This is a topic that I've realized I don't really understand.

For this question, I'm gonna redefine an overbet as a bet greater than the geometric bet size. This factors in SPR and negates the "its the only way to get stacks in by the river" argument. It's a more useful definition for my purposes.

Ok, so why do you overbet? What are the fundamental underlying reasons, in theory, that would make overbetting a preferred strategy?

There are a few moving parts to consider:

- Range & nut advantage, and the polarity of your range
- The elasticity (price sensitivity) of the ranges
- The value of denying equity (including hands that would have outdrawn you later)
- The value of getting called by worse hands with the top of your range
- Rake

None of these factors are good enough metrics on their own though...

• There are plenty of spots where one player has the other player's range crushed and will still bet small.
  
• There are plenty of high rake cap spots that use or even prefer overbets
  
• Denying equity is offset by folding out worse hands that might have called a smaller bet size, and it gives you a worse price on your bluffs

So is there a more fundamental model that explains when and why you'd want to overbet? I know it ultimately comes down to EV, but how do we model which situations are more likely to result in overbetting being higher EV?

Definitely has to do with how equities are distributed. Bluffs with robust equity lead to bigger sizings. Value hands that want to deny/can get immediate value but not necessarily future value aswell

Math thats why

It's betting theory - it doesn't matter the size.

The bet that puts the highest % of opponent's range in 0 EV spots will be used.

Overbets OTT put most of opponent's flopped top pair in indifferent situations.

Consider two situations.

a) UTGvsBB Q62r board - we 1/3 range flop turn Ace. Overbet spot. All Qx is indifferent OTT

b) UTGvsBB Q62r board - we 3/4 flop. turn Ace. Now we don't OB OTT. We use 50% sizing instead.

We use smaller sizing now because our range isn't the same and neither is our opponent's.

The 3/4 cbet made us less Ace heavy OTT since we mostly check Ax back when using 3/4 sizing. So we use smaller sizing now since we still want to bet our Qx but our Qx doesn't want to OB.

In the 1/3 example we still have all Ax in our range - so now more of our range wants to OB and he folded some Ax OTF so there is more range asymmetry.

With the 3/4 flop cbet - once we get to the turn. Now all Villain's 77-JJ/6x is indifferent OTT vs 50% sizing and maybe even some weaker Qxs.

This is something that doesn't get talked about enough. The strength of your bluffs has a huge effect on sizing. You need way more nutted hands to make villain's top pair or w/e indifferent if your bluffs have 0equity compared to if your bluffs have lots of drawing power.

"The bet that puts the highest % of opponent's range in 0 EV spots will be used."

This is a guideline that I've been using for a long time now, and it's a metric that intuitively gauges what the right strategy is. When we make our opponent indifferent (between continuing/folding), that part of their range will be 0 EV. I like to imagine we are "capturing" the EV of that indifference region.

But maximizing the % of their range that we make indifferent isn't necessarily the best strategy. It's a damn good metric, but it's not a hard law of game theory. There may be a smaller or larger sizing, that more efficiently captures EV.

Ok so let's move away from overbets and talk about sizing theory in general. How do we formalize the theory of the optimal sizing? Hand-wavy idea's welcome!

Why do you think it isn't necessarily the best strategy?

I don't think it is a metric at all - it is game theory at work.

The best tool we have to study the game is solvers. Once you work backwards from a solver's solution you begin to understand why they choose the sizing they do.

It's the same reason why you overbet on a T94r board but use small sizing on an Ace low low board. These aren't metrics. They are solutions.

If you bet small on T94 you put a bunch of Ax and ~66 hands in 0EV spots. If you overbet on A37, you put a bunch of Ax and 7x hands in 0EV spots. This way of thinking isn't very logical if you ask me.

One thing I've learned while studying sims is to not project too much meaning onto what I'm seeing. Sometimes the interpretation is right and sometimes it's not or it's incomplete.

For example, you can justify T94r overbets as a betsize that makes your overpairs able to go for 3 streets on almost every runout, while any smaller would make overcards kill some of your action

Not sure how checking is supposed to fit in with your theory.

