I'm not sure about even calling the turn vs a seemingly competent reg with no real reads as we have reverse implied odds for the river. Sthief09 covered this well with lots of math in his playbook series, but basically pure bluff catching turns without strong reads on how the opponent plays the river is -EV.
Sauce123 also presented some info/math on this in response to a question about a hand in one of his videos:
Could you talk about the J9ss hand in your latest video (20m mark) where you check and call a flop of Th9d2s and say that you'd fold a turn 2d without a read. I've been having some trouble reconciling turn play in these situations. As a consequence, I sometimes get into tough river spots as one would imagine.
Could you explain to me why this isn't one of those 'just because it's tough doesn't mean it's wrong' kind of situations and why you're folding here? What are you looking to see or have a read about to allow you to confidently check and call a turn. If the board is T944x, do you think it's more of a justifiable call because you can CR bluff the river some of the time (probably a poor example of a board, but you know what I mean).
Anyway, this has been one of my largest problems in hold'em and everyone I ask has the standard answer for me that doesn't help. How can you apply math here and what are the implication for a turn call here? It seems to me that most of the time you're committing yourself to a showdown on a lot of rivers against a player betting 100% of his river range on some river cards like that 6d."
There are lots of ways to answer this, some more precise than others. It seems to me from reading your post that you are confused about how to represent the value of reverse implied odds with math. The most precise way to answer this question in this case where we have a known hand vs an unknown range would be to go through and calculate our equity vs the known turn betsizing and then our equity vs all sets of river decisions/betsizes. I'm not gonna do that. The second best thing would be to go and calculate our equity vs some set of the most likely river ranges/betsize combinations our opponent is likely to use. I'll do some of that.
The problem with both of these approaches though is that they don't really shed any light on how to represent reverse implied odds in general. So my plan is to first find a rule which is pretty easily applicable at the table and applies reasonably well to any case where you have to deal with reverse implied odds holding a bluffcatcher, but sacrifices some precision to do so.
Basically, what you need to do is treat the multistreet reverse implied odds situation (in this case the turn + river) as a species of overbet. To make this work we need to add a few assumptions though, first, that we are checking our range on 100% of future streets (which in this case we are, except on maybe a T or 9). Second, we need to assume a few variables, a) our opponent's range and b) our opponent's river betting frequency and range. It isn't vitally important that these two variables are incredibly precise, however, the more precise you are in estimating them the better your results will be using this model (in other words, you can justify any play by tweaking these input variables in a dishonest or mistaken way). In this case, I gave our opponent a turn betting range of K9+ for made hands, any OESD, any FD, and 35% of his gutshot and overcard combos (whether this is at all accurate is for you to decide). Against that range our turn equity is 36.5% and our pot odds require 31% equity to call, making a turn call ai +ev. To represent our reverse implied odds, I suggest multiplying an estimate of our opponent's future street betting frequency times his future street betsizing and add this onto the initial turn bet to get our 'real' pot odds (aka how much we have to call on average to get to showdown). In this case I used 65% riverbet freq times 3/4th psb (.65*15=9.75) making our real pot odds 24.25:16.25, where we now need 40% equity to call, which isn't enough. The problem is that smuggled into this calculation is the assumption that opponent bets all rivers equally frequently and that we always call and the only situation where this is particularly useful is when the board is super dry and we hold a bluffcatcher.
I'm not sure, but I think this overbet model + some scaling works OK to model spots where our equity takes different sorts of distributions on the river. Let's take 3 cases as a thought experiment. Assume the river cards are equally frequent and that our opponent's range has 100 combos and that we hold a determined hand..
Case 1- assume we have 33% equity vs a range, get 2 to 1 and there are 3 river cards, one of which we win on and 2 of which we lose on. Assume we make perfect decisions. We can call one and win on one river and check fold on 2 rivers no matter what opponent bets thereby realizing our equity.
Case 2- we have 33% equity, get 2 to 1 on the turn and there are 2 river cards, one of which we have 20% equity on and the other of which we have 46% equity on. On the river where we have 20% our opponent can pot with his whole range and we are forced to always fold. On the river where we have 46% equity our opponent can pot it with the 54 combos which beat us as well as 17 combos of bluffs such that we have to fold. This leaves us beating 29 combos at showdown such that we win (.50 * .29 = .145) or 14.5% of the pot making our turn call very -ev.
Case 3- we have 60% equity, get 2 to 1 on the turn and there are 4 river cards, one of which we have 100% equity on and 3 of which we have 47% equity on. One river we shove and our opponent folds or we check and he checks and on the other 3 rivers we check and our opponent shoves for 20x pot so we have to fold (needing 48% equity), making our turn call -ev. as we only win 25% of pots.
So from these three cases we can infer the more general rule that what matters in reverse implied odds situations (if we don't give ourselves the option to bet/raise) is on what frequency of rivers we think we can continue and what our river call win% is on those rivers, with the worst case scenario being infrequent rivers where we have 100% equity vs. frequent rivers where we have a flat equity distribution vs opponent's range such that we are forced to fold to our opponent's betsize. Knowing we have a 46 card deck, it seems like getting 2 to 1 on the turn we can call anytime we have > 55% or so equity. E.g. if we have 55% equity vs a range on the turn the worst case scenario is when we have 100% equity on 15/46 rivers and a totally flat equity distribution on the remaining 31. We can then model our total win % as 46*.55= 2530%. We can model our river equity distribution as (100*15) + (x * (46-15) = 2530, where x= flat equity distribution equity, which solves out to x= 33% such that we are indifferent. In this worst case then, we win 15/46 = 33% of total pots with 55% turn equity assuming pot sized river bet. I'd characterize the J9ss on T922dd example as a situation where our opponent puts us on the hand we have and value bets rivers quite well against us (although the maximally exploitive betsize is an overbet AI). Assuming pot sized river bets though, I'm guessing we end up making what are either slightly incorrect folds or slightly incorrect calls on something like any non 9 non offsuit J river card. Meaning we win very frequently on 4/46 river cards (I'll assume 100% of the pot to partially account for implied odds for ,085%) win probably 60% of the time he checks on 35% of remaining rivers for another 19% equity and end up making breakeven or worse calls on the other 60% of rivers making our turn call worth around 27.5% of the pot on this board.
Against a top 25% CO open range betting turn with K9+ made hands, any FD, any OESD, and 35% of gutters/overs then I'm thinking we need a made hand > A9 or K9 to call turn- in other words we have to beat some small portion of his made hand vbetting range and can't have a pure bluffcatcher. I'd also like to note for those not reading the whole post that this is in no way a recommendation to check/call all bluffcatchers midpair2ndkicker+ in this spot since our bluffcatching strategy is incredibly sensitive to assumptions about the CO's turn betting range. E.g. if the CO bets JT+, 40% of his strongest flushdraws and 10% of his overs then we need to be folding turn with hands as strong as QT (and this might be reasonable for some very tight 50nl regs). At high stakes hu in certain situations we might run into a reg who barrels made hands 88+ and 100% of his gutshot combos and 35% of his pure air which makes J9 a pretty standard call/call any river. Occasionally you might even run into the super polarized turn bettor who barrels only QQ+ on the turn and adds 35% of his gutters and all of his draws in which case you have to call turn and play a mixed strategy on the river.