Tyler, you mentioned Titpon's HU book at start of this video.. just curious if you have learned the majority of how Game Theory applies to poker through books? and if so, which books do you think offer the best information on this?

(I know the following authors have written books on GTO Poker: Will Tipton, Matt Janda, Bill Chen and Jared Ankenman, and Philip Newall.. have you read the books by all of these authors? and are there any other authors that you know of that have made a good contribution to this area? )

The mathematics of poker is the best. The other books are all good. I read them all because even .01 bbs increase pays for the book.

That's a great summation! I was surprised at how much the betsizing oscillate on small equity differences. Adding an EV calculation into the excel spreadsheet would be of good value.

Justification for the smaller betsize is that we could bet smaller with our nut hands, get called gain value, then bet bigger with our bluffs and get folds. So the right play here is for our opponent to fold to any betsize. From a practical standpoint, I'd just go all in, because it makes me seem more aggressive and can never be wrong and is unexploitable. AND its far bigger mistake for them to call the all-in bet with a 1-A strategy, then to call a small bet with 1-A strategy. Let them make big mistakes!

incredible video. In this example, is jamming despite the fact that we dont have sufficient bluffs better than just the size that balances our value to bluff ratio?

Yes, because if our opponent chooses to call, he now loses money (when we bet bigger than size that balances our values to bluff ratio). If we only balance he breaks even, so calling isn't a really a mistake.

so if we have a value hand that wants to bet 75% pot, but we only have enough bluffs to support a 50% pot bet, we should bet 75% of pot regardless? is this because either way, we win the whole pot (bc calling 50% bet has 0 EV and folding to the 75% bet has 0 ev)? so we freeroll that he by mistake might incorrectly call the 75% bet?
And is this the maximum EV for our whole range if its EV is the entire pot on the river? (bc if our EV cud get any higher, our opponent cud just better their expectation by folding their entire range?)

sorry for all the questions but wud really appreciate if u cud comment on this question above ^. Appreciate you taking the time to answer everyone and so promptly. Im just really thinking about this bet sizing strat and I think without adequate bluffs in my range, ive been sizing down for the reason to be 'balanced' when makes no sense to do so.

This is a cool graph. If you bet >= 50% pot in your example then your EV against optimal strategy is always = 1 pot. That optimal strategy is always fold at > 50% pot (and call or fold when bet = 50% pot). However if our opponent plays suboptimally calls any betsize sometime our EV > 1 pot at bet > 50% (and our EV keeps increasing as we bet bigger) and EV = 1 pot when bet = 50%. So we are really freerolling our opposition by betting 75% pot here. Either we win 1 pot against GTO or we win > 1 pot against Anti-GTO.

perfect thank you

I thought this opponent was tight enough to only have suited fours, whereas I felt I was playing K4o, Q4o, 64o and 54o in this spot. Given board texture, my range has a real advantage on the 5 and he checks close to 100% of the time in position. This makes betting very appealing (its also usually bet as a bluff because he will rarely call the turn and river here with AJ and that's close to the cutoff for a profitable bluff. ).

Yeah, I think the turn lead is good but iirc you chose to donk bet flop as well. I think in real time you misread the flop action.

You are right I did misread flop action and yes its bad to bet the flop. It was a mistake :(

Its pretty incredible what small perturbations to the system can do :). Shows just how essential good hand reading is (and tricking your opponent into misreading your own hand).

I think it makes more intuitive sense if you think of betsize as being more related to 1-(equity) when equity gets closer to 1

Thanks flux!

You're right Paul. Good Catch!

good old PEMDAS

I like your logic and I'd like this post twice if I could!

