Is something not clear or did I mis-speak? I've seen your other posts, seems like you should know this right?

If you want an elaboration on the math: It's quite common in math to have non-constructive proofs of the existence of objects with certain properties, that is you can prove for example there is a number that has a certain property without any hint in the proof of what that number is. Nash's proof that every game has an equilibrium is non-constructive. It is actually based on a very simple idea. First you define a function f that takes a strategy pair and changes it slightly so that every action that is a better response has slightly higher probability of being taken (i.e. it is an improved strategy). Then you apply a result calls brouwer's fixed point theorem that says that for every continuous function f, there must exist some input x, s.t. f(x)=x. (For a given f, we have no idea how to find its fixed point x, that is why it's non-constructive). Therefore there must be some strategy that cannot be unilaterally improved.

if, by some strange, unforseen circumstance, we ended up hu with the limper to our direct right who donk bet the flop, we would want to have 1 bluff for every 11 value raises, (for simplicity's sake we'll assume the small blind went down the rake hole), since the pot odds we would be offering the villain is equal to 11:1.

if we use combonatorics, then raising for value with these hands:
[AA, 99, 88, AJo+, AJs+, A9s, A8s, QT, JT, and 89s] = (54 combos). we can round up to (55) to keep it simple and that means we would need (5) combos of bluffs here. from what part of our range should we choose hands to bluff with? in this specific spot, as gross as it may seem, we should bluff with the top of our folding range [66, 77, KTs-KQs(no flush draw), KJo+]. from this folding range the top is 77 (6) combos. bluffing with 77 every time we were in this exact scenario would have us bluffing a little too much, so we'd have to remember to fold 77 every sixth time we were in this spot.

this is my very basic understanding of correct bluffing frequencies.

Simply put, a game of finite set of players and strategy sets has at least one Nash Equilibrium. Poker very possibly falls into the category.

maybe bluffing with 77, even though it's the top of our folding range, isn't the best idea. we might rather bluff with something that can become the best hand later on. KhTh, KsTc, KsTd, KhJh, KsJc might be better.

Thx Dougl and silver bail. I see now said the blind man.

I'm not sure I agree.

1. How do you know the equilibria are stable nodes?

2. Even if you could prove the existence of a stable Nash Equilibrium in some sort of continuous distribution, it may be problematic implementing it practically because the number of hands available is quantized. That is, let's say you need to bet the top 35.7% of QQ+. Betting all AA is 33.3% (too little) and including KsKc is 38.9% (too much).

Combined with problem #1, this may lead to the ridiculous conclusion of, "If he only bets AA, I fold 100%, if he bets AA plus black kings, I raise 100%."

You're having much more problems remembering a strategy that complex than in randomizing the hands. Especially using computer science to brute force approximate optimal strategy, you're pretty likely to derive something you can't remember. Still, if all of us feel players decide to bluff 0% and our algorithms say 9%, we know a leak. If we go the other way and cbet enough that we bluff 50%, we have a huge leak.

What do you think the donker is folding on the flop and turn? I ask because I think you're talking about setting up a river bluff on the rare occasions that you actually get it heads up against the donker, who probably isn't folding anything before the river. So now you're investing 3 big bets in the hopes that he folds the river enough to make the investment worth it? Seems very optimistic. Also, I don't think this is a spot where you could make the donker indifferent even if you tried.

You seem so sure that a bluff here is correct, but you haven't even suggested which hand to bluff, which matters a lot.

I'd raise any flush draw and or open ended straight draw on this flop and I think any other attempt at a bluff will be -ev.

even simplified versions of heads up limit poker have not been solved ( GT wise) nor do they have much chance to be in the near future. Poker is way too complex of a game. throw in the wetware and it gets even more dicey. but the models we do have are awfully helpful

only add in KsKc a fraction of the time

sensibly stated

I'm fairly certain that you aren't taking into account the multi-street component of holdem where you can continue to bluff with, or cease to bluff with, certain hands. So think about what if you bet all 5 of your bluff combos on the flop, all 5 of your bluff combos on the turn, and only 1/5 of your bluffs on the river. Your opponent can't know this. Under your logic he is structuring his calling range so it will eventually call river under the assumption that you have 5 bluff combos but because the game has multiple streets, he arrives at the river with an incorrectly structured range if he knew you were betting river with only 1/5 of your existing bluff combos.

tl;dr you use a river betting range composition on the flop and the flop does not play like the river.

what i am doing is using the bluffing strategy from phil newall's book "further limit hold 'em." i believe i am using it correctly. what i did wrong in the post you quoted was i chose the wrong hand to bluff with. i corrected it in a later post.

because the action changes street to street along with the odds we are offering our opponent in a hu pot, we use a flow chart to understand how to construct our ranges.

has a player already bet?
if yes, then bluff the top of folding range
if no, then
are there more cards to come?
if no, then bluff the very bottom of range
if yes, then
are you last to act?
if yes, then bluff the very bottom of range
if no, then a.) try to take the most value through to the next round and b.) bluff with stronger hands as the pot gets bigger

it seems to me that on the turn, we start over asking ourselves these same questions. and yes, some hands we will continue to bluff with and some hands we will not, because of how the turn and/or river card(s) affects our hand/range. example: on A98 after our lone opponent bets, (the other opponents have been arrested and dragged off by security), we raise KJ, villain calls. turn: [A98] K. villain checks to us and now we refer to the chart above and realize we are not at the bottom of our range. our bluff has now turned into a bluff catcher. thus, we no longer have a bluff and we check. what to do on the river depends on villain's action and the river card.

the optimal strategy isn't static. after bluffing the flop, we're not going to just barrel off KJo regardless of the turn and river. we will continue to bluff at the optimal frequency derived from the pot odds we offer our opponent. we choose which hands to bluff with based on 3 things, 1. the betting round, 2. our position, 3. whether our opponent has already bet.

I appreciate your response and will have to reread it a few times. I'm from a NL background where you have to take into consideration that you can bet anywhere between 0 and effective stack sizes rather than just a BB. Thanks for illuminating the difference in approaching properly formulating a ratio of value bets to bluffs in LHE.