First I'm not a pro here but I'd like to comment anyway.

Read through the thread real fast and I'm with you on this one Jim (as usual). Shoootaa keeps saying that we should "steal the pot when our range is this strong". Have no idea what he means. In my mind shoving can only be good if villain will be calling with worse hands or folding a J which feels unlikely. Betting small as you propose (which was my first thought as well) has the benefit of getting called by worse but also the benefit of villain turning worse hands into bluffs. The drawback here is if we are betting small and then considering folding to a shove which it seemed like the OP was hinting at. I personally wouldn't although we are getting a negative freeroll. Which we by the way get anyway if we shove.

Then again he says shoving is the GTO play which I will have to think about for a while. Would love to here our own Gametheory's thougths on this one.

thanks rob.  I guess I should clarify, I'm more than happy to hear thoughts and comments from anyone.  I was just particularly interested in what the lead pros (especially people like GameTheory) might have to say on this one.

Extremely disagree with shootta that shoving is the best play or even that it's closer to GTO.  No way do we have enough bluffs here so I think he's going to be correctly exploiting us really hard by folding all his non Jx.

I'm kinda tired so I don't feel like hashing out the whole answer/ reading all the posts on 2+2 but I'm pretty sure that if you watch this VIDEO from roughly 17:20 it should explain Shoota's POV to you. Might be easier to watch the whole video but 17:20 onwards should probably do it.

Cool video.  It makes sense to me now and I have to change my mind and agree with shootaa.  Betting small is purely an exploitative play in that we can easily balance our range such that villain should still always fold because our range is just too strong, we're just trying to entice him to make a mistake by calling.  The GTO play is actually just close our eyes and shove 100%.  Really interesting.

I guess the part I don't understand, is doesn't the theoretical problem discussed by Chen (and this hand) make some assumptions about what Y/villain is holding that we can't reasonably assume in this OP hand?

IE Shootaa is assuming that villains checking range on the end has a mixture on non-Jx hands, some Jx hands, some AJ.  By shoving 100% of the time with our super strong range (basically a couple busted club draws, maybe a set TTs, and some Jx/AJ) we maximize the amount of time we win with our bluffs, allowing us to win the pot the greatest portion of the time.  But couldn't theoretically our opponent also have a stronger range than us thus such a shove would actually be a reverse freeroll?  IE our greatest equity in the hand would be to never bet?

lol I still don't exactly understand how this theoretical example applies specifically to OP.  I feel like it relies on assumptions about villains range that I'm not sure I agree with.

I think the main point to take away from that thread is that hero can and should call turn in theory with some weak hands, as he can play some of them +ev/0ev.

Basicly with clubs, you cant only think if you have potodds to call, its more complicated than that. Maybe some AT, A8 (what ever) combos are a call as well. Not because they are doing well on the turn, but they play so well when river is a blank and its checked to us (bluff).

The highest EV line isn't neceseraily GTO 

Yes but when are we actually at the equilibrium and if we ever end up being there how do we know we are?

From what I've read, shootaa/the thread makes the following assumptions:

- The opponent is very good/plays near GTO

- His range is stronger than our range on the turn, more AJ/Jx/sets/twopairs

- His river checking range contains few AJ/Jx combos

- We don't want to turn Qx or better into a bluff on the river

- Our opponent will not fold Jx to any size

Then based on these assumptions, shootaa concludes that it is optimal to bet $311 into $128 with a range that consists of all our air and our AJ/Jx.

Lets look at our range first, assume that we call between 40 and 60% preflop to the 3-bet.

In both cases we have 14-15% air hands, missed flushdraws. Note that it is assumed that KJ and JT always bet the flop, if we check those with some frequency we even have less air. Other hands that we might have as air are hands like A(c)T-9. Also since both the T and the 9 are clubs, we can't have a missed flushdraw with a T or 9. Considerations like these could increase our air up to 20%.