Yeah but that's when you look at your own range on T94. None of our range wants to bet small.

The highest frequency used sizing combines what your range wants to do - to what part of your opponent's range you want to target.

Overbetting on A37 doesn't make any sense since we don't put a high % of our opponent's range in tough spots. Mainly his K high/Q high hands/J high hands.

Also we need to look at our own range. There's a ton of hands that don't want to put any money in the pot on Ace high boards. So if these hands do bet - it makes sense they would bet very small.

I guess everyone think's about poker differently though. This way of thinking helped me a lot but maybe it doesn't help you.

In general you just check hands that don't want to put money in the pot. Or if your range checks a lot - you check back some strong hands to not be exploited.

So 54s get's checked back a decent amount on 542r because our range wants to check a ton.

Same thing with AA etc.

The goal is to make a complex topic like bet sizing - accessible and easy enough to understand.

Words like robustness/equity distribution aren't really helpful when discussing this topic. You need practical examples.

Well everyone has the same tools. I think of a solver as an open book test, except the book is thousands of pages.

Someone at the micros can have access to the same solutions as a world class player. But they are interpreting the data much differently.

That's why you need to actively study. It's very easy to plateau in this game.

Let's be honest here - Pretty much everyone is looking at the same solutions and just coming up with some Ad-hoc justification for why it does what it does. It's not exactly a rigorous process.

Complicated topics often need to be described with complicated language. The truth is there probably isn't some pithy phrase that sums up how to gauge different bet sizing strategies.

That's why I want to look deeper. I want a more fundamental model for bet sizing theory. I want to be able to look at a board, look at the equity distributions, and predict with rigorous accuracy what the preferred bet sizing would look like. Do you guys think that is possible?

That's a fair assessment.

Every book I've ever read about poker has swung and missed completely when it comes to this topic and most of these people are very smart.

You'd probably need to study a superbot like Pluribus - but a few iterations more advanced. Then talk to the someone like Noam Brown to really understand how the superbot is playing. It would be a Herculean task though.

So I haven't dove deep into this, but I heard that for a fully polarized range (our opponent only has bluffcatchers, and we ONLY have value or bluffs) we want to bet as much as possible to maximize our EV. So overbetting makes sense in that context that we can incorporate it in real situations in poker.

In a one street situation where villain can only call or fold, you should always bet your strongest hands as much as you possibly can so long as: (1) they beat all of villain’s range, (2) you have enough bluffs to support the value bet. Bigger bets require more bluffs, but never more bluff hands than value hands… if you have some non-nut hands that you want to value bet (for less), those have to be supported with bluffs as well, which may take away from some of the bluffs you needed for your nut hands…. balancing bluffs and value across multiple bet sizes is really hard, so most just decide on a single sizing most of the time for a given situation, but that’s definitely not optimal. The whole thing gets even more complicated when villain can raise and there’s multiple streets, because then you start worrying a bit more about keeping strong hands around to protect weaker ranges.

Perfectly polarized ranges use geometric sizing to end up all-in with the last bet, which is not overbetting the way OP defined it, unless I misunderstood. A 1/4 pot turn jam would be an overbet, though...

You can always count on browni3141 to pay attention to the details!

So the perfectly polarized player should always bet the geometric bet size to get stacks in by the river, using equal pot% bets each street. That's proven in The Mathematics of Poker.

However, I've defined an "overbet" as a bet that's larger than the geometric bet size. In other words, you bet more than what was required to get stacks in by the river. This definition is more useful for my purposes than the arbitrary "greater than pot" definition that is commonly used, for reasons I outlined in the OP. I actually think this is just a better definition in general, but that's just semantics.

So, a perfectly polarized player, using my funky definition, would actually NEVER overbet. That leads me back to the original question - when would you bet more than you needed to, to get stacks in by the river?

We may want to overbet (compared to geometric sizing) when our strongest hands may not be strong anymore by the river.