I should qualify my thought with check a3 and a6 here to strengthen my range, but also be willing to overbet hands clearly over the top of his range (AT, 66, 33).

i agree with this analysis but a few things: 1. TT seems to be a very reasonable check back ( i do it a lot as a defense versus ur strat and the fact that standardly vbetting 3 streets wont work well with blocker effects) 2. AA can check at a certain frequency as well on flop.
So when we start blasting in overbets, we start to run into these combos on occasion when we have 33/66 (not so much with AT bc of blockers obv), but i doubt this is a huge concern.
I do agree though an overbet leading strategy can work well, and balance it with the lots of straight draws/FDs we will have without SD value that want to realize their equity.
We arent leading any other hands though right? I think an overbet lead or a check is best.

We arent leading any other hands though right? I think an overbet lead or a check is best.

Agreed. We almost always want to be check calling top pair here, so we have very little incentive to construct a normal size leading range.

The EV of check raising don't come from the fact that he will not always cbetting his air hands on the flop and can delay bet them at a certain frequency turn as this card hits his range pretty well ? We can put pressure on his As x by check raising large and overbet river and include the appropriate ratio of bluffs. I wouldn't bet too too much because as you said he can have some traps (TT/AA sometimes), and if he knows our strategy it incentives him to check back a lot of other trap hands
If it goes check check we can always bet large on the river, not too large imo as he can easily check back his weakest As x turn and bluff catch river no ? I would probably bet pot or small overbet and include some bluffs.

I"m curious why Garrett and FBB both like over betting turn compared to x/r'ing. It looks like a turn where we're going to face a bet from a wider range than we're going to get a call from when we bet, so wouldn't our value region be gaining the most from check/raising? I would think Ax+air will make up more than Ax, Tx-3x but I could be miscounting there. I guess I should also add that we likely make more vs his Ax by taking the x/r line rather than over bet turn, bet river.

Yes, exactly, Caesar, it assumes villain is clairovoyant and will always shove any hand better + indifferent bluffs over our bet.

The key issue is the small perturbations in the system causing large swings in betsizing, so the numbers themselves will never be terribly valuable. What is valuable is realizing that betting 1/3rd pot with a nut hand is or overbetting a "range merge" value hand is a big mistake the vast majority of the time. I think about this video as more of a lesson in how NOT to bet.

I also wanted to point this out with a couple more thoughts to the question.

Since we can't valuebet every single hand with its optimal betsizing against a good opponent we group our value hands in different betting ranges.

Consider this example: we decide to have two betsizes in a river spot where a straight is possible. In the smaller betsizing we can have some 2pair and sets and a small % of straights + bluffs. In the bigger one we have only nut hands+bluffs. So when we bet for the smaller betsizing should we:
a) bet a size that is 100% accurate to the weakest hand we valuebet;
b) bet a size that is not optimal for any of our value hands but is somewhat inbetween.

If option b) is the right answer then it becomes really tough to calculate which betsize we should use for our range. In many spots if we do an in-between size we might risk of going over the valuebetting threshold for our weakest hand, resulting in a bad bet with our weakest value and lower than optimal with our better value hands that comprise that range. It seems to me there's room to possibly expand Tipton's formula to include these considerations to explore optimal range betsizing.

Cloud nailed the answer.

Good Catch Seth!

Using one or two betsizes for entire ranges makes a lot of sense, but its more demonstrative to show a hand vs range situation. However, you can easily extend the logic to range betsizes.

Thanks FBB, Go bigger, because we always win > pot. (If he correctly folds, we win pot). If he calls incorrectly, we win > pot.

And yes there is some strategy choice against a bad player, where going allin with bluffs and betting small with value bets is the best play. But I have rarely met this sort of player and if he starts foldign to small bets and calling big ones we now lose money in this situation. Seems like a big mistake to take a spot where we win > pot and turn into a -EV situation on the off chance our opponent meets some weak tight specifications.

Every movement toward GTO betsize will yield improvements in EV. I like to think about more as evolution rather than multiplication. In evolution, every animal slowly adapts better to its environment (moves close to GTO). But in multiplication 7 X 7 = 49 is always true and if I wrote 7 X 7 = 51 it would be wrong (Its either right or wrong).