So given that we only want to bet AJ/Jx for value and that we have so litte air, it is certainly not optimal to bet all-in with our range. There are three main reasons for that:

1. Our opponent has a fixed amount of Jx in his river checking range that he will call no matter what, hence our air loses more if we increase the bet size. While at the same time Jx doesn't gain any more value by increasing the bet size.

2. In order to make KdKs indifferent (or even losing) to calling, betting $43 (1/3rd pot) is enough, since we only have 20% air, so he will be calling $43 to win $128+$43.

3. Our opponent still has AJ in his range, we need to defend our range against a shove. Firstly we will fold all our air to a shove. Assuming we have 20% air, when we bet $76 our opponent is risking $311 to win $204, hence if we call less than 40% (=$204/($204+$311)) of our entire range our opponent auto profits with any hand that could not have called profitably. That means that out of our 80% value range, we should call a 40% range, hence we must call 50% of our value range.

From the PPT screenshots, AJ was between 16 and 30% of our value range. That we raise AJ some of the times on the turn should makes this percentage even lower. Hence we need to call a decent portion of our Jx combos after we bet $76. For instance if we have AJ 16.7% of the times we bet for value, we need to call another 33.3% of our 83.3% Jx range. Hence we would call all our AJ and 40% of our Jx combos.

Another issue is defending against our opponent shoving Jx to fold out chops. If he would just call the $76, he would have $140 equity in a $280 against our Jx range, and $0 against our AJ range. By shoving, firstly he always wins against our air. But moreover he is risking $235 to win our $140. The EV of our opponent shoving Jx as compared to calling =

P(we have AJ)*(-$235) +P(we have Jx)P(we fold Jx)*($140) = $0

(1/6)*(-$235) +(5/6)P(we fold Jx)*($140) = $0

If we have AJ with chance 1/6, then by folding Jx with 34% we make him break even. Hence we need to call 66% of the time with our Jx combos, which overrides the earlier result of 40%. The larger the bet we make, the more our opponent can win if we fold to his shove, the more we need to defend with our Jx. Thus betting larger will just result in a betting and folding in larger amounts and a higher probability of getting stacked!

My conclusions

- In order for shoving to be correct, we need to bluff more hands than just missed flushdraws. More weak one pair hands need to be bluffed with. Otherwise we gain too little and lose too much.

- Auto calling all your Jx combos to a shove after betting $76 is not optimal and not needed to defend your range against bluffs and Jx that try to fold out chops. The more AJ we have in our range, the less frequent we have to call our Jx.

- When our opponent still has AJ/Jx in his range, it is no longer optimal to make large overbets as in the example from Chen/Ankenman. Especially him having AJ in his range makes large bets costly, since we need to defend our range against raises.

- The quote from shootaa: "If the cat is good at poker, then this is a clear all-in spot." is most likely false. Good cats protect their river checking range against overbets and normal bets much better than bad cats. That is, unless he has a different perception of good and bad cats than I do.

wow! such great analysis.  That is basically all of the things I was trying to understand in my post, but didn't quite know how to put it in to equations.  I'm going to reread that post multiple times just to understand the theory for this example.

that post alone is worth my $10 this month.  many thanks!

let me make sure i'm understanding the PPT sytax correctly...

in the 40% example:

row 1: A top 40% hand with clubs that does not contain a K through J

row 2: A top 40% hand that contains a J, but does not contain an A, K, or T

row 3: the number of combos of AJ in a top 40% range

so to summarize: row 1 is our air, row 2 is our Jx, and row 3 is our nuts.  correct?

awesome.  I would be happy to link this thread to the 2p2 forum is you guys would like me to (and 2p2 would allow it). it would certainly be free advertising for RIO, but obviously I don't want to step on anyone's toes and put up a link in a way that the RIO pros/owners wouldn't appreciate.

let me know...

Wondering if this situation could be a spot where we want to overbet our whole range to simply win the pot more?

http://weaktight.com/6010742

We get to the river with alot of strong hands but villain will also show up with KT KJ AQ given his line. Would it make sense to turn our weaker sdv into a bluff and overbet the river with our whole range? 

