Semibluffing for implied odds

I was answering a question in private that looked interesting enough to post publicly, so here it is:

Given the nature of NLHE can it be more profitable to semibluff the turn with no fold equity purely for the extra value we gain by being able to value be the river larger when we hit? What varying amounts of fold equity yield what profitability?

Theoretical spot:

We have against a megafish who never folds and never raises.

The board is and we expect villain to have a set at this point. We have %18 equity.

EV is simply measured as the sum of all possible outcomes multiplied by the probability of each outcome happening.

So for events A through C, the formula would be:

Total EV = Probability(A)*Outcome(A) + P(B)*O(B) + P(C)*O(C)

Assume $100 pot on turn.

$75 turn bet, which gives us sizing for $200 into $250 on the river.

As the control, ie what we're measuring our different EV calcs against to see if they're more profitable, we have to measure the EV when we check turn and try to spike the river. Assume that when we check the turn, we value bet for $75 into $100 on the river.

EV when we check turn:

Total EV
= [EV of checking turn and valuebetting when we hit river] + [EV of checking turn and missing river]
= [Probability of hitting*likelihood of getting called*outcome of hitting]+[Probability of missing*outcome of missing]
= 0.18*1*175+0.82*0
= $31.5

On average, we make $31.50 by checking back and value betting river when we hit.

Now let's try an EV calc for %0 fold equity and the assumption that we bet turn.

EV when we bet turn:

Total EV
= [EV of betting turn, hitting river and valuebetting] + [EV of betting turn, missing river and giving up]
= [EV(turn bet)+EV(river vbet)] + [EV(turn bet)+EV(river ch)]
= [-$75+(P(hit river)*P(call)*outcome)] + [-$75+0]
= [-75+0.18*1*375] - 75
= -7.5-75
= -$82.50

So with %0 fold equity on the turn, we clearly lose money.

Solving for least FE needed for play to profit more than checking back turn:

To find out where the breaking point for EV is vs checking back, we set our breaking point to $31.50, and solve for the fold equity needed to reach that point. Any more fold equity than that, and we have a good bet. Gets a bit complicated:

$31.50 < EV(turn fold) + EV(turn call, river hit & we bet) + EV(turn call, miss river)
$31.50 < P(turn fold)*O(turn fold) + P(turn call)*O(turn call) + P(turn call)*P(river hit)*O(river bet) + P(turn call)*P(river miss)*O(river miss)
$31.50 < X*100 + (1-X)*100 + (1-X)*0.18*375 + 0
$31.50 < 100X + 100 - 100X + (18-0.18X)*375
$31.50 < 100 - 67.5X + 6750
$31.50 < 6650 - 67.5X
67.5X < 6650-31.5
X < 6618.5/67.5
X = 100.7

I redid the equations several times with this post, and as of right now I'm assuming the place where I went wrong would be in this last one for solving for X. If someone can point out any mistakes that'd be great, otherwise the conclusion is simply that a bet can't be profitable without %100 fold equity, which doesn't make any sense.

Ev when we bet turn is wrong already. Will do the math later if needed. Drinking sAint-Joseph right now.

correct

FYP

Iirc, FP had done a similar thread quite some time ago where he computed we should more or less always bet turns with equity provided we thought villain would not raise turn.

+ 1

Turns should usually be bet/fold or check/fold (unless you were fairly certain villain would fold to a bet but bet if you checked. Then you could check/raise)

Bet the turn.

if we expect villain to have a set, we should just overbet jam the river imo

I've always wondered about the point of building a pot on the turn for betting the river when we hit. Never really calculated it though. I did know that checking is always more +EV because 30% (off my head) of whatever we bet will always be -EV as a bet in itself. If I'm reading the above correct it's still +EV to bet anyway assuming we know we'll get called on the turn and river? And if we have any kind of FE we should always be betting a draw?

so is this when we are OOP or does it not matter whether we're in position or not.

Very very nice. I'd like the last equation solved too (beat: I'm not knowledgeable about statistics to do it myself) for the breaking point of estimated FE needed where betting becomes more profitable. I guess it depends on how much dead money there is in relation to our bet size?

Seems very important because it applies to all the hands when you want to barrel I guess, even if you're drawing to top pair and have 3 outs

No it's not wrong imo, it just ignores the dead money. It's the EV of the turn bet itself and only that. You don't need to put the dead money into the equation because we're trying to see if it's +EV to bet with 0 FE , not if it's +EV for the hand overall, preflop to showdown, when we bet.

The dead money only becomes important when there is FE, I guess that's why you need it to be profitable betting.

What the hell kind of drink is a saint-joseph? Sounds classy and good

My assumptions/the situation are really not very realistic at all, I'm just trying to figure out where the breaking point is. Obv if he has a set and never folds, we just ch back turn and overshove river when we hit, give up when miss. Smartypants.

Position is completely irrelevant here, I'm just trying to figure out the breaking point for where betting the turn with zero fold equity becomes profitable based on the increased pot size for a river value bet.

Belgo: tbh I ran over this so many times yesterday that my brain hurts and I can't even make it halfway through your FYP part without failing to understand something. Would be interested in hearing a clarification for somebody who's very hung over.

And yeah, FP did this but I couldn't find it. I assume his math will be much better and clearer than mine.

Should we be more inclined to barrel/semi-bluff when our outs are obvious scare cards and more inclined to check back when they're hidden (ie. double gutshots)

$375 when we hit

lose $75 when we miss

Win $100 when we get a fold

FE*(100) + (1-FE)(375*.18 - 75*.82) = 31.5

100*FE + (1-FE)(6) = 31.5
100*FE + 6 - 6*FE = 31.5
94*FE = 25.5
FE required = 25.5/94 ~= 27%