This post is in regards to a hand in which we defend from the BTN vs a CO open. Villain cbets the flop, we raise. The turn brings a scare card which connects better with our flop raise range than Villain's call range, and we decide to make an overbet of 2x the size of the pot (which for the sake of simplicity is the exact amount of Villain's remaining stack):

Hero is on the BTN and flats a CO open. CO has a very high cbet frequency and we decide to raise (with air) expecting to take it down here often. CO calls. Villain checks on the turn and we decide to make an overbet of 2Xpot, putting villain all in.

(Q1) Please correct me if I am wrong, the fact that we are putting Villain to decision that involves all of his chips means that we can look at this turn raise as if it were the river and therefore calculate our Value to bluff ratio in the same way as we could on the river?

(Q2) If the above assumption is correct, we are giving villain 2/3 odds to call, meaning that he must win 40% of the time to make his call +EV. In order for us to be balanced and to make our opponent indifferent to calling we should (in a vacuum of information on an opponent or vs a good opponent) be bluffing at that same ratio of (40%)

(Q3) If however, we realise that because of Villain's tendencies, or the way in which his range connects with the board, that he is likely to be folding too much how do we adjust our bluffing frequency? And how do we calculate this? In regards to a particular hand that i have in mind, I have come to the conclusion, through post game analysis using Flopzilla, that due to how poorly Villain's hand connects with the board, he could not possibly continue anywhere near to what MDF (34%) would require, and is more likely to be folding around 65% of his turn range.

I'm not entirely sure that I am asking the right questions, or maybe I am over complicating things? Please let me know either way.

Thank you

If you want to slavishly follow MDF calculations, then a shove of 2x pot requires a value:bluff ratio of 60:40, so your 40% figure is correct.

In reality, MDF is kind of useless. Board texture (and how ranges connect with it) is a thing. You can't calculate (or look up) a specific bluffing frequency number without knowing the equilibrium solution. In some spots your EV might be maximised by bluffing 40%, in others it might be 20% or 80% or any number between 0 and 100. In most common spots, I think the frequency will be somewhat close to the MDF number, but each situation has its own unique equilibrium strategy. In short, frequencies aren't massively useful. You should just bluff more if you think villain overfolds in a particular spot and do the opposite if he doesn't.
I mean, you know how you c-bet certain flops more often than others (because of range asymmetries), well the same goes for turns and rivers. There's no optimal "one size fits all" frequency for betting on any street. Poker isn't the AKQ toy game.

A MDF of 34% would mean the villain can fold 66% of his range, no?

How is this not backwards? Doesn't one need to know specific bluffing frequencies to eventually arrive at an equilibrium solution?

Real poker has asymmetrical ranges that kind of co-mingle, by which I mean both players can have some of the same hand strengths. A player can bet a straight for "value", but get called by a flush, or he can bet two pairs and get called by a range that includes one pair and sets. In effect, one range isn't entirely pure value hands and bluffs, and the other isn't all bluffcatchers. Since you can't even strictly define hands as value bets and bluffs, you can't calculate a ratio for how to divide up your range.
To use an extreme example, you might bet a full house on the river and think "I have lots of boats in this spot, so I'm going to bet a lot of bluffs". And then villain snaps you off with quads.
If he has some combos of quads and boats in his range, that can mean that some of your "value combos" aren't pure value hands, because they don't always win, so you can't calculate an exact number of bluffs to match the 'value' hands.
Unless you've got access to villain's entire range and strategy, you can't calculate exactly which combos you should bet, or in which ratio. i.e. you need to see the full solution before you've got some numbers to look at.

MDF and the value:bluff ratio as per pot odds only apply to nuts:air vs bluffcatcher scenarios OTR.

OTT your bluffs have equity (unlike OTR) which lets you have more bluff combos.
But your value bets aren't solid locks either, which means you need ...less bluff combos.

Also when the situation is not nuts:air, think about what happens to villain's pot odds / required equity when we are thin value betting as per what Arty said


don't get me wrong, MDF is a good baseline, but if you do not adjust to account for the above, you'll get crushed.

I don't think it matters if our value is pure value or not when calculating bluffing frequencies.

Let's say we bet full pot on the river. We are betting 10 combos of pure value, which always wins when called. How many bluffs should we bet in order to make the villain indifferent to calling with their bluffcatchers? (I say 5 combos.)

