Modeling this will be quite difficult & time consuming (but not impossible). The approach I would use (solution design only - I haven't actually ran this analysis to date).

You would need assumptions to find villain's turn starting range derived from a first principle build up of:

- min raise range

- call 3b nai range

- call flop cbet range

Then you would need to map villain's price sensitivity (net of relative hand strength vs perceived hand strength of various bet sizes) for his various holdings in his turn starting range. Ideally you would use a complete distribution, but in order to simplify to discreet measurable categories I would probably categories villain's hand distribution into something like:

- strong nutty made hands (eg. 2pair +)

- strong non-nut made hands (eg. top pair)

- weak made hands (eg. 2nd or 3rd pair)

- strong draws (eg. nut flush draw or FD+OESD)

- weak draws (eg. gutshot)

You would then calculate the expectation of betting size B, where B= f(P,S); P = pot size at start of turn, S = effective stack

so X is bound by

- 0 where you check; and

- (S-P/2) where you bet your entire remaining stack

including the expected river hand distributions & actions across all potential river cards for the cases where there is still money left behind at the end of turn play. That is, to model the turn in isolation, you need to also incorporate into your decision what will probabilistically happen on the river.

Conceptually I would expect you could then run a sensitivity analysis of value vs protection benefit sub components of net expectation as a function of a two dimensional array spanning villain's hand type on one axis and our turn bet sizing on the other.

Based on intuition, I would expect that the price sensitivity function for villain facing any overbets will significantly contract his continuing range. That is, even though at equilibrium, we would expect that villain's continuing range would be a function of bet sizing to stop hero being able to overbet bluff ATC, I would expect most villains to perceive our overbets to be quite imbalanced and thus villain will respond in a binary fashion - always calling or always folding as a function of the particular hand type he holds and how it fares vs our perceived imbalanced overbetting range.

So you could simplify the problem by assuming villain only calls a turn overbet with particular hands that lie above some treshold (eg. 2pair +). Then you could calculate the value of the equity you fold out by overbet jamming turn vs the value of keeping more hands in villain's range making a smaller, more "standard" bet size vs villain's continuing range. We would still need to consider how we expect he will play his various holdings on the turn and the river for the full distribution of river cards. But once you know the distribution of villain's range, how he plays the subsets of that range vs various bet sizes and structurally how often dynamic holdings improve it becomes a relatively simple problem to solve.

Hi cdon3832,

Your analysis is very good and it is certainly true for a standard, rational opponent, but many opponents are not standard or rational. 

I didn't go in such a deep range analysis myself but my decision is based on two pieces of information that OP gives:

    - "I've also got a note on villain that he likes to attack rivers if I show weakness."

    - "If I check villain will almost certainly bluff and leave me stuck in a tricky spot."

This info if enough for me to make drastic changes to my gameplay and "adjust" to my opponent. The play is highly exploitative and it would be wrong against standard ranges and standard villain, but my decision is based on the "read" that villain is a player who bluffs a lot and this changes the situation. Folding strong top pair against someone who bluffs a lot when a perfect bluffing or "scary" card comes on the river is not a good play in my opinion.

To make a proper analysis and to construct villains ranges I would need to have info on some other villains tendencies and stats (min raising %, fold to nai 3b %, fold to cbet in 3b pot %...) but unfortunately this piece of information is missing in the post, so my decision is based solely on those 2 pieces of information that are available.
 

Without this additional info it is hard to say if a villain bluffs on the river more than 29% of time. He also sometime jams value hands that we beat (weaker Queens, pocket pairs JJ-55, and even some 4x occasionally) so we don't actually need him to bluff 29%, this percentage is much lower. 

The way you played pre-flop, flop and turn is quite standard. No real mistake there. Another option is to size your bets bigger on every street so that your turn bet is all in. This would make your life easier as you won’t need to play the river…

Playing 2 streets instead of 3 post-flop when out of position should generally benefit the player in BB…

The real question in this hand is how to play the river. 3 people gave 3 different answers (check-fold, check-call, bet all in…). Which play is the best?  It is very hard to say without knowing villains tendencies. I could imagine each of the 3 solution to be good versus different type of players:

1. Against a tight villain who doesn’t get in this spot with enough air: CHECK-FOLD would be best.

2. Against a loose-aggressive player who does have enough air and likes to bluff: CHECK-CALL would be best.

3. Against a loose-passive player who likes to hero call river bets:  BET ALL-IN would be best.

I'm convinced it's a pretty clear c/f for the reasons I provided. 

The only counter arguments offered by other posters was that villain will bluff us when we check to him. That's valid if villain gets to the river with enough missed draws in his range or if he's capable of turning a weak show down holding into a bluff - neither of which are true in this case given the action & board run out within the context of typical villain tendencies in 3b pots. 

I expect we will be ahead of his check behind range, behind his jamming range when we check to him and he doesn't call with a worse hand enough to merit a bet on the river. A check fold with our bluff catcher here is exploitable if villain bluffs too much. But if he's constructing his preflop, flop and turn ranges such that he gets to the river with a ton of hands to bluff with and exploit us, he playing a strategy which deviates quite far from the villains I've played against.