I don’t have the final final answer either as I’m not sure how complicated it really gets due to all hands having different equities against each other, but one simple way to think about it is for example if you think the villain’s all-in bluffs are some sort of flush draws w/ ~20% equity, then he’s really risking only 80% of his actual bet amount with each bluff. So you could adjust your formula to something like b/(P+0.8b). (I’m pulling this from the top of my heads so don’t count on the formula being 100% correct, but I think the principle is valid so the actual formula should be relatively simple to derive for both optimal folding frequencies and value-bluff-ratios.)

Teppo

P.S. I do believe overbets appear more often in optimal play compared to how often most people use them, but personally my game has been slowly moving to the opposite direction simply because I think I’ve had a little better success with smaller bets. Being too willing to fire overbets with both strong hands and draws will limit your profits if you are able to outplay your opponents or if they are more likely to make bigger mistakes when there is more money to be bet at later streets. This is opponent dependent though, so against SmokinAl for example I might benefit from using more overbets in my strategy.

Actually I feel kind of embarrased that I’ve payed poker for several years without doing any of these simple calculations for myself!

True when you’re the bettor, which I guess was the case in your original question. I was just thinking of it from the perspective of the player who is facing the bet.

For example if your opponent is on the button and shoves the turn, how does he play his drawing hands? If he always semi-bluffs w/ his draws and gives up his complete trash, you’d need to call more often because he is not risking his full bet with his semi-bluffs, and if you folded "normally" he’d make immediate profit from his bluffs due to his extra outs. On the other hand if he always takes a free card with draws and bluffs with his complete trash, you can afford folding more because he’s no longer making immediate profit from the bluffs since he doesn’t have the extra outs when his bluffs are called. (In true equilibrium strategy you’d have the right way to play without having to know how your opponent plays, but when the true equilibrium is unknown and we’re trying to use the simplified tools for figuring out the correct folding ranges etc, I’m not so sure if that’s possible without knowing or assuming something about how the villain plays.)

From the villains point of view the same principle would mean he could bluff more often when the bluff’s are actually semi-bluffs than if they were pure bluffs, again because he’s not really risking the full bet size anymore. Similarly I think you are correct that the optimal bets are larger when the (semi)bluffs have more equity (this assumes that your equity does not change much based on how loose or tight your opponent’s calling range is, and therefore it applies primarily to drawing hands on early streets but not to river decisions when all the cards have already been dealt).

(Disclaimer: I thought I had figured all these out a long time ago, including the optimal folding formula I wrote down from the top of my head in the earlier post, but now that I’m writing it all down again I’m not so sure if I got it all correct. I’d have to go back to the drawing board and verify if what I wrote is in fact correct, but I’ll have to leave that for later and when I have a little more time. I will get back to this though once I’ve done the work to know for sure what I’m talking about.)

Yeah I was making a false analogy, having nuts x% of the time and air the rest is not equivalent to having x% equity against a range, because in the range case villain has additional strategic options (he has some control over x). I like your semi-bluffing modification to the nuts/air case – I’ll try to think that through.

I only half read this on mobile, but I don’t think this is right.

The intuitive way it works is that the more you overbet the pot, the less you can bet for value, but the more hands *per value hand* you can bluff.

So if you have nuts or air, then you maximize your value by shoving your air plus the right # of bluffs.

In the case where you have overlapping hand ranges, it’s more complicated, and I don’t think you can calculate it as a function of equity (since "50% equity == symmetric range" is completely different to "50% equity == 50% nuts, 50% air").

The maths to work it out exactly in common spots isn’t too difficult (but it’s pretty fiddly), but generally the rule of thumb is that the more polarized your range is, the bigger you can overbet (and then the more hands you can bluff with).

I’ve been cataloguing a few ways the top players have used overbets, for dork interest (and based purely on watching the odd big hand!):
 – Spirit Rock used to overbet shove the river against players who capped their range by checking behind for pot control
 – Durrr likes to overbet against people who check-call down against him (again usually capping their ranges)
 – Isildurr seems to do odd things on the turn to construct nuts/air ranges on the river so he can overbet

Sorry if I’m just repeating what’s in earlier posts!

I think SmokinAl is correct in all points he made about polarized vs. non-polarized hand ranges, assuming they are "made hands" and not "drawing hands", which probably means the logic applies primarily to river decisions. (And I think he said that too.)

In the non-drawing cases the non-polarized ranges would suffer from overbets because larger bets make the betting range tighter, and therefore they would have to check significant number of hands they would have been able to bet for value if using smaller bet sizes. A very polarized range on the other hand might be able to overbet the whole value-range and not give up on any value on the times he does not bet.

In my earlier posts I was thinking of drawing hand situations only, not polarized vs. non-polarized ranges. And I actually generalized my intuition to situations that were too complicated for my simple analysis. I think my logic (and probably the folding formula I had) applies well for preflop situations, because then both players will have some pot equity even if they end up getting the money in with the worst hand. For example if the preflop "bluffer" gets called and still has ~20% equity, or if the caller defends with a weaker hand and ends up having ~20% equity, then the optimal betting and calling ranges are looser due to the fact that neither of the players are risking the full bet but only ~80% of that amount.

The above may be a lot more complicated in situations like my turn example where a real drawing hand has lot more equity than a made hand that calls when already behind.

I believe Al was referring to a general case, where regardless of your hand range if you do overbet you can do that with a fewer number of hands, meaning fewer value bets and fewer bluffs (in absolute terms). Relatively speaking your bluff range gets larger though, because your value bet range shrinks more than your bluff range.

An example would be pushing all in preflop with big stacks. For that play to be profitable you would only be able to do it w/ Aces and maybe with a semi-bluff hand like A5s (depending on how deep the stacks really were). You can profitably play a larger percentage of your hands only if you bet smaller amounts. And with non-polarized range like that you should bet a lot less because otherwise you wouldn’t be able to play many of the otherwise profitable hands at all, which is what I meant when I said that non-polarized ranges suffer most from overbets.


Sounds like you are thinking of the hand range after the bet? When I say non-polarized range I mean the hand range before the bet is made, even though it will turn into a more polarized range after the bet, especially after a large overbet.

So I do consider draws as the bottom part of the polarized range after you have polarized your range by making the overbet, but the analysis starts from the hand range you hold before you have made the bet and are deciding which hands to bet and which not to bet, and by how much to bet when you do bet.

And the earlier in the hand you do the analysis for, the less polarized the hand ranges generally are, mainly because there have been less changes of polarizing your range (one standard sized post-flop bet makes the hand range less polarized than firing 3 barrels).