If this doesnt make sense could someone briefly explain where i missed the point.

thanks guys

Hey everyone! I just wanted to come on here for the first time and make a few things clear about my point of view, since it looks like a lot of what was said and quoted are a bit out of context. It's a cool problem and a fun spot. BTW, the cat spot was joking around with another 2p2er. I'm not an asshole :) We are just having fun!

I did not purport that we had to bluff with all our air, just that we should bluff in proportion to our Jx/AJ combinations. At the very start of the thread where the original poster asked about value betting, I commented that it might just be because he isn't floating the turn and/or turning weaker made hands into a bluff often enough (what most would call floats anyway since few hand combinations correctly check back flop that call turn with any decent equity against Jx). Obviously, if we can get away with it, turning something weaker into a bluff that has no showdown value probably works out better than having an overly widened turn calling range. My point is that we take into account river possibilities when calling the turn.

The river ranges listed above seem quite a bit off to me. As I'm interpreting the spot (and others are welcome to share different opinions), our range is much richer in Jx than our opponent's. I believe that was another above listed assumption of mine that isn't the way I laid out the issue and believe that it should be laid out. There are several reasons for discounting our opponents range including pre-flop play (we call more Jx than our opponent 3bets), flop play (we probably check back more Jx than does our opponent), the small affect of slow plays on the turn (our opponent may check some straights to us on the turn in order to avoid exploitation), the mixed strategy of betting and checking Jx on the river (our opponent wants to check some 'nuts' to us to avoid being so exploited), and card removal, which in this case is a very valuable source of information given the narrow value ranges. For that reason, I'm fairly certain that our value range is polar enough (necessarily best if called by bulk of bluff catchers) relative to our opponent's possible calling range (and possible range as the hand plays) such that we can value bet Jx and bluff in proportion (optimally speaking) in order to win the pot with a high frequency and steal the most ex-showdown equity. For more, definitely check out The Mathematics of Poker.

Of course, those ideas refer to optimal play. Others have mentioned that optimal isn't necessarily maximum value winning. If your opponent calls a small bet size with a worse hand often enough, then it may very well be the case that you can get away by betting a smaller size and enticing a weaker range to pay off. Because OP is worried about value betting Jx, I did mention on 2p2 that was probably a bit optimistic of an assumption. That's when the cat gifs came into play! Probably a better way of exploiting an opponent (instead of a smaller size than all-in with these stacks) would be to bluff the river with a higher frequency than optimal. Frequency manipulation is more difficult to read and the bluff catching player is already in an impossible spot wherein check-calling an over bet. His best response, if both players are playing optimally is to fold to hero's all-in. He is exploited in a way, but the minimum amount he can lose is by folding. Again, there's a nice example of why this is the case and why smaller bet sizes are exploitive in MoP. 

To clear up the combination of Jx points (mentioned on TwoPlusTwo http://forumserver.twoplustwo.com/56/medium-stakes-pl-nl/400nl-hu-there-value-1359663/index4.html), I would estimate that after all the discounting of our opponent's range based on the best plays he can make given the board run out from earlier streets that our opponent may have around 3-7 combinations of Jx (NOT 63!). Also, I would certainly think that given the pre-flop odds that we can have more Jx than some discounted combinations of KJ, some QJ, and JT. At least throw in J9s! There's definitely points on either side of the small bet/shove argument where someone might attempt to skew these combinations to prove their point and I think I've listed out each opportunity to do so, but I certainly think ignoring discounting the bluff catcher's combinations of Jx at all these points (or any of them) isn't a good way to go about creating the best approximation of the bluff catcher's river checking range. I've tried to be fairly pessimistic with my discounting for that reason. 

So yes, there's an inflection point whereat betting smaller with Jx/AJ makes sense. For instance, it's almost certainly not wise to shove 70 times the pot with ranges as estimated. Wherever that point is, and I would love it if someone wants to take the time to grind the math; but, I'm fairly confident it's well beyond a bet size of 2.437 times the pot.

