Hello everyone!

I've been watching poker videos and reading poker books and forum posts and as I try to work out the advice the instructors give on my own, questions arise. The answer to this question is super fundamental but I don't see it written explicitly in poker books/forum or said in videos (except for one of Ed Miller's limit books).

One poker advice I'm trying to understand is the reason why we bet : "We bet to make better hands fold or worse hands call." One question that I asked myself is: "If for some situation, better hands wont fold, but worse hands will call, how much of those worse hands need to call in order to make betting the right play over checking?"

A lot of people reason that a bet is for value if worse hands call or if you are ahead more often than not. Before today, I used to bet just because I thought a few worse hands would continue. But these definitions are too vague and neglects one important factor: # of players in the hand.

Example:

Let's create a simple Hand vs Range scenario. Hero is IP on the river versus a semi-loose passive. Because the villain is passive, he will only check/call the river as opposed to check/raise or bet. Also since he is semi-loose and not a complete calling station, he will check/fold his worse hands.

The actual hole cards and run out of the board doesn't really matter because all we care about are how many combos are ahead or behind, and how he will play those holdings. So for simplicity's sake:

Villain's Range
# of combos that are ahead and will check/call: 3
# of combos that are behind and will check/call: 2
# of combos that are behind that will check/fold: 5
Total Combinations : 10

Lets also say that the Pot is 10bb and the Bet amount is 5bb.

Now let's compare the Hero's EV of a Check to that of a Bet.

If the Hero check's behind, he will win 70% of the pot because the Villain has 7/10 combos that are behind.
<Check> = Win% * Pot = 70% * 10
<Check> = 7bb

If the Hero bet's, 50% of the time the villain will fold and win the pot, 20% of the time the villain will call and the Hero will win Pot + Bet, and 30% of the time the villain will call and the Hero will lose his Bet.
<Bet> = Fold% * Pot + Win% * ( Pot + Bet ) - Lose% * Bet
= 50% * 10 + 20 % * ( 10 + 5 ) - 30% * 5
<Bet> = 6.5bb

In poker, we never want to choose the play that is just +EV, we want the play that is the most EV. And its clear that Checking is the superior play in this scenario.

I've heard or read that it would be a good spot to thin value bet since when the villain calls, you are winning 40% of the time and it only needs to work 25% of the time because of pot odds. I may be missing something, but if not, it seems like there's a more specific reason to bet here.

How much of his worse hands does he need to call?

It turns out that of the times the villain calls, the Hero needs to win more than 50% of the time to make betting the right play (Villain must have more worse hands than better hands). At 50%, the EV's are the same and surprisingly, its the same regardless of the Bet size. Further, this percent is dependent on the number of players in the hand. So for n players in the hand, Hero must win more than 1/nth of the time when he is called in order to justify Betting over Checking.

How should we size our bet?

One other important idea is sizing your Value Bet. I touched upon this earlier, but one misconception is that you want to size your bets so that the success rate is higher than the percent given by pot odds. In the example above, a half pot bet needs to work 25% in order to be profitable. This is definitely true, a half pot sized bet will have a positive EV when the Villain calls. But Betting wont have the highest EV, Checking will. What's important is that you need to choose your bet sizing so that the villain will call with enough worse hands such that Hero wins more than 1/nth of the time.

I hope that people on this forum may discuss my findings and help validate or invalidate anything that I have said here to further my understanding of poker. If what I have said is true, I hope that this helps other understand value betting and bet sizing a little more as it did for me.

It's actually fairly well known that one definition of a value bet is "a bet that wins more than 50% of the time when it gets called".
i.e. We don't bet for value because we're beating half of villain's total range. We bet for value when we beat half of his calling range.

A fat value bet would be one that wins much more often than 50% of the time when it's called. A thin value bet is much closer to a 50/50 proposition. Any bet that wins less than 50% of the time when called isn't a value bet.

"How should we size our bet?"

So that [(P villain calls with worse)-(P villain calls with better)] * (bet size) is maximized. As you might have figured out already, if P villain calls with better is greater, this will always be negative.

Making the assumption that better hands don't fold. Add in fold equity times pot size to the sum if there's a chance of that happening, but generally when you value bet, you'd never assume better hands to fold because then how could worse hands call?

The key to thin value bets is your hand reading skills. Being able to bet the river with one and two pair hands when you slightly ahead in hold'em.

I actually think these marginal hands are the key to the game. Its easy to win when you have the nuts. Can you maximize your wins and minimize your losses with an over pair or two pair hands.

Thank you! It must have just been me and my poor reading comprehension. And thank you for spelling out the difference between thin and fat value bets because for the longest time, I thought getting some worse hands to call is thin value betting and getting a lot is fat. I didn't compare it to the 50% success rate it required to be a value bet.

In addition to the 50% u also have to consider that you need more than 50% if Villain has the option to raise and you have to fold vs the raise (and Villain is capable of bluffing)..

Therefore, thin value have to be stronger vs agressive and more component players..

