In my previous thread on protecting checking ranges the consensus was that we should protect our checking range for the following reason:

If we don't protect our checking range, villain can bet more often and bigger for value into our checks. If he's betting with a balanced range this means that he'll win the pot more often than he otherwise would if we were protected. If villain starts doing this, suddenly the EV of our strong hands become higher as check calls than as bets. So, we start check calling our strong hands, but then villain adjusts by betting less often and smaller into our checking range. We re-adjust by check calling less of our strong hands and the process of adjustment and re-adjustment continues until an equilibrium is reached - the point at which the EV of check calling a strong hand is the same as the EV of betting it.

So, my question: how do we know where this equilibrium is? Is the only way to find this equilibrium with a solver - whereby millions of strategies and counterstrategies are played against each other? And i'm not just referring to the equilibrium of protecting our checking ranges but the equilibrium of all spots (for example, the EV of betting small vs betting big with a certain range).

If yes, then a follow up question: how did people create winning strategies pre-solvers? I recall a video from Doug Polk in which he explained that he used to keep spreadsheets of his opponents frequencies as he was coming up through the stakes. If you know your opponents range, frequencies can show you whether someone is over-bluffing or over-folding a spot, however, they tell you nothing about whether a checking range is balanced or whether a smaller or bigger bet will be higher EV?

To the first question, I've seen some toygames and depending on the range distribution what you're trying to do is making thin value bets indifferent between betting and checking, so around 1 trap for every 3 bluff catchers if we're protecting against a pot bet.

To your follow up question, you don't need to balance in order to have a winnning strategy, specially when your opponents don't have any way to know whether you're balanced or not

Where does this 1 trap for every 3 bluff catchers number come from?

And yes, it's true you don't need to balance in order to have winning strategy. However, you do need to know where the equilibrium is so you can understand if your opponent is deviating above or below it (in order to exploit them).

Before solvers came along, people spend a lot of time and effort deriving "Caveman GTO" strategies for oversimplified toy game versions of poker. It's kind of a dying science since it's so much more efficient and accurate to just solve this with simulation. This comes down to solving toy games and doing a lot of math.


Here are some books to get you started:

• Expert Heads Up No Limit Hold’em, Volumes 1 & 2 by Will Tipton
• Play Optimal Poker, Volumes 1 & 2 by Andrew Brokos
• Mathematics of Poker by Chen & Ankenman
• Applications of NL Holdem by Janda

So those books are for the 'caveman' GTO stuff? Cuz I already own the last two

Check out Tipton's books. He's got lots of info on solving more advanced spots by hand.

Btw I'm still really interested in how aner0 got that '1 trap for every 3 bluff catchers' number if anyone knows? I can do the EV calculations by hand but I'm wondering if there's a quick way to find it like for bluff frequency and minimum defence frequency?

Will do, thanks.

How do we know? We can't. The current love-frenzy for GTO and other attempts to "solve" the game bypasses the fact that poker is and always will be a people game; a game of incomplete information. So affecting such a decision will be a host of variables that we can't possibly quantify even remotely accurately unless we have somehow been sitting on the shoulder of our opponent and been watching him for the last ten years. And no, all those online stats don't help, because you only see the hands they've shown down and simple frequencies don't tell you much--it's like trying to figure out a slot machine's payout percentage after fifty pulls.

How did people create winning strategies in the Stone Age? They compiled a mental database of what worked more often than what didn't. They learned to characterize people based on their mannerisms, tone, demeanor, hand-playing frequency, hands they saw, and overall success. Note that I list the intangibles first because that information is always more frequent and accessible and the latter group of data has to grow pretty large before it's reliable.

I realize the urge to crunch numbers and say, "Aha! He bets the turn 39% of the time but the optimal frequency is 37%! Therefore, I call his ass! Bwa ha ha ha haaaaa!" But sample sizes for this type of calculation, even if it's valid in the first place, will always be too small to be anything more than GIGO.

I fully acknowledge that the above is blasphemy, and you all can express your umbrage by finding me on stonetheunbeliever.com.

I think this only applies to the river. Here's an oversimplified toy game that demonstrates the concept.

OOP has A & Q (traps and bluff-catchers)
IP has K & J (Value bets and bluffs)
Pot-sized shove allowed, highest card wins.

The trick to solving this one is to make IP's value bets indifferent between betting and checking.

• OOP should trap A often enough to make IP's K indifferent betting and checking behind.
• IP's K needs at least 50% equity when OOP calls to justify value betting.
• OOP checks 1 trap (A) for every 3 bluff-catchers (Q), so their checking range contains a total of 1/(3+1) = 25% traps.

OOP checks, IP shoves.

• OOP will call half their range facing a pot shove (MDF), and therefore half of their calls will be traps that beat K, and half will be bluff catchers that lose to K.
• K therefore has 50% equity when called and is indifferent between betting and checking behind.

If OOP has excess traps then they should develop a leading range. If OOP doesn't have enough traps to make K indifferent, they should rangecheck.

Poker is more intuition than calculation in practice.

The problem with feel play is that the feedback loop is broken in poker, so good plays are often punished and poor plays are often rewarded, which makes improving through intuition and experience alone a long and difficult process. So GTO strategies were developed to find the best strategy in a vacuum without knowledge of your opponent's strategy.

The point of toy games isn't to memorize some arbitrary numbers, rather it's to extract fundamental principles.

In the example above, if you trap too much then IP's should check back and stop value betting the K (lowering the value of your slow-played traps). If you don't trap enough, then IP should always value bet K and bluff more, (lowering the value of your bluff-catchers). Identifying that principle is arguably more important than the arbitrary 3:1 ratio of this toy game.

You're fighting windmills my dude, no one here disbelieves in live reads, intuition or experience. It's just we additionally believe in math.

If you want to talk about statistics in poker, I'd recommend not talking from the intellectual framework you've built, but instead getting your feet dirty and putting in the work, maybe that way you would develop a "feel" for it

tombos21 do you have any thoughts on Modern Poker Theory by Michael Acevedo? Does it fall into the same category?