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Ok let's first guess his PF range.

44% opening range
62% fold vs 3bet, 10% 4bet

If he uses a polarized 4-b range, he 4bets top 5%, calls next 28%, and 4bet bluffs next 5%.
Top 33% of his opening range, excluding the top 5% of his opening range, is ~12% range

This is my guess on what his range could look like preflop.
66+, A9s+, KTs+, QTs+, J9s+, T9s, 98s, 87s, 76s, AJo+, KQo

Now, let's try to estimate his equity on the river vs our hand.
Board: 8h4d9c5d9h
Equity Win Tie
MP2 41.03% 41.03% 0.00% { JJ-88, A9s, J9s+, T9s, 98s, 87s, 76s, AdQd, AdJd, QdJd, QdTd }
MP3 58.97% 58.97% 0.00% { KdKh }

Note that I assumed he was defending close to optimal GTO frequencies on Flop and Turn, didn't really look at his stats.

Anyway we have <60% equity which clearly means that we cannot value bet here. So we have a very clear check.

Do we c/c or c/f? I think it's probably a c/c. We beat A:dQ:d, A:dJ:d, JTs, Q:dJ:d which should be enough busted draws.

If we assume he's going to call with JJ and most TT (as he should) then you can make an argument that betting is better although we have only 43% vs his calling range.

EV (bet) = 0,28* (76) + 0,72 * ( 209 * 0,43 – 66) = + 38

Check - call
Check back worse = 15 combos -> 38,5%
Bet better = 16 combos -> 41%
Bluff = 8 combos -> 20,5%

EV (check) = 0,385 * 76 + 0,205 * (76 + 66) – 0,41* (66) = + 31

Correct?

oh yea perhaps, this argument applies to being OOP
of course IP we would always prefer checking. Maybe betting could be better if you had >60% equity edge, though I do find it hard to believe that value betting > check/calling.

Now let me read your calculations...
Total: 39 combos
Busted draws: 8#
JJ/TT: 12#
9X/straights/boats: 16#
87s: 3#

EV(bet) = (11/39)(76) + (12/39)(67.2+76) + (16/39)(-67.2)
= 21.4 + 44.1 - 27.6 = 37.9

*This looks right*

EV(check/call) = (15/39) (76) + (8/39)(143.2) + (16/39)(-67.2)
= 29.2 + 29.4 - 27.6 = 31

Yes your calculations are right, PROVIDED your assumptions are all true

one thing I would like to point out:

Villain defending TT/JJ might be a little too loose.

His bluff catching range = 37#
Against a 88% pot size bet, he should defend only 53%
19.6# = 9X+, 4 combos of JJ.

If you assume he defends all 12 combos of TT/JJ, that's way too loose. Whether he actually does that or not is another issue altogether.

If you say you have 43% equity vs his call range when you shove, why do you want to shove? I agree that ev of 66bb bet (shove) is better than 0bb bet, but why dont we use bet size for which we have > 50 equity vs his defend range? EV of 1bb bet size should be > than 0bb bet size (check) and EV of 10bb > 1bb, and so on. So you simply find max bet size such that has max EV.
If you want e.g your hand to be better than e.g 55% of his defend range, then you first estimate what is the percantage of that defend range from total river range, from that perc you get pot odds you give him with your bet size, and from pot odds you get bet size with which you have around max ev bet size.

You are assuming that villain doesnt have the option to make a raise.

If he raises here, are we calling or folding? Of course blockbetting could be good here but we would probably have to use some sort of balanced frequency to make vikkain indifferent between bluffung and not bluffing.
Or unless we have a read and know his tendencies against a blockbet

If villian raises balanced we are indifferent of calling or folding so it is like we lost pot and bet on river (folded). So we can count also bluffs that raise river as winning hands, and still we can find bet size for which we have X% equity vs his defend range, max EV bet size.
Off course in some cases shove is max EV, but it has to be proven, and not only play check or shove because these are standard bet sizes.

Good discussion.

Just wanted to throw in that turn is a check. Unless you think he checks back nitty (like QT checks and 9x checks) or shoves over a bet loosely (shoves JT, 9x), checking turn to check shove should be highest EV.

Originally posted by lnternet
Good discussion.

Just wanted to throw in that turn is a check. Unless you think he checks back nitty (like QT checks and 9x checks) or shoves over a bet loosely (shoves JT, 9x), checking turn to check shove should be highest EV.

I play as standard on drawy boards c/shove with overpairs that have value, because I expect most villians to bet almost whole range, some hands he has to bet for protection and some for value or bluff. I dont see what can villian play check behind? Would also on flop play with some AA c/r for similar reasons as on turn.

What do you think, is small bet on river better than check? If we bet 10% Pot, villian has to defend around 90% of his non-zero equity hands and so we would have clear value with bet, which should be higher EV than check (assuming villian doesnt do anyting bluff heavy when we check). Off course we can bet also larger, until our bet has > 50% equity vs his defend range. When he raises us with balanced range, he can use even less bluffs in range (which are all winners having EV as when we always fold vs raise) comparing to when we play check and he shoves because we have better pot odds when he we bet small and raises us.