, or how we got from the first line to the third line.

Is P the pot size or the probability?

How can P(1 -c), which is the frequency that X folds be the same as c, which is the frequency that X calls, unless it's 50%?

The core of the book is excellent. I was trying to add to the book. MOP's GTO approach is essential especially when you don't have much read on you opponent which is often the case.

Someone should write a version of this book in layman's laguage and also eliminate errata. It should go further into the implications of GTO for practical play which MOP doesn't do. That's what active poker players really wants I think. I was trying to do a little of that. GL with it.

page 112

P=pot size, not probability
c=frequency (proportion) of time X calls
1-c= frequency (proportion) of time X folds
bluff bet= size of pure bluff bet =1 unit always

MOP is trying to determine how often X needs to call to breakeven with Y versus a pure bluff bet by Y when pot size=P. Say the pot is 5 dollars. Y bets 1 dollar with total trash so that when X calls, Y loses 1 dollar all of the time. If X folds, Y wins 5 dollars. Then MOP says X should call c=p/(p+1)=5/(5+1)=5/6. If X calls 5/6 of the time do the payers breakeven?
verifying from Y's view EV=($5)1/6+(-$1)5/6=5/6-5/6=0. Yes

in general then for EV=0, you would want ($P)(1-c)+(-$1)(c)=0 then dropping the dollar signs and applying algebra in baby steps:
P(1-c)=c
P-Pc=c
P=c+PC
P=c(1+P)
c=P/(1+P)
hope this helps

Thank you for that explanation.

I think I was mostly getting lost going from the first step to the second, but I found my error.

(pot size)(frequency X folds) = (bluff bet)(frequency X calls)

so
P(1-c)=bc
P(1-c)/b=c
but since we know that b=1 unit we can simplify to
P(1-c)/1=c
P(1-c)=c

In this toy game, if we didn't have the best hand 50% of the time, would this somehow change the formula?

The proportion of bluffs to value bets is always 1 : (P+1)

No, if I understand you and the MOP right. When MOP says "frequency" they generally mean the top proportion of hands in a range. X is assuming that Y is bluffing some hands and it is a GTO proportion of hands. So if the game was simply matching single cards 2 thru Ace, X may be obligated to call with as low a card as a 5. In the example above, where Y bets $1 into a $5 pot, according to the formula X should call with the top 5/6 (83%) of his hand range which would includes 5 thru Ace and but he should fold 2 and 3. 4 is on the border. Even though 5 is in the bottom 50% of his overall hand range. This is because the pot is large in ratio to the bet and Y is bluffing with a 2 or 3 or 4 card sometimes.

Hello,

I have been wondering about following. I re-read it now (after 6 months) and still feel that there might be flaw in logic. Please comment!

Example 21.1 (page 123)
Both player have equal stacs of S units
each player receives a number uniforly from [0,1]
player without button (call him defender X) post a blind of 1,0 unit.
player with button (call him attacker Y) posts a blind of 0,5 units and acts first.
attacker may eiter jam (all in S unit) or fold
if there is a showdown lowest number wins.

...

The expectation of folding for Y is -0,5
(my view: This I agree with)

Since X will never call with hand worse than y,
(My view: X will never call with hands in region (y,1] since Y would have better hand and win if X call ... right?)

X allways wins when he calls.
(start of my view: WHAT?? WHY??. As I read the example each player know only his number... right?? There is no statment that Y is clairvoyant .. right?? ... Soooo.. my reasonig would state that when X calls:
Y:s hand [0,y]
X:s hand [0,x] where x<y.

When X calls and Y:s hand is within region [0,x]each player has eqal equity. Propability that X:s hand is within this region is 100% (by definiton). Probability that Y:s hand is within this region is -1*( (0-x)/(0-y) ).

When X calls and Y:s hand is within region (x,y] Y has 0 equity.