Hand read well and understand your own range. I know this sounds fatuous, but these really are the prerequisites to application.

Actually the equity number accounts for chops (it adds 50% equity hands into the range for all chops). We don't want to end up in situation where we are only called by 4x and 64s. Then our bet actually becomes negative -- chop with 4x and lose to 64s.

Its a good question, and ultimately the answer is that the average range equity wins out. If there is not enough bluffs in your hand range, you will take a hand with showdwon value and turn it into a bluff (at an optimal sizing for your hand range. (Hand equity was for example only). This means all your bluffs will be +EV and your showdown value hand will be equal EV between checking and betting. Its actually a super awesome spot to be in. :)

aren't your bluffs zero EV in a balanced range? if we dont have enough bluffs, we take worse hands in range (probably thats wins lets say 5% of time) and bluff them but how do they stay at +5% of pot in EV? It still loses to all their bluff catchers. So you expect ppl to slightly overfold to keep it at around +5% of pot EV?

Hi FBB,
If we bet twice pot and don't have enough bluffs in our range, then our opponent should always fold. Our bluffs are now worth more than zero. They are worth 1 pot. So we should always bet our bluffs, because checking them is worth zero EV and betting is +EV. If we continue with a large range with some showdown value hand, we should add bluffing with hands that have less than 1 pot EV in this example, because when our opponent folds we make 1 pot EV. Eventuallly our opponent will settle on a range that makes the strongest "bluff" (hand with showdown value) indifferent to calling. All bluffs lower than this will be positive. I'd encourage you to construct some ranges in a free version of piosolver and try this out. Seeing is believing.

"eventually our opponent will settle on a range that makes the strongest "bluff" indifferent to calling." Can you expand or elaborate this point? Im not sure I understand, and i feel its the crux of your point.

Otherwise we wouldn't bluff enough because our bluffing range of 0 EV bluffs ( betting are strongest bluff would be negative so we would check ti) this would be less than enough to make him indifferent to calling so he would always fold. Which means we would gain vaue against by bettig our strongest bluff. This oscillation is indicative of exploitative play.

oops my bad... after checking some numbers.. if there is money behind, villain raises his nuts along some bluffs denying our share of the pot making it inferior to x/back and showdown our hand.

Holdeq it's a crev addon

Solid - I think its a good starting point for a beginning strategy. It doesn't do anything obviously preposterous.

Hi Niz,

Thanks for the compliments! We should compare against his bluffing catching. In the first spot with A3 both KqQd and KdJd are bluffcatching hands. Most of my bluffs will be low flush draws/gutshots so those hands actually have decent equity against my range. In the A5 hand of course we have 87hh, JhTh, QhTh+ but those are only 4 combos, so QJ only has 10% against my range (which makes it an autofold).

I appreciate your thoughts. Well said!

Sort of this is semantics, but I win when he folds here also. I'm never folding so I win this pot 75% of the time. I win much less often when my bet is called.

Hi Like,

Getting called by chop nets us 0 dollars from our bet (Win 50/Lose 50 = 0 EV bet). This means that to make money when we often chop with our opponent, we need to bet smaller. The only money will really win is against weaker hands that our forced to call.

In your example, likely optimal strategy is to bet smaller with Kx and to jam AK with balanced bluffs.

We also win money, if we FOLD out a chop. If we bet large enough, opponent will be forced to start folding kings. Whether our strategy should prefer extracting value from weaker hands or folding out chops with overbets is another thing. Better example would be wheel straight 2345, where only one player reps 6, but aces are very likely for both players. Thats a pretty common spot for overbets which are supposed to fold out chops. And the formula you are using is gonna recommend way too small bet, because it's gonna act as if we are likely to lose often due to much less than 100% equity (even though we can't actually lose). Yet our strategy should be huge overbet with both aces and sixes with the intent of folding out chops.

