I'm newer to the GTO style of play so sorry if this is a dumb question.

In the following hypothetical:

Pot is 2, I all-in for 100. Half the time I have the nuts and win, the other half I have air. Pot odds are basically 50/50 for you, so if you’re unable to call every exact time I’m bluffing, you’re losing EV no?

Intuitively I don’t believe this is true, but mathematically I can’t figure out why.

Does the importance of pot odds decrease dramatically as you progress through streets? How should I be calculating how often to bluff the river with an all-in?

In your hypothetical, yes, you are correct in saying that if your opponent is unable to perfectly identify when you're bluffing and when you're holding the nuts, they will lose EV (Expected Value) over time. However, it's worth mentioning that this is based on a very simplified assumption of the game that isn't representative of the depth and complexity of real poker scenarios.

In this scenario, pot odds are indeed 1:1 or 50%, as the opponent would have to call 100 to win a pot of 200 (your all-in 100 + the initial pot 100). However, let's discuss the situation from the perspective of GTO (Game Theory Optimal) play.

From the GTO perspective, it's about being unexploitable and making the most +EV decision in the long run. GTO doesn't always mean calling correctly every time, it means making the decision that gives your opponent the least benefit. If you are perfectly balanced with your bluff/value ratios, your opponent cannot exploit you.

Let's assume your opponent knows you're 50% value betting and 50% bluffing. Given the pot odds of 50%, they should theoretically call every time. But if they know you're only bluffing 30% of the time, even though the pot odds are the same, the EV of a call decreases.

However, in real poker, your opponent won't know your exact frequencies. GTO play aims to mix value hands and bluffs in such a way that makes it impossible for opponents to exploit your play, given the size of the bet.

The amount of times you should bluff with an all-in on the river, or any street for that matter, can depend on a lot of factors. These can include things like the size of the pot, the cards on the board, your opponent's likely range, your own perceived range, your table image, etc.

A common strategy is to base your bluffing frequency on your value-betting frequency and the pot odds your bet is offering to your opponent. This can be mathematically calculated using the formula:

Bluffs = (Pot Odds / (1 - Pot Odds)) * Value bets

So, for a simplified example, if you're betting pot size (giving your opponent 2:1 odds), and you have 30 value hands, your number of bluff hands should be (1/2) * 30 = 15 hands.

Remember, in real play, things will be far more complex and variable than these simple examples. But these principles provide a starting point for understanding GTO strategy. The aim is to make your decision frequencies balanced and hard to exploit.

Finally, pot odds don't necessarily decrease in importance as you progress through streets. Instead, as the hand progresses and more information is revealed, you must constantly update your understanding of the situation, and adjust your strategy accordingly. Pot odds are always an important factor in deciding whether a call is profitable in the long run.

Some articles on bluff frequency:
https://upswingpoker.com/what-is-bluff-to-value-ratio/
https://www.runitonce.com/nlhe/calcu...-overbluffing/ <- Shaun Pauwels has a good note here for some quick calculations you can refer too similar to quick pot odds calculation.

Hey, really appreciate the in-depth reply, thanks for taking the time to write this up! (I've had some trouble getting straight answers from like reddit and friends lol). A couple of quick follow up questions:

In the simplified example to demonstrate the equation Bluffs = (Pot Odds / (1 - Pot Odds)) * Value bets, ultimately you end up doing the same math as my original question right? i.e. Bluffs = (1/2 / (1-1/2)) * 30 = 30. Just wanting to confirm my understanding and make sure I'm not missing anything.

I do realize now that there's basically no situation where you can make pot odds that are 1:1 so I guess my original example is slightly flawed. Also, I appreciate the resources!

Last follow up question: is this how GTO players decide what frequency to Raise/Bet/Fold? I've been wondering this for a while now as well, seems counterintuitive to start with bet sizing and then decide frequency of action from there though. Or are the GTO wizards just literally memorizing situations from solvers and following a massive chart in their brains?

Thanks again for your help!

Yes, you can polarize in a perfect nuts vs bluffcatcher spot to capture the entire pot evwise. Real play, though, is not so simple. One thing is there are almost never pure nut vs bluffcatcher spots in game -- opponent can retain some traps, for example, and even when there is a nuts vs catcher spot, there are very limited nut combos and we may want to move some of them into smaller betsizes (to strengthen those ranges and to have a calling range when we bet smaller and get jammed on). The concept in general is correct, but the pure application is pretty limited.

In terms of frequency, there's more to consider because in real play ranges are more robust -- generally speaking on turns and rivers, we have to figure out which combos we want to bet for value (which is almost always not exclusively supernut hands), and then we add bluffs until we make our opponent's bluffcatching region indifferent. Deciding how many hands to bet for value, as well as the region of villain's range we're targeting for indifference (different betsizes target different regions), depends on factors like the range matchup and the board runout.

In general, people who study solvers recognize patterns regarding the thinnest hands we're looking to vbet in given situations as well as the types of hands solver draws bluffs from. At a high level, players can also think about how many vbets our range is going to have, and then decide how many bluffs to add. Then of course there's also the exploitative layer, which might lead someone to over or underbluff any given situation. While toy game examples do a great job of illustrating the concepts, real play has a lot more complexity for exploitative reasons, because we rarely have the pure polar vs catcher distribution, and due to blocker effects in the calling range.

These ideas are definitely beyond the scope of BQ, but it sounds like you're learning some of the fundamental gto math and you're on the right track

Gotcha, ty Duncelanas! Random question, what's BQ? I tried checking the abbreviations dictionary on the beginner FAQ thread and it didn't come up.

Also is it really just studying solvers to recognize patterns? Like there's not way to calculate a rough percentage of bluff/c-bet/v-bet/check/fold preflop and flop purely based on ranges?

Appreciate the thoughts and help everyone!

This subforum is called Beginner's Questions

You can have some idea of percentages from studying if you understand how the ranges matchup on certain kinds of boards. But pretty sure nobody is trying to crunch numbers in their head and come up with precise frequencies on the fly while at the table. Rather, you heuristically recognize the flop as favoring a certain sizing and frequency, and then consider where your hands fall within that framework.

For example, a decent player might see a flop like AK3r and realize you have very heavy advantage throughout the ranges and a ton of hands that can sustain a small flop bet, and think that it's a good spot to rangebet small. A stronger player might see that flop and think we have no incentive to get protection with thin vbets and very large nut advantage, and so decide to push that with a flop overbet at lowish frequency and retain a lot of checks. These are both heuristics pulled from solver study.

You see the flop (or turn, river), consider how the ranges matchup, and build a heuristic strategy based on what's there. But basically nobody is out here doing precise combinatoric work on the flop -- maybe some people are in more narrow range spots like river play, after x/rs, and so on.

Ah got it, this makes so much more sense now with people talking about thresholds and bottom or tops of their ranges! Great explanation, thanks for everything!