This is in contradiction with the books following statement

<Y, jam|y>=(defender calls) (-S) + (defender folds)(+1)

Verbal summary: I would say that Y has more equity since he will win some of the times when X will call.

)end of my view

Please comment!

p. 123
The authors don't say it explicitly, but X can see Y's number. Y can't see X's.
X's only "strategy" is about as basic as you can get: call when ahead of Y's number. X must post a higher blind, so that may be why he gets this advantage. Despite this aid, the authors show that X will still lose money when stack sizes are 1 or 2 as long as Y chooses the right (game theory optimal GTO) threshold to jam with. With larger stack sizes Y is in trouble as you would expect and just tries to minimize losses.

Mathematicians tend to write like they are using their own blood for ink, as terse as possible. This book is written usng implicit assumptions, conventions, and notations that produce gaps for many readers.

p.116
[0,1] Game #1
Y can either check or bet.
X has to call a bet.

It says underneath Figure 11.1:

Half the time, Y bets, and half that time, he gains a bet, making his overall equity from the game 1/4.

This doesn't seem right to me.

Y is betting his top 50% of hands, or 0-0.5. X is calling everything. So when Y bets, the average strength of his hand is going to be .25, and the average strength of X's call is 0.5. How is Y only winning this half of the time?

I would argue that this is not the case. P. 124 when X equity is discussed there is a consideration that X might lose. This is an contradiction to X beeing half (or full) clairvoyant.

Hmm.. as quite familiar with math I would say that mathematicians more often include all assumptions within text. This particular book had quite many print error in first two prints. I would also say that this book is quite friendly (thou it will take hours to go through) to casual reader when compared to text intended for other mathematicians.

I'm trying to figure out the clairvoyant inconsistency on page 123 as well.

p 116
Seems correct to me. Y bets only with a .5-1 range. When X has .50-1, Y is losing 1/2 of the hands. Specifically:

50% (2/4 of time overall ) of the time Y doesn't bet and so gains 0.
When Y bets and he is up against 0-.49 half (1/4 of time overall) of the time and he always wins $1.
When Y bets is he is up against .50-.1 the other half (1/4 overall) of the time and has net gain of 0.

So overall ev=1/4

[QUOTE=Monsieur;8516705]I would argue that this is not the case. P. 124 when X equity is discussed there is a consideration that X might lose. This is an contradiction to X beeing half (or full) clairvoyant.

Upon more careful reading, I think you are correct. I take back my post about p. 124. The line in the text is misleading. My best guess is that the clause on p. 124 " ..,X always wins when he calls" should read something like "...X will have a positive expected value when calling".

Lower hands are better IIRC.

So if Y bets with a 0-.5 range, he's going to win everytime against X's .5-1 range (remember X has to call every bet) and he's also going to win half of the time against X's 0-.5 range. So 3/4 of the time he bets he wins, and he obviously doesn't win anything when he doesn't bet.

EV=0(.50)+.75(.5)=.375

What am I doing wrong here? FWIW I think the book is mistaken.

Y will also lose 1/2 of the time versus that range to cancel out the wins.
So when he bets, Y's net gain versus X's 0-.5 range is really 0.


yes I happened to invert the ranges in my post.

I'll have to check which print I got, I just know I bought it recently and it's riddled with mistakes. Also, it is a pity that it is quite unfriendly to fellow mathematicians.

How is Y going to lose if he's only betting better hands???!!!

Remember, the lower the hand the better.

So if Y has a hand that's 0-.5 he is going to win 100% of the time against X's calling range of .5-1.

Y is still going to win half of the time against X's 0-.5 range.

Yes he'll win (+$1) half of the time but he will lose (-$1) the other half of the time so his net=0 versus X's 0-.5 range.

Okay I found my mistake. Thanks for the help again.

Hang on.

If he has 0 EV against X's 0-.5 range, and 1 EV against X's .5-1 range, shouldn't his total EV be .5 overall?