We want to get the money into the pot ASAP before:
1) We get drawn out
2) We can't get action by worse
3) We get bluffed out

For example, I saw a situation where the solver preferred to shove 3x pot on the flop in a 3bet pot, BB vs BTN @ JT8ss

If you add draws to the perfectly polarized scenario. If draws have +ev call even vs value hands, then sizing that either folds out draw or makes them 0 ev will be higher compere to the geometric sizing. It dose not have to put them to zero maybe bigger bet just lowers EV of strong bluff catchers, for example when opponent has lot of pair+draw type of hands.

There are also spots where you have value bet that are too strong to only bet once but too weak to bet twice. Then you can "cheat" and bet big ott with plan to check river. This is often the case in BUvBB cbet turn when you OB strong top pair which will almost always check otr. Maybe this never gets bigger then geometric...

An easy counter example would be UTG vs BB SRP. AK2r board.


Solver immediately starts over betting in UTG. Which is definitely not due to BB being able to bluff easily on later streets.

Is UTG overbetting relative to geometric sizing on AK2r?

Just ran it and gave it options of 33, 75, 125, 150, 200 on flop and turn.


When given the option, it will choose overbets relative to pot size. It could use 125% on flop and turn to geometrically have a pot sized river shove. However, it still chooses 150% at a very high frequency and ends up with less than pot size by river.

One could argue that it's still trying to stay close to a geometric sizing to get money in by river.



Regardless, solver doesn't shovel money in due to scare cards that we may have to fold to.


Let's use another example. BTN vs BB SRP. Ah Kh 9d 7d. Double suited board. After a flop and turn overbet, the calling frequency vs a shove on a flush completing card isn't much different. Even if we took a smaller bet sizing on flop and turn........absent a jam for a pretty big river overbet by the BB.....we aren't folding out a large enough % of our hands to correlate to the suggestion that on a flush completing river the opponent can bet and win the pot.

For that to be true, we would have to fold a significant amount of our hands if we A) didn't use the overbet sizing to "get money in now) or B) when we do "get money in now" and get called and then face a river bet on flush completing cards.

What about this example: BB vs BTN 3b pot -> BB cbet shoves for 3x pot on JT8tt.



As I mentioned in post #19, I think "not getting bluffed out of the pot later on" is part of the reason.

We also want to deny equity and get the money in while we still can get called by worse.

Toy game: shoving turn for equity + bluffing denial

The board: A 2 2 Q
Stacks: 1x pot

Villain (OOP) has 1 combo of A A + 9 combos of flushdraws (let's call it 20% equity with his flushdraws).

Hero has K K and villain checks to us on the turn.

Hero has the option between shoving and checking back.
If we shove, villain does not have the pot odds to call his flushdraws, so will only call with AA.

Should we shove despite only folding out worse and getting called by better?

****************

If we shove:

90% of the time villain will fold and we take down the pot.
10% of the time villain will call and we lose 1x pot.

Our EV of shoving turn: 90% * 1 - 10% * 1 = 0.8x pot

If we check (let's assume river is checked down first):

20% of the time, villain hits the flush with his 9 combos of flushdraws: 20% * 9 = 1.8 combos
He will also always beat us with his 1 combo of AA.
So on average he will have 1.8 + 1 = 2.8 winning combos on the river (28% equity).

Our EV of checking turn: 72% * 1 - 28% * 0 = 0.72x pot

Conclusion: Shoving turn purely for equity denial is correct in this example.

***************

If villain is allowed to bet the river:

However, there is one more reason we want to bet the turn: "Bluffing denial".
We want to deny villain the ability to bluff us of the best hand on the river!

20% of the time, the flush river will come and we have to fold 100%.
But that's okay, in this example, because on the flush river we never fold the best hand anyway.
(Of course we should have shoved turn, but that mistake is in the past).

80% of the time, the river is a brick and villain will value bet his 1 combo of AA and balance it with 0.5 combos of bluffs. We are indifferent between calling and folding so we call 50% of the time (MDF).
He will check 8.5 combos of air.

When he bets (15%) our EV is 0 and when he checks (85%) we win the pot.

So our total EV when villain is allowed to bet the river is:
20% * 0 + 80% * 0.85 = 0.68x pot

In conclusion: It is correct to shove the turn, not only for equity denial, but also for "bluffing denial".