Now let's change it so that we still bet 10 combos, but now the villain has some stronger hands in their range so our value hands only win 60% of the time when called. These are still valid value bets, since we beat more than 50% of the calls. But how many bluffs should we add in this new scenario, if we want to make the villain indifferent to calling with their bluffcatchers, i.e. hands that beat all our bluffs but none of our value? (I still say it's 5 combos.) Note that we can never make the villain indifferent to calling with their best hands - they will have a profitable call no matter how we divide our range. At least this is my understanding of a balanced strategy - we want the caller to be indifferent between calling and folding with the worst hands in their calling range. But maybe my thinking is overly simplified

Depends on how much equity our bluffs have, and how much equity the opponent's calls have against our value. If the worst hand in the opponent's calling range has 0% equity against our value and 100% against our bluffs, then we need to be bluffing 40% of the time as you mentioned. If the opponent has 5% equity against our value hands and 80% against our bluffs, we can calculate a bluffing frequency B which makes the opponent's EV zero:

EV = B*0.8*3*pot + (1-B)*0.05*3*pot - B*0.2*2*pot - (1-B)*0.95*2*pot = 0
2.4pot*B + 0.15pot - 0.15pot*B - 0.4pot*B - 1.9pot + 1.9pot*B = 0
3.75pot*B = 1,75pot
B = 0.467

So we'd use 46.7% bluffs and 53.3% value in this scenario.

In theory, if the opponent folds too much it's profitable for us to bluff with every hand that we have. In practice people tend to notice quickly if we're always betting in a certain spot (especially if it's a common situation). So it may be more useful in the long term to increase our bluff frequency in a less obvious way.

Hero's primary motive is to maximise his own EV. Maths is not my strong point, but I think hero should bet less often (or choose a smaller size) when he has less of a range advantage, because his opponent can adjust his strategy to one that punishes hero for having too many bluffs. (i.e. villain will call more often vs a weaker range, and stop hero's bluffs from breaking even).
I'm probably missing something (I don't have my spreadsheets on this PC), but if hero's 10 value combos only have 60% equity, and his 5 bluffs have 0%, doesn't that mean his entire range only wins [(60%*10)+(0%*5)]/15 = 6/15 of the time (against a villain that always calls)? If hero pots it, to maximise his EV he wants to win 10/15 (two thirds of the time), not less than half.
If my (probably flawed) maths is correct, I think hero's strat would be to bet the 10 combos that have 60% equity, and balance with just 3 airballs, because 60% of 10 is 6, and 6:3 is the same ratio as the 2:1 required for a PSB.

I think statmanhal has an equation for working out how often to bet, (or the optimal size to choose) given the frequency that each combo wins (or loses) when called, so he'd be a better person to ask than me.

This is a good point. As villain's range gets stronger, we can bet fewer hands for value. Or we use a smaller size to force the villain to call down with a wider range. It makes a lot of sense.

Yeah, 6/15 times we win 2x pot (original pot + villain's call), 9/15 times we lose our bet (1x pot). So our EV when called is 6/15*2 pot - 9/15*1 pot = 0.2 pot. You're probably correct that full pot is not the most optimal size for hero in this scenario.

In the quoted example the opponent has a range of a small number of strong hands which beat some of our value (which always call), as well as bluff catchers (which are indifferent between calling and folding). The point was that our value still has 100% equity against the bottom of the calling range, even if we lose to some of the stronger calling hands.

This sounds really interesting.

Here is villain’s EV equation for hero making a lead bet with a polarized range designed to make villain indifferent to calling or folding. Don’t yell at me, Arty made me do it!

EV_villain_call = Bl*((1-Be)*(P+B) - Be*C) + (1-Bl)*((1-Ve)*(P+B)-Ve*C) = 0

Bl= hero bluff frequency (basically what we’re solving for)
Be = hero bluff equity (can be > 0%)
P= pot before hero bet
B = hero bet
C= villain call amount (usually = B)

Ve= value equity (can be < 100%)

The first term is villain’s EV when hero bluffs and the second term is his EV for the hero value bet.

I don’t solve for Bl directly but have this equation in Excel and use its Goal Seek function to get the solution.

Example: Pot is 100. Hero makes a pot size bet with estimated value equity of 80% and bluff equity of 10%. To achieve indifference, he should bluff bet 19% of the time for a value to bluff ratio of 4.3 to 1.

Here is a graph showing results for various combinations of value and bluff equity for a pot size bet, taken from a draft book titled Hold ‘em Poker By The Numbers

Since this equation is linear in BL we should be able to easily solve for BL in terms of the other variables.

If I did the algebra correctly, I find:

BL = [VE*(P+B+C)-(P+B)] / [(VE-BE)*(P+B+C)]

Somebody may want to verify this equation.

Verified.

I have a program template that uses Goal Seek for other more complicated problems so I was lazy and used that instead of doing the relatively straight-forward math.

Those scary equations make me want to run for the hills, but I'm glad you posted them. (My earlier maths was quite wrong, I think). Nice work as usual, fellas.

Awesome.