Not trying to attack anyone's assumptions so much as clear up what was said about my own. I'd rather just have a discussion; otherwise, ego tends to get involved where there should be logical discourse (not a quip directed at anyone, just what I notice happens in poker forums a lot and something I try to avoid). Thanks to GameTheory for organizing spots for unknowns. Let's fill it out together and solve for the inflection point at which the value of betting a smaller size is equivalent to the value of betting all-in, and accounting for the respective frequencies of our ability to do each. That would be a great exercise. For anyone following along, I am discussing what is in Chapter 14 of MoP. Page 149 discusses some of the points I'm referencing.

Before I get into estimations of ranges for each player, I wanted to note a fairly common outcome in hands like this hand: the opponent is not check-calling with worse than one of a particular absolute strength. I realize that idea has the potential to get us away from a purely GTO solution, but it's worth considering because its by far the most likely in-game response of most players at this buy in level, and probably most buy in levels. Perhaps then we need two answers, what's optimal, and what's maximally exploitive if he's never calling worse than Jx when we bet, though I don't believe that the answers will be different for our relatively "small" over bet.

There's an equilibrium amount of Jx combinations to be reached based on the probability and adjustments of either player's "ability" to hold a jack. Ability accounts for likelihood a player holds Jx based on rational play up until the river decision. I'll discount both players Jx combinations just a bit based on their need to capture the most value on average, and not simply to balance for this scenario (Hero has Jx, Villain might have Jx - the reality is both players have Jx with some probability and make oscillating adjustments to the other's strategy until equilibrium is reached) as I don't anticipate it being an overly strong 'force' on the amount of Jx combinations, I'll use a 15% discount multiplier. If anyone can make a more accurate multiplier or show math about creating this equilibrium, that would be great to see and very much appreciated! So in that way, the solution for us knowing that we hold a jack is different than the actual solution, if that makes sense. Because we know we have a jack, we can remove several of villain's Jx combinations. To assume that Villain is balanced with perfect knowledge of us holding a Jx hand is not correct, again giving credence to shoving being a better play because further discounting Villain's checking range (fewer combinations of Jx) increases Hero's distinct polarity advantage. Anyway, on to the combination estimation based on the fact we hold a jack. Maybe there should be two multipliers? One for Hero, given knowledge of his jack, and one for Villain given no knowledge. I'm just using the 15% across the board for now.

I've said "Varied as needed in proportion to XYZ" a few times because those combinations should be predicated on the value combinations. Both players, at equilibrium, have some balancing of their bluffs to do, so sometimes I say "Varied as needed in proportion to XYZ" to account for that.

Villain - Checking River Range (11.05 combinations of Jx)

A - AJ combinations: 4 * (1-0.15) = 3.4

B - Jx combinations: 4 KJ (discounted given flop), 1 JJ, 4 QJ (if he checks all of them on flop) * (1-0.15) = 7.65

C - non-Jx made hands: Varied as needed in proportion to Hero's turn calling range

D - air: Varied as needed in proportion to Hero's turn calling range

Hero - Value betting River Range (35.7 combinations of Jx)

A - AJ combinations: 10 * (1-0.15) = 8.5            <--- maybe we raise turn some of the time for value

B - Jx combinations: [4 J7s, 4 J8s, 4 J9s, 2 JT, 12 QJ, 4 KJ] * (1-0.15) = 27.2

C - non-Jx made hands: Varied as needed in proportion to value range

D - air: Varied as needed in proportion to value range

I used the top 40% of hands that may call the 3bet, as GameTheory suggests. We could have a few more Jx combinations, like J9o; but I just omitted them. What do you guys think so far?

This is a really interesting spot. 

The discussion, although good, is very difficult to learn much from. One of the most interesting parts of reading the discussion is that pretty much everything being said by everyone makes sense, even when advocating completely different ideas about how to play a spot like this.

I disagree with Reid's assumption that an exploitative play is inherently sub-optimal. "Betting smaller than all-in is therefore sub-optimal (exploitive)."

