There are two ev sources for bluffcatchers in equilibrium on the turn:

a) free showdown ev
b) draw value

"bluffcatching ev on the river" is not an ev source, because hands that can only beat a bluff are 0ev by definition in equilibrium, with the exception of hands with good blocking qualities, which are slightly +ev.

so to calculate (a) when facing a bet:

opponent's river check frequency * pot * my equity vs checking range = (free showdown ev)

to calculate (b) when facing a bet:

% hit hand * equity when hit hand * pot = (draw value)

the sum of (draw value + free showdown ev) = total ev of bluffcatching the turn in bb. If we set (pot = 1) then the value will be a fraction or a percentage of the pot.

Please let me know if I missed anything. Thanks.

Depends if your playing poker with criminals that would shoot you when your facing a bet. changes the ev.

You missed the fact that in the second equation, after you hit your hand there could still be more bets going in on the river.

It's way more complicated than that. Like vernon said there are more bets to go in on the river and different river runouts affect the EV of your turn calls. Some rivers your missed draws will be profitable bluffs (either a bluff raise or a bet when checked to) and on others they won't be. There's too many factors for such a simplification to be useful imo.

I attempted to do an EV analysis with future river betting making some restrictive assumptions to simplify somewhat. Presumably the decision point is calling a villain turn bet and having villain either check or bet the river. The EV equations without any assumptions on hand characterization or river equilibrium are as follows assuming hero checks back and no hero raises:

EV Call v turn bet = P(v river check)*(EV v river check)+ P(v river bet) *( EV v river bet) – Turn Call $

EV v rchk = eq_chk * Pot ….. (v rchk = villain river checkl, hero checks)

EV v rbet = eq_bet * (Pot + 2Bet) – Bet (hero calls)

where

Pot is the pot before river action
Bet is villain’s river bet amount

Example: At the turn, pot is 100 and villain bets 50. If hero calls, pot before river action is 200. Villain will go all-in for 250 if he bets the river. Hero estimates his equity versus a villain river check and hero check is 45% and his equity is 25% versus a villain river over-bet with a call.

EV v rchk = 0.45* 200 = 90 (hero checks back)

EV v rbet = 0.25*( 200 + 2*250) – 250 = 175 – 250 = – 75 (hero calls)

Then, if Pv rchk is the probability villain checks the river, we have

EV call turn bet = Pv rchk * 90 + (1- Pv rchk)(-75) - 50 = Pv rchk*165– 125

Thus, the break-even case (EV=0) for the turn call is for Pv rchk > = 125/165 = 76%

The likelihood of villain checking at such a high frequency will depend on the turn card potential for both hero and villain and on villain tendencies. Of course, you can estimate a villain check frequency and solve for bet size or even use more complex models such as assuming check frequency and bet size are related or consider the possibility that hero bets facing a check or hero raises facing a bet.

While the above addresses a turn call and future river action, I’m not sure how closely it relates to OP’s specific condition of bluff catching in equilibrium. Also, as indicated by others, the inclusion of future action is a complicated issue which I only partially addressed with my example.

Thanks guys. I was thinking that the type of hands that would be found closest to the 0ev turn call margin would be those that remain bluffcatchers even if they improve, thus rendering the future action calculation unnecessary. In other words, if a hand has such good draw value that it’s receiving implied odds on the river, I think it would follow that the turn call isn’t marginal, with the exception of pocket pairs which only have two outs to improve.

Regarding the additional ev sources mentioned above:

River bluffs in position are indeed profitable in equilibrium, but I think the idea of bluffing the river is at odds with (unimproved showdown ev) to the effect that (river bluff ev) decreases as (unimproved showdown ev) increases.

Also, bluff raising is 0ev in equilibrium on the river with the exception of hands that have good blockers. Hands with good blockers and good draw value typically can call the turn profitably; these hands are not near the 0ev turn call margin.

The profitability of bluffs is only limited by SPR. If stacks are deep enough for a bluff-raise to get a fold, then those are profitable. When SPR dictates that any bluff will be nearly all-in, then that bluff is profitable AND elimiates any profit to the caller (the call is by definition 0EV).

Is all of this true on the river? I'm working under the assumption that betting the river causes the bettor to call a raise at minimum defense frequency, which would cause the player facing the bet to be indifferent to fold/call/raise* with the top of his folding range.

I don't think the bold is true, except for hands that can only beat a bluff.

*since fold is 0ev, call and raise must also be 0ev.

Hmmm. What I meant was when a bluff happens, and it puts one player all-in, there is no more bluffing going to happen. So, that bluff is profit, even vs a gto caller.

So by taking away my ability to raise, you eliminate my ability to bluff, which is auxiliary profit for your bluffs?

Bingo presto. River SPR is everything.

Wouldn’t that ev be transferred to your value hands that would have to fold to another raise? I don’t think it transfers to your bluffs.

I don’t follow what you mean. However the action goes, the last bet put in is the only bet that turns a profit, GTO vs GTO.

I have value hands that can beat part of your value range.

And I call at MDF. Thus, your value earns you nothing.

No matter how complicated the action, the entire +EV of a hand of holdem is the bluff frequency of the last bet placed, multiplied by the size of the pot when that bet is placed.

If my value hands can beat a portion of your value raising range, then my value hands profit by calling the final bet.

This makes my bluffs 0ev, but my value hands will profit when you call, otherwise why am I betting?

This is true of the (nuts+bluffs) vs (pure bluffcatchers) solution, but I don't think it's true of real poker because my range isn't (pure bluffcatchers).

This has nothing to do with bluffcatcher or air.

Draw up a hand of holdem, and follow it to every endpoint in the tree, folds or showdowns. The EV to each player is the bluff freq of the last bet.

This statement doesn't consider the price of the call, which should be subtracted from the caller's ev to find the actual value of calling:

Sure, when I call a pot sized bet with a hand that can only beat a bluff, my immediate ev in equilibrium will be 1/3(your bluffing frequency), but after subtracting the investment my ev is zero.

When I call a pot sized bet with a hand that can beat 1/4 of your value range in addition to beating all of your bluffs, my ev after subtracting my call is 25% of (pot + bet).

The bettor’s value range is a range of hands, that will get called by a matching range of value from the caller for zero EV at equilibrium, in GTO.

You don’t *know* which combos will actually win at showdown, simply that they are value bets.

So, why does the caller not overfold and call with only the exact raw equity needed to break even, or turn a profit?

Because the bet also comes with a corresponding portion of 0EV bluffs.

This portion of losing calls at the bottom of the caller’s range is exactly the same as the bluff portion of the bet.

If this is the end of the action, then the action has +EV in favor of the bettor, and this is precisely the bluff frequency.

This is your thread Bob, I am not wanting to derail it.

It's fine, but I feel like you're not really addressing nor disproving the thing's I'm saying, which I believe to be true.

This may be true of symmetric range stuff, but poker on the river is asymmetric.

I've never heard of any equilibrium rule that states that the bettors value range and the callers range match in any way. Of course, there will be some overlap and matching combos in both ranges, but it's not a wash.

Let's go back to this.

Once you raise my bet, yes you cut down my ev, but that doesn't negate my profits that I earn when you call and lose. If my profits are negated, then I shouldn't have bet in the first place.

I only call and lose, with -EV, because you bluff. No bluffs? Oh, then I only call with raw equity needed (I overfold).

Again, these are ranges. Do you think the bottom of your value bet range is going to win often?

You should never make a -ev call.

I believe the bottom of my value bet range should win often(not vs your raise, but vs the sum of all of your calls and raises), or else I shouldn't bet.