I agree with what you said here. I'm trying to figure out if the other part of this equation needs to be included. That is if we also need to consider maximizing (P villain calls with worse) * (Pot). Since it is generally the case that (P villain calls with worse) is greater than [(P villain calls with worse)-(P villain calls with better)] and the Pot is greater than the Bet size, would it make more sense to maximize the calling percentage? In other words would maximizing (P villain calls with worse) * (Pot) be the main way to maximize the EV of Hero betting and Villain calling? I haven't studied this enough so I'm not sure.

Ah I see your logic. A competent player wouldn't fold their better hands and also call their worse. Thank you for this!

the first part you need to internalize is how often the size of your bet makes them call

so do a simple scenario:

river is $100, villian check to you, you are ip. you have the best have 80% of the time, of the 20% that beats you it only calls--never XR (we can add this aspect later). So how much do you bet?

maybe you'd find some things useful itt:
http://forumserver.twoplustwo.com/58...-quiz-1570476/

This is pretty much the same scenario as my example, except I have the best hand 70% instead of 80%. I agree that we need to play around with different bet sizes and estimate how often they will call with each bet size.

The answer, thanks to Vanhaomena, is when [(P villain calls with worse)-(P villain calls with better)] * (bet size) is maximized and greater than 0. This maximization guarantees the highest EV play.

Mmk, so whats the answer?

No no, there is no estimate. That is why I said the first part of what I did. Which I figured you don't understand and what that thread I linked goes into.

I looked through the thread and although I am a little confused, please point out any errors in my analysis. It seems like our bet size can dictate his calling %. For example, If we bet $150 dollars, our bluffs need the villain to fold 150/(250) = 60% to make our bluff break even. Thus he would have to call 40% of the time, 50% of the time he calls, we win and the other 50% we lose. But this will only get us the same EV as checking. So after doing some more calculation, it looks like betting about $58 dollars is what maximizes our EV.

60$$$

Nice, yeah, $58 is correct.

Right yeah this is what I meant by there is no "estimate."


When I was first working on this stuff I'd build excels with the math in there so I could see all the #s, like this--just type in where its red and it spits out everything else.



Facing a XR 10% of the time?




Usually the threshold for the amt to bet stops around 50% of pot, ip on the river.

Can't emphasize enough how good working this stuff out is. Even in the day of solvers etc, its still really important to have good understanding of the basic maths.

Thank you for this! I learned a lot through this exercise. I haven't come across any books or videos that teaches you this. Where did you learn this?

Does this fall into GTO? We were able to get his calling percent because based on the bet size, he needed to call enough of the time to make our bluffs break even. This is similar, as finding the indifference point. So villain needed to call with an optimal frequency so that he doesn't get exploited?

Tiptons books cover some of this, Mathematics of Poker cover rest.

Yes exactly.

Np, some from books, talking with friends, just diving into the work trying to figure it out. The actual math isn't that hard but figuring out how it all applies isn't that easy. I can't stress enough how important getting the math down is for building a solid foundation for poker.

the 2 doctor877 mentioned and also "applications of no limit holdem" by Matthew Janda.

Also, there's a youtube series that I stumbled across where the guy does a really good job of going over GTO math concepts for poker. Starts a bit slow but there's like 30 parts and gets deeper. Would recommend checking it out.

Bump for this awsome excel work...

Tbh this is easier to understand/follow then any softwares that are currently out there (from my POV). I'm assuming this scenario is river only and it only solves for 1 bet size?

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Really interested in how to create one in excel to calculate EV's of a bet or a check, if anyone knows the formula's/algorithms used to making one i'd appreciate the help. I'm pretty sure it's not the complicated but i'm not good with excel/creating these sorts of things...

@ Toocurioussa i msged ya a month ago but you never got back to me haha so i guess ill bump this thread just in case?

The EV of checking the river is the frequency at which we win with the best hand.

Let 'f' be the frequency at which we lose when be bet 'B' pot-sized bets on the river. Assume our hand is at best a breakeven bluffcatcher when we get raised.

When be bet on the river, the villain must defend at a frequency of 1/(B+1) if he wants make us indifferent between bluffing and checking.

This formula gives the EV of betting.
EV = B*(1/(B+1)-2*f)+1-f

The optimal bet sizing on the river is a function of the frequency at which we will lose when we bet on the river. The fraction of the pot which we should bet is given by:



I won't go through the derivation, but it involves using calculus to find the local maximum of the formula for the EV of betting.

Okay...

Right so it's just plug the EV of betting formula & EV of checking of checking formula for IP into the excel box things and then start plugging numbers in and then it spits out an answer?

So confused on how he created this chart/EV calculator...

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Also anyone know the calcs for EV of checking OOP?

I am quite uneducated about GTO but I have cople questions
You sad "just type in where its red "and bet size is red and that is what we want to figure out.My question is How you get 58 number ?You just keep changing number until it hit EV max or there is some formula?
And we check with 7% of our value hand right?

Correct me if I m wrong.If B is % of the pot we bet (50% then B=1/2) then when we bet B we should have B/(2*B+P) of bluffs where P is size of pot and 1-b/(2*B+P) of value.Right?And V must call B/(B+P),then EV would be

EV=B/(B+P)*((1-B/(2*B+P))*(B+P)-B(2*B+P)*B))+(1-B/(B+P))*P