IMO - VERY few players would be capable of being balanced in your opponents' spot here. I would bet/fold vs. some players, bet/call vs. some others, and occasionally check back if I thought it would induce my opponent to try to adjust to my check back in a way I could exploit (likely to be very rare). I'm having a hard time coming up with a way of figuring out who I would call a shove vs... as I'd probably have to be there in game to really know for sure.

The point about you not having the nuts seems to be really important in deciding whether or not to overbet here. I would expect that against a lot of players an overbet would actually be -ev in that you would get called by better more often than you get called by worse. That might change if your opponent thought you were capable of bluff shoving here.

I guess I get lost when someone says they're not trying to find the solution that would win the most.

"Of course, those ideas refer to optimal play. Others have mentioned that optimal isn't necessarily maximum value winning. If your opponent calls a small bet size with a worse hand often enough, then it may very well be the case that you can get away by betting a smaller size and enticing a weaker range to pay off." - Reid Young/Shootaa

Calling a strategy optimal that isn't trying to win the most seems to be inherently wrong by definition.

To find the solution that would win the most means that we should be taking in information about our opponents to predict how they would react with various parts of their ranges. That would be exploitative play. How that's not thought to be optimal is beyond me.

It seems like this is the type of spot that's going to generate a lot of heat between the GTO/exploitative ways of thinking. Pick what's right for you, and go with that.

x-post from 2p2:-

Someone correct me if I'm wrong, but as I understand it, there's a relatively easy way we can calculate the % of the time we can be called by a better hand that makes shoving 0 EV.

Z = Maximum % of time we can be beaten at when we bet for 2.43x jam to be optimal.

EV = X[1/(X+1) -2Z] = X/(X+1) -2ZX

dY/dX = 1/(1+X)^2 + 2Z = 0

Substituting 2.43 in for X*1/(3.43)^2 + 2Z = 0

Z=.0425

So if we're going to be called by AJ more than 4.25% of the time we shouldn't shove.

Can someone point out if I've gone wrong somewhere in that calculation?  Also, I'd like to adjust that to calculate the inflection point where stack sizes are such in relation to the pot that shoving is 0 EV, but that depends so much on our assumptions about how often V has AJ/Jx that it seems kind of meaningless.

re: Michael Dolle's comments above:-

There's always going to be 'heat' between GTO/explotative analysis, but the way I look at it is that if we can come up with a GTO strategy (i.e. highest EV strategy against a perfect opponent), we can improve our understanding of spots so that we can both:-

1.  Play perfectly against complete unknowns; or
2.  We can apply either specific or 'herd' exploitation to deviate from an optimal strategy.  

It's not just about 'picking what's right for you', it's a matter of improving our theoretical understanding by analysis so that we can make better decisions in game.

Fwiw, I've done a very similar analysis, but I gave Imfromsweden a 4-bet range preflop, and he defending a total of 45% of hands (instead of 40%). But it's still pretty close. 

Beyond on that, I took the time to put together what I would think is some decent ranges for flop and turn from a GTO standpoint... I spent a fair bit of time doing this and I've had a lot of practice in the past, so I'm guessing that it's going to be better than the ranges that you guys are choosing. You can goto the end of the post where I give some of the major decisions I made on earlier streets. 

RIVER RANGES BEFORE ANY ACTIONS:

FLOP ASSUMPTIONS:

Opponent: 

- His frequency for checking AJ & AT is only proportion to balancing his check-raising range... in other words, contrary to what Imfromsweden said, AJ would not be used as a check-call. The gut shot is too strong to play as a check-call on the flop unless it's with the intention of ALWAYS check-raising the turn when bet into, in which case our opponent would have slightly less AJ in his checking range. AJ is one of our better nonmade hands, and it'll turn into the nuts quite a bit, so it should not be played for high card showdown value. This means that the Villain will have ~9.6 AJ in his flop checking range -- which actually seems very similar to what GTO thought. 