I'm sure we can contrive some strategy where his entire range is chops and our entire range is chops + nuts that we should now overbet our chops. In general though when ranges include other hands then smaller bets with chops make sense. You can test your hypothesis in Pio. I'm happy to be wrong.

It actually means the formula is not applicable for some reason. It only works within the constraints and 100% equity isn't included within range. With 100% equity, you should always go all-in without range considerations.

Yes exactly. Think of it like a 20bb cap game where we have 4000 bbs behind. We'll only calculate the math based on that 40bb pot.

Equity = Hero's equity
Bluff Catcher = Villain's hand

I'm using bluff catcher as a hand that beats our bluffs but not the value hand. We know that our opponent folds some % of the time to our bet, so a hand that has 90% equity against the entire range and bets pot will only win 80% of the time against hands that will call the river. We can describe this change as (90% equity - 50% Folding Frequency / (50% Calling Frequency).

Hi Tyler, please, can you answer my previous questions?
Also, are you sure you retranscripted well Tipton's formula?
I get strange results (with Equity = Hero's hand Equity vs Villain's calling range).
For example, with S2P ratio = 4.33
If Hero's equity = 99.39% -> B* = 4.33pot (which is OK)
But :
If Hero's equity = 98.48% -> B = 21.78pot (which is NOT OK)
If Hero's equity = 97.56% -> B = 7.52pot (which is NOT OK)
If Hero's equity = 96.33% -> B = 4.09*pot (which is OK)

Any idea of what's happening here?

Are you sure that :
Tipton 's S = stack / pot
Tipton 's T = 1-Equity
?

thanks

1. Equity should be against the whole range
2. Assumption for the formula to work:
   a. You have a good understanding of your opponents range and your own range.
   If we know both ranges then we can use PIO (insert multiple betsizes into the solver) to find an optimal strategy with optimal betsizes. The more betsizes we use the closer the solution will be to GTO for NL.
   b. If we don't have PIO, use the median hand equity in your value betting range as Tipton optimal betsizing hand. Its an approximation, but it will hold up well against exploitation and will rarely be awful .

@Betsizes -> any betsizes over stack size is considered out of range in Tipton's formula. All the funny numbers are telling you that we should be betting over stack size here (or all-in in human speak).

P.S. I actually get -128 at 99% here, which means we are clearly out of bounds.

Just a sanity test, getting -128 would infer a shove correct? I thought when optimal bet size was negative it inferred checking > betting?

Yes, it's shove.

1. Its pointless to calculate equity against hands that will fold to any betsize and don't beat my bluffs. They aren't in the handrange that we calculate equity against because they are always folds.

Sorry for the confusion.

So Im still confused on the topic.

We calculate our equity vs villains defending range on the river to derive a optimal bet size however, villains defending range is a function of our bet size, so how can we accurately assign them a bluffcatching range if we dont know our bet size in the first place?

I guess from what I understand then is we are calculating our range vs villains range up to that point in the hand minus any hands in villains range that would not be able to beat our weakest bluff?

secondly regarding adding bluffs,,

I think this makes sense, " maybe the formula is made so that we only have to take into account the equity of our Value range and then we can build the bluffing range corresponding to this betsizing?"

Otherwise if the change in equity vs villains range when adding bluffs is to be considered, when could we stop iterating?

Thanks Bubu!
Yes you could definitely sum this up as an exploitative adjustment :)

Look at 2nd-to-last comment just above your first post.

Cheers!

This is a bit of rabbit hole. If we can profitably call a raise, that will make our bet more profitable which will increase the optimal betsize.

If they can bluff raise us, our betsize falls as their bluffing frequency goes to GTO. After the bluffing frequency gets bigger than GTO than our profitably will increase again and so will the optimal betsize.

Any hand better than your bluffs. If they fold too often, then we aren't playing optimal anymore, so you'll want to choose a different approach.

Thanks Truepower!