Let's assume its roughly correct to defend enough on the flop so our opponent can't bluff any two cards. I understand this is a simplification and it often won't be the case in a lot of situations. But suppose we open the button and get called by the bb. We check behind the flop, and face a pot sized bet on the turn. Given that we are now likely at a range disadvantage, since the stronger portion of our range would have bet the flop, do we still need to defend 50% of our range?

Also how much should it depend on % we decide to c bet on the flop? For example if we are C betting 80%, we shouldn't have to defend as much of our check back range, since it will harder to villian to exploit us by calling any two cards preflop in hopes getting into that auto profit spot on the turn. If we cbet a more reasonable range, like 50%, than we likely have to defend more of our check back range on the turn?

Steve

I'm really not a fan of pre-canned strategies. If you happen to see a flop that is going to give your opponent the edge then you are better off knowing when to cut your losses rather than thinking you need to defend this spot some percent of the time.

The largest mistake I see players making is when they overplay a situation that turns a small loss into one many times larger because of some notion that they're not defending enough when they may just be getting out flopped more than normal due to variance.

Every decision should start within the current context of the hand. If you want to think about balancing, it should be thought about in the context of balancing for similar situations.

You are out of your element Donny.

No idea of who your talking about.

On boards that hit my opponent's range harder than they hit mine, I check back a mostly junky non showdown bound range. On boards that hit my range harder than they hit my opponent's, I check back a very showdown bound range.

The probability that your opponent holds a threshold bluffcatcher on the flop is very low, thus striving to make this hand indifferent seems wrong to me.

In a [0,1] game, the probability that your opponent holds a threshold bluffcatcher (or any other distinct hand) is 0. Still, the indifference equations hold.

As for the original question, after you check back the flop, the ranges are not likely to be symmetrical or a pure bluff catcher situation, so can't assume alpha as the guideline, imo. If you check back only 32o on AKQo flop, no need to defend half of those on each street. If you check back AA and JT only, you'll likely want to defend more.

Which kind of begs for the question: how should we construct the checkback range, then? Should be perhaps aim to symmetry vs villain's preflop calling range? Can/should we allow for a slightly weaker range due to positional advantage? Is it a function of how often we check back this particular flop? Function of how often this particular flop flops?

Agree with the indifference equation still being applicable. I got what I think is a pretty good explanation somewhere else, and although it was long and a little complicated, the idea was that it will depend heavily on our flop c bet percentage. If he wants to exploit us on the turn by firing with any two after we check back the flop, he has to start calling preflop with any 2, hoping we check back flop and he gets into that profitable turn betting situation. So if we cbet a ton, it won't matter if we rarely defend after checking back flop, because he'll have to check fold so often on the flop with the bottom of his range. But if we have a more reasonable c bet frequency, like 50 percent, now if we start defending say way less than 1-alpha, we open ourselves up to being exploited, especially since even the worst hand in his range will have equity vs our range. Im not really capable of the math that would show the solution for different c bet percentages, but this line of thinking made a lot of sense to me.

Not according to "Further Limit Holdem" page 54. The out of position player checks and folds at a rate higher than that which would cause indifference.

I might be misapplying a concept here, but isn't this more less the same idea as us needing to call enough to make our opponent indifferent between betting and checking with his bluffs, considering that his bluffs have EV as a check? So its unlikely that we ever have to defend enough that it wouldn't be +EV for his to bet his weakest hands. I think thats sort of why we defend more in position than we do OOP, because villainn can check back in position and realize some portion of his equity with his "bluffs." OOP he still gets to the turn some percentage of the time, but he will often have to just check fold when he faces a bet from us.

I don't think so. You're talking about the table on page 54? Did you consider that IP will bluff some hands (p56) that would have +EV as a check (the range between 5.9...14 in the 1BB case)? OOP doesn't try to make the IIP's bluffing threshold hand 0EV, he tries to make the threshold hand indifferent (which doesn't mean 0EV).
EDIT: this answer was to Bob148

You are thinking about the general relationship between c-bet frequencies and fold-to-turn-probes correctly.

For a zero-equity hand (which doesn't exist, of course) that can only make money from probing the turn and will wave the white flag otherwise, assuming HUNL with 2.5 bb opens and pot-sized postflop bets:

1) against a 50% c-bettor, the BB's probes have to take it down >80% of the time to flat profitably pre;

2) against an 80% c-bettor, the BB's probes have to take it down >125% of the time.

I'm tempted to invent a "toy hand" with a simple set of properties, the idea being to simulate a junk hand: 1) it has an EV of 15%*pot when it faces a c-bet; 2) 40% of the time when villain checks back, it has an EV of 50%*pot; 3) 60% of the time when villain checks back, it always probes, functioning as a semi-bluff that, when called, has an EV equal to 10% of the resultant pot.

This particular "toy hand" needs its probes to take it down roughly 65.68% of the time to flat pre. It has a (preflop) expectation of -0.40 bb against someone with a f2tp of 50%, despite those probes being +EV on the turn. But I'd be careful about drawing any strong conclusions from this (the hand is completely fictitious, after all).

(And for the captious among us, no, I am not implying that the BTN must be folding this often at equilibrium, or that the BTN should design a pairing of c-bet and probe-defense strategies specifically to make junky hands indifferent between flatting and folding pre. )

Wow great stuff, thanks Rei.

It is not true that in all poker situations when there are more cards to come that you want to defend vs a bet or vs a raise enough so that your opponent can not bluff you with any two cards on that betting round.

This is not always true from an exploitive stance.

This is not always true from a GTO stance.

One simple example could be given from pot limit low ball where a pat hand that was a very slight favorite over a draw should check and fold if the draw (being played by an expett) turned his cards face up and said "what do you want to do?"... If both players had deep enough stacks.

I don't think anybody in this thread anyway is trying to say we should in all situations, at least vs a bet. However I think you are mistaken that we don't need to defend at that frequency vs a raise once we've already bet.

How is "a nearly 50% draw" related to "any two cards"?
Anyway, interested in seeing the math behind the spot.

in to reread later

Yeah that explains it. Looking a little deeper into it I also see that in the bigger pot the oop player value bets thinner, so his checking range is weaker and thus the ev of checking back the threshold hand is higher than in the smaller pot, which should explain the differences in the % increase of folding frequencies relative to bet/pot.

I wasn't aware this was the case, are you talking about all streets or just the river, as far as the OOP player value betting thinner the bigger the pot gets?

It's a one street 0,1 game with one raise, but I think the general rule of value betting thinner the bigger the pot gets holds true for multistreet solutions.

Do keep in mind the examples were from FL... it's about the bet size vs pot size. Nothing special about the absolute pot size that causes thinner value bets, just the betot ratio. (Well, bet:stacks ratio too, at least in terms of what kind of model should be used to solve the situation).
Also, IP and OOP betting ranges both widen as the pot grows (in FL), but indeed, OOP ranges widen more.