- Opponent will be checking all his Qx type hands. I think this is a pretty fair assumption, I could see some arguments for betting flop and turn, and then checking river with AQ. Especially if we wanted to bet the flop small, and then the turn bigger with the intention of folding out some of the gut shot hands our opponent could have when the turn blanks. I didn't decide to do that, and it doesn't effect things much, but I'm just putting it out there. 

- Opponent will be check-raising all his sets. I think this makes sense, since the opponent will want to have a check-raising range here and he can't cap his checking range too much (even though it is a 3-bet bot). Even though he'll be check raising into 16 combos of JT, Imfromswedends next best hands are going to be 99, KQ, so he's doing well against those type hands. 

IMFROMSWEDEN:

- When checked to, Imfromsweden will bet ALL his AJ hands. I just can't really come up with a good reason why we'd need to check-back any AJ. The only reason that you'd want to check-back AJ is if we're going to have to bet-fold them to the check-raises. But this is not the case, they're going to be part of our bet-calling range. And they also don't have enough showdown equity to win with Ace high. This means that Hero has 0 combos of AJ on the river. 

- If people totally disagree with me about the AJ above, I did have Imfromsweden checking back ALL his QJ hands. This is probably a stretch. I could see a very sensible argument for betting flop and betting turn and checking back river with our QJ hands. However, if we do start betting a lot of our QJ hands, then our flop betting range is going to be wider than our flop calling range, and the Opponent is going to have a large incentive to not have a betting range on this flop and just check-raise a ton. (fwiw, this flop hits Imfromswedens range pretty hard and there's an interesting side discussion about how often the Villain should be betting the flop anyways... but that's for a different thread). 

- Imfromsweden will bet ~66% (pulled that out of the air) of his JT straights on the flop. Hero won't bet all his JT on the flop, otherwise is turn is going to be capped and Villian will start over-betting the turn large, until it's going to be higher EV for Hero to check back the nuts. I believe this would go back and forth until the EV of betting and checking JT will be the same, and GTO action will be a mixed strategy. 

- For both players, I'm having both of them bet ALL there back door flushdraws which aren't pairs. I think this makese sense because these will be the best bluffs which will have a chance to approve. I think they're probably better ideas than trying to make a small pocket pair a bluff. But I could see an argument for turning weak 9x into bluffs which I didn't do. My only point for really pointing this out is that I don't think either player will have many flushdraws on the turn because they both checked the flop. 

TURN ASSUMPTIONS:

GENERALLY:

In the hand, the Opponent ends up betting the turn. The problem with this scenario is that the Hero's range is quite a bit stronger than the opponents on the turn, and while he can't have any AJ in his range, Hero's range is nothing like bluff catchers. I don't think there's a lot of incentive for the Villain to bet the turn to be frankly honest, but I'm going to go ahead and give him a betting range so there's a range for the river. This would usually mean just making it super polarized like straights and better, but then this makes for a super simple river situation and not a very interesting one. So with the goal of having an interesting river discussion I'm going to give the opponent a turn betting range which isn't super polarized. 

Opponent: 

- Similar to the flop for Imfromswedend, the opponent will both bet and check his AJ and Jx hands on the turn. The reason is the same, he'll have an incentive to not overly cap his checking range on the turn, since there's so much stack depth left with only two streets to go. 

- Opponent will bet TopPair or better on the turn. Fwiw, more than 50% of Imfromswedens calling range on the turn will be weaker than TopPair, if that makes anyone feel better about the Opponent betting top pair on the turn. 

IMFROMSWEDEN:

- I didn't give him a raising range. This could be open for discussion. The Hero doesn't have any AJ in his range, so that's not great. But I could see an argument for both raising and calling some % of Jx. In other words, from a GTO stanpoint, Hero probably would raise some Jx and call with some Jx. But I'm not giving him a raising range on the turn. 

- The rest of his range is pretty normal.

The proportion is to the value combinations. It's not that they're any more complex to discover (I hope!). Once we agree on the value combinations, then let's find the bluffing proportions. 

I just wanted to make sure we can come to an agreement there and move forward as a group with more information. Our solution should be able to hold any number in place of any variable to account for differing assumptions anyway.  

