I understand why you should bluff your weakest hands math wise, but can someone explain what it means when the online pros say "you have too much equity in this hand to bluff", do they just mean you have enough showdown value that you can call profitably against his range so bluffing is unnecessary? I think that they aren't looking at the change in image equity caused by both actions (calling with some sort of hand, as opposed to bluffing and potentially opponent not seeing cards/you get a more aggressive image) also, can someone help explain these things?

Yea you have the right idea. You want to maximise your value from your entire range. To do this we shouldn't be bluffing with hands that can win at showdown. We should bluff with hands that have little to no chance of winning unimproved(ie semibluffing).

General principle is that if you have showdown value, then turning your hand into a bluff is not a great idea - you risk paying off hands that have you beat, and aren't folding, but don't get any money from hands you already have beat, which aren't calling.

^^ perfect explaination right there

One should be careful to consider whether their hand really has 'showdown value' though. Too many people revert to level 1 thinking in this sense - second pair has showdown value if there's only been one or two bets postflop, but if you miss a pair + flush draw draw, after check-raising the flop and leading the turn, you do NOT have showdown value. Seems simple, but a surprising number of people forget this.

Something that's occurred to me right now - if villain's range is polarized, and we're on the river, and hero is OOP with a hand that is between the two poles, but the distribution of villain's hands is not sufficiently weak that hero can c/c, and villain will always fold the bottom end of his range to a bet and call or raise with the top end, is it ever worth betting? (EDIT: Assuming the bet sizings are the same for hero's bet and villain's bet)

If we bet, what do we get value from? Villain folds everything we're ahead of and calls/raises all hands that have us beat. (Unless I've misunderstood the question.)

The bet gets value from nothing and bluffs out nothing, but prevents villain from making a bet we can't call. In essence, I think our actual hand strength doesn't matter because when we check, we lose the pot (because villain bets 100%), and when we bet, we lose 100% if called (so, if you like, our hand is part of a merged bluff range). The difference between betting and checking is that when villain has air, we win if we bet and lose if we check. But now I think I see the answer - given that we can't c/c, and we always need better success rate to bluff at a pot than to c/c the same size bet (because villain's bet adds money to the pot), betting is the worst play here assuming non-varying bet sizes.

What about merging our range?

Won't that get calls from weaker hands that think our range is polarized?

Basically this.

Though, there are times this doesnt apply - When its more profitable to bluff than take your SDV, you bluff. These situations are rare at the micros however. (its rare that you will have enough hands on a guy to know that he can fold TP, for example)

Another (very general) example to note is the following:

You 3B AcAx OTB vs a CO open who then calls. He looks half decent. Were 100bb deep and all sizing thus far has been standard.

Flop is 642ccc, you CB, he calls. Turn is a Qx. He checks to you. Do you fire again for value?

Well... Ask yourself, how do you feel about being ch/shoved on OTT... it sucks badly. Calling isnt ideal and folding all that equity suck even harder.

Thus: We have a large deal of SDV(we could be ahead now). We have tonnes of equity to improve with the nut flush. A decent player may not pay us off often even with a flopped OP(plus draws are rare - and were not scared of it hitting. It would be somewhat disapointing if it hits, however). And a decent player wont be ch/shoving worse all that often/ever.

We could check back the turn with the intention of calling off a river bet. It keeps his range wide: allows him to bluff missed draws (kc, Qx eg) and value bet worse OTR (eg: jj).

What are we scared of when he check/shoves the turn? Our Ac blocker is HUUUUUGE here. The only made flush hand he can hold basically is KQcc, given we're 100bb deep. If he's good he has semi-bluffs in his range, and when he flat calls we're almost certainly ahead. Not betting the turn there would be horrid IMO (and I know you said 'could' but you 'could' also fold AA pre in cash).

Only reason to check back that turn is because you want to induce a bluff and feel that vil just can't call anything, so you will get more value by not betting. The only hand vil is repping on that turn is precisely QQ which makes no sense given his line so far. If vil is c/r his flopped flushes on the turn then make a note as his flop c/r range will be quite weak.

There is one very underrated (and underused, by me included) aspect of this, which is that if you have a bluffable situation, but a hand with showdown value, then you should go ahead and bet for value. Not enough value to bet by a hand that will call based on value, but enough for hands that will call a bluff but can't beat your hand. A typical example would be if the board is KT732 and you have AT, normally you would just show this hand down because there's not much use in betting if you're only going to get called by K weak kicker or better. But if you bluff having the K, then there are lots of hands worse than AT that will call you if they think you are bluffing.

Not sure I've understood you right, but you're basically talking about thin value betting, yes? The other way I saw this was that you could've been saying that 'a hand which is too strong to bluff should always be bet for value' which is plainly untrue.

Interesting, thanks for the comments. I think some of the people are misinterpreting my question as a way of asking a way ahead/way behind question than what I'm talking about here (to be fair both are similar in that a hand that is too strong to bluff will get call/raised)

So then, let's say you have a deck of three cards. A deuce, a three, and a four. The highest card revealed at showdown wins, and there is one street of betting where the first player has to automatically check. There are two players in the game. How should you play this game, what is your strategy approach?

Sounds fairly similar to Kuhn poker so this will have been solved. Are there any blinds/antes? Can the first player c/r? The only thing that's obvious is that we never bet if we're IP and we hold a 3, and always bet if we hold a 4. Therefore, we need to bet at least some of the time when we hold a 2, to prevent our strategy being completely exploitable.

This is just another version of the AKQ game (only 3 cards in play, A, K and Q). There is a solution to this game, specifically (assuming a pot of 0 before this game starts, things change if the pot is larger):
second player always checks K, always bets A and bets Q 1/3 of the time
first player always calls A, always folds Q and calls with K 1/3 of the time.

The second player has an EV of 1/18 of a bet purely through the fact they get to act last.

Any deviation from this by either player can be exploited by the other player.

Example from "Mathematics of Poker" by Chen, Ankenman.

that's right! i recently heard about it and thought it was interesting that we bluff our weakest hands

When you think about it that's the only way we get value from them as they beat absolutely nothing otherwise. Our weaker but non-total-air hands still beat total air so they still retain some value by checking, air has absolutely no value by checking.

It can be correct sometimes to bluff to prevent an opponent from making a bluff that you can't call. There's a detailed example in Concept 4 in the Concepts and Weapons section of Sklansky and Miller's "No Limit Holdem: Theory and Practice" which is a good book. The short version is that you're on the river in a $100 pot and if your opponent bets $50 you won't be able to call, so you bet $30 instead so you pick up the pot when he would have bluffed.

Betting with the intention of stopping your opponent betting is commonly called a "blocking bet". If you intend to make a bet like this you need to be clear about your reasoning. The purpose of a blocking bet is to prevent your opponent bluffing in a situation where you can't call. Bad players misuse them when they don't trust their evaluation of the opponent's range. If you think your opponent bluffs enough that check calling is profitable, do that. If you are beaten so often that you should just check fold, do that. Thinking "well I'm pretty sure I'm beaten but I hate folding so I'll blocking bet and see a showdown" isn't acceptable.

Good post Chris, I would also like to add that you could potentially bluff if a scare card comes on the river; this is assuming that you know your opponent is capable of folding to a raised pot with a scare card and you have played the hand in a similar way, if not the same way, that you would've if that scare card actually did help you.