Hi 2p2, first post here so pardon me if the post formatting is off.

Game was at a local casino on a Friday night. I do not consider myself a regular here but I do make the occasional stop on weekends. The table was 9-handed, and there was a younger mid 20s Asian player making preflop $15 raises around 50% of the hands and being quite aggressive post flop. For the actual hand, he isn't as important, but he's what I based my preflop decisions on. We'll call him V2. The main villian, a mid 30s Asian, V1, has been pretty quiet the whole night, playing only about 1 hand an orbit, and while he hasn't shown down a hand, I don't believe he has gone out of line at all.

OTTH

Hero is UTG ($350) with AKhh after being card-dead for about 2 orbits, and limps hoping for V2 to raise so he can reraise (thoughts on this?). 2 MP players call, and V2 (~$600) raises to $10 from CO. This set off some alarm bells as this was the first time he made a raise to under $15, but Hero's plan is still to reraise if it comes back to him. Folds to V1 ($240) on the SB who 3-bets to $31. BB folds, and Hero tanks a bit here, as this is quite unexpected, and decides to flat. Both MP and V2 call. 5 players to the flop.

Flop ($157) Ad Jd 8h

V1 checks after thinking for about 15 seconds. Hero now tanks with TPTK and BDFD, and bets out $75 (thoughts?). Reasoning behind this bet was, I put V1's 3-bet range preflop primarily on TT-AA and AJ-AK, and I believe all his AJ-AK combos bet this flop OOP, and maybe even JJ and AA. His check looks like TT, QQ, or KK that got scared on this A-high flop, and I didn't want anyone else to get a free card with 2 broadways or 2 diamonds. Both MP and CO fold, back to V1 who snaps allin for a bit over $200 total.

Hero?

Assuming action, prolly folding to Small Blind's JJ 'cause he sat there thinking for about 15 seconds deducing if anyone has a flush draw.

Other options in this hand: fold pre-flop, four-bet/re-evaluate pre-flop, check/evaluate flop.

snap call against described V.

too bad he had JJ

You have 2 ways to play this hand.

Either fold pre, or get all the moniez in OTF with TPTK.

Dangerous board. Best case Ax, 1010, QQ, KK. Worst, AJ or JJ. Is villain willing to stack off with 1p + draw or a pure draw is what I'd be asking myself. You can get too invested in TPTK, it's situational dependant. A crying call at best... I don't think a fold in this spot is horrible.

You should gii with TPTK when the SPR is 2. You're getting 3.5:1 to call. You beat 8 AQ combos, flip KQdd, chop AK, and lose to 6 AJ, 3 JJ, and 1 AA. You're completely committed to calling when you bet the flop

You have TPTK and the flop SPR is under 2.

Folding is not an option.

Your flop bet committed you to AI.

It happens. I hope you got a chop out of it!!

This form of 'hand-range' analysis seems erroneous to me insofar as we should assign percentage weights to the likelihood of each of these possible hands. If we want to get pedantic/academic, we can express this notion via conditional probability, and Bayes' theorem in particular. p(a|b)=p(b|a)p(a). e.g. where a~sb has KdQd and b~sb's line: Probability of KdQd given small blind three-bets pre-flop and check-raises on the flop is equal to the probability of the small blind three-betting pre-flop and check-raising on the flop, given small blind has king queen, times the probability small blind has king queen. A lot of analysis seems to do this job of counting the number of possible hand combinations, which is good, but then just stopping there and saying OK so there is 8/26 chance of SB having AQ here, 1/26 chance of SB having KdQd here, 6/26 of AK, et cetera. There might only be three combos of JJ, implying relatively lower p(a), but seems that p(b|a) is relatively high for JJ. AQ, AK, and KdQd flat pre more often than JJ, check-call more often than JJ....

without tons of reads and history, it's going to be generally impossible to get an accurate assessment of the likelihood p(b|a). So whether people realize or not, their use of a completely uninformed prior is probably not terrible. though in this case you definitely have an argument for weighting down AQ and AK somewhat.

Seems epistemologically equivalent to suppose a range of hands in set {a} and to estimate p(b|a) over set {a}, "without tons of reads and history."

LOL thanks. You can tell that to the dealer and other players while they're waiting on you to act at the table.

I think flop is a check. AP price is crazy but you really beat nothing. I would folding getting a worse price but I'm calling here. Hate it tho and this may just be a fold

I guess my point was that just choosing p(b|a) = 1 may have benefits in the form of robustness and ease of calculation that make it preferable. I am not sure that most players are consciously making this choice, and they should think about specific situations where the prior should be shifted meaningfully one way or another from uniformity. Your post certainly got me thinking about this. This might be one of them.

In theory, every hand except ours and the board cards exists in our villains range. The hand ranging above is making a shortcut by assigning some likelihood cutoff and discretizing to {0, 1}. In that likelihood, we are considering p(b|a) for sure. So the question is what cutoff are we using. Then, given that cutoff, what is the worst case equity based on this range. This analysis would be sick.

How badly this estimation can be is clearly situational. I'm not sure if you had some insight into how bad this approximation is or if you were just pointing out a gap (the latter is interesting, the former would be really telling and would probably sell a ton of e-books).

Tldr: I think your point is valid for sure. I think epistemologically valid too

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Good post. Thank you. Seems we are on the same page with this. Notice some hand equity calculation software programs on the computer where one clicks all hands as equally possible in an opponent's range, i.e. finding set {a} for {a} where p(b|a) = 1, and then just crunches the equity. So I think there is a systemic error in forgetting Bayes' Theorem in the poker community and especially in hand range analysis. Post #5 and Post #6 and perhaps OP seem to be using this methodology. And this method helps sway opinion toward a call, whereas I believe this should be a fold.

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The stack size / odds element of the hand, as it factors in our decision to call or fold, needs to consider some subjective meta-game attributes imo. How many chips we have behind to add-on, how much longer we feel we can play for, whether we expect small blind to still play for a while if we fold and he gets a $470 stack, whether we expect small blind to still play for a while if we call and lose and he gets a $670 stack, how a $470 stack versus a $670 will change his style of play, et cetera. If the night is young, we have many buy-ins behind, or small blind would not hit and run or tighten up with a $670 stack, we can call here and get showdown information on small blind and reload to like $300 which is enough to play good poker in position against a $670 small blind. If small blind might tighten up with $670 or hit and run with $670, we might consider folding more seriously, leaving small blind with $470, and we can reload from $250 up to $300 and still have a good stack size to grind at small blind in position.