I used the multiplier to avoid abstraction back from several streets. If we continue to go back (beyond the turn IMO) then the number of variables to quantify is pretty nuts as far as optimal play goes. I'm also going to say that I'm much more of the opinion that solving for when Jx can can call and other hands will not call is much more important that how often either player has strong or weak bluff catchers. Again, this highlights a difference between solving for the optimal answer and the maximally exploitive one. The good part is that the maximally exploitive answer should be much more simply because of our knowledge of our jack.

Reid, I don't like the way you are arguing here.

imfromsweden made a thread about which river strategy we should use after our opponent, on which he has some reads, checks. There you argued that shoving all-in for 2.4 times the pot was the optimal way to play. I disagreed with your conclusion, and I made a long post where I showed with calculations that betting smaller is better under certain conditions.

And I keep repeating that your interpretation of The Mathematics of Poker about bet sizes is completely wrong. The best way to convince you about that is to disprove your conclusion, while using your own assumptions.

You came here to defend your conclusion. But what you did was mostly write very longwinded wordy arguments, and asking me to fill in your assumptions. It is your responsibilty to state your assumptions clear and explicitly to draw your own conclusion, not mine. 

dodgybob

Someone correct me if I'm wrong, but as I understand it, there's a relatively easy way we can calculate the % of the time we can be called by a better hand that makes shoving 0 EV.Z = Maximum % of time we can be beaten at when we bet for 2.43x jam to be optimal.EV = X[1/(X+1) -2Z] = X/(X+1) -2ZXdY/dX = 1/(1+X)^2 + 2Z = 0Substituting 2.43 in for X*1/(3.43)^2 + 2Z = 0Z=.0425So if we're going to be called by AJ more than 4.25% of the time we shouldn't shove.

Where do you get these formulas from, they seem wrong to me.

Consider the following range distribution after our opponent checks:

Opponent: {1 combo AJ, 19 combos sets and two pairs}

imfromsweden: {14 combos Jx, 10 combos air} where the air consists of Ac[8-2]c and [8-2]c[7-2]c, all no pair missed draws.

For simplicity of argument I'm ignoring card removal effects, also since they are a small effect in this spot. If imfromsweden shoved 2.5 times the pot, our opponent needs to call 2.5 to win 6 (=1+2.5+2.5), hence he needs 41.7% equity. Thus if we shove all our 14 combos of Jx and all our 10 combos of air his pot odds are exactly 10/(10+14)=41.7%. Hence he is indifferent between calling and folding his bluffcatchers.

In order to make us indifferent between bluffing and checking with our air, our opponent must call 1/(1+2.5) times with his total range. This is since we risk 2.5 to win 1. Hence he calls 28.6% of his entire range, or 5.7 combos. Which means 1 combo of AJ and 4.7 combos of bluffcatchers.

In order to calculate the game value of this solution, since our opponent will be indifferent between calling and folding his bluffcatcher, the game value stays the same if he were to always fold his bluffcatchers. Therefore can assume that our opponent always folds his bluffcatchers, meaning we win the pot 19/20 times, hence our game value is 0.95.

(Of course as a strategy always folding is not correct, but this is a nice trick that is often used in game value computations.)

Now the case where we bet 0.5 times the pot instead of 2.5 times under the same assumptions.

Again we must make our opponent indifferent between calling and folding his bluffcatchers. We offer 3 to 1 pot odds, hence we must bluff 1/3rd times 14 combos, which equals 4.7. In total we bet 18.7 combos. Again, our opponent must defend a fraction of his range that makes us indifferent between checking and bluffing with our air. We risk 0.5 to win 1, hence he must defend 2/3rd of his range, or 13.3 combos.

Also note that now our opponent has the option to raise his AJ. If he shoves, we must call 2 to win 4, hence our opponent must bluff in a 2:1 ratio since he raises exactly the pot.

Therefore he will defend by shoving 1 combo AJ, 0.5 combos bluffcatchers,  and calling 11.8 combos of bluffcatchers, and folding the last 6.7 combos of bluffcatchers.

With his bluffcatchers, he had an breakeven call when he decides to shove, hence he is risking 2 to win 2, thus we must call 50% of our entire range, for a total of 9.3 out of our 14 value combos.

We use the same trick again to calculate the game value for us, we assume that out of our opponents calling and folding range, all hands fold. And we bet fold always.

Thus our EV becomes:

1*P(we win the pot after betting) +0*P(we lose the pot after checking) -0.5*P(we bet and get raised) = 

P(we bet)*P(we don't get raised) -0.5 *P(we bet)*P(we get raised) = 

(18.7/24)*(18.5/20) -0.5*(18.7/24)*(1.5/20) = 0.69

In conclusion, in this extreme example where our opponent held 95% bluffcatchers and 5% AJ, while we held 58.3% Jx and 41.7% air, betting 2.5 times the pot was best.

When betting 2.5 times the pot our game value was 95% of the pot and when betting 0.5 times the pot our game value was only 69% of the pot.

Of course I don't think this range distribution is accurate at all, but it shows what imfromsweden was saying: that the "gto bet size" strongly depends on the range distribution that is caused by the actions on previous streets.

GameTheory, you miscounted value combinations for both players quite a bit. The math, in the way it is set up, looks correct. The variables are not correct.

You need to account for card removal and for post-flop play. The re-raiser's range gets discounted for all possible Jx hands in several ways that Hero's range does not. We need to account for these when considering the polarity difference between the two players. 

Without abstracting several streets back from the river, creating 100% accurate variables for input in your math is difficult. I used multipliers and logic-based assumptions about post-flop play, rather than altogether ignoring the issue - that is why your results are different than mine.

When we are talking about a value range that is essentially one card, the idea that "For simplicity of argument I'm ignoring card removal effects, also since they are a small effect in this spot" is not correct. Card removal accounts for a very large portion of the combinations and skews the result of your math quite a bit. Ignoring card removal makes game theory calculations much easier to do, but the answers, especially in the case of narrow value ranges like these, are going to be inaccurate. 

I think you're correct given your own assumptions, they're just assumptions that overlook much of what is the case and prevent an accurate description of this hand.

EmptyPromises

Fwiw, I've done a very similar analysis, but I gave Imfromsweden a 4-bet range preflop, and he defending a total of 45% of hands (instead of 40%). But it's still pretty close. 

4-betting preflop has indeed a marginal effect, many 4-bet hands would bet the flop anyways. Altogether these differences in assumptions are small so I would agree that it is close. The best way to make these assumptions accurate could be obtained from imfromsweden giving his own estimations about both his own range and his opponents range.

FLOP ASSUMPTIONS:

Opponent: 

I think your assumptions are quite extreme here, for instance check raising all sets seems quite off to me. Especially since imfromsweden will check back pair + gutshot often, just like he did with hi actual hand.

IMFROMSWEDEN:- When checked to, Imfromsweden will bet ALL his AJ hands

This is again quite extreme. And it will be rare that it is optimal to bet all AJ combos in a spot like this. Not onlt since it will complete cap our checking range, but also since we should expect most better hands to call or raise.

- For both players, I'm having both of them bet ALL there back door flushdraws which aren't pairs. 

Again, I think this is very extreme, that means that we have no 8c7c or Ac3c in our range on the turn. This makes our range range strongly capped on many runouts. And I disagree that they are the best bluffing hands on KQ9r, hands like J9 and T8 are much better than A3s

TURN ASSUMPTIONS:Opponent: 
- Opponent will bet TopPair or better on the turn. Fwiw, more than 50% of Imfromswedens calling range on the turn will be weaker than TopPair, if that makes anyone feel better about the Opponent betting top pair on the turn. 

So you have imfromsweden calling many weak hands like AT, Q8 etc? That makes this spot favorable for overbetting.

Recall that in my first example I had a value (AJ/Jx) to bluff (missed flushdraw) ratio of 4 to 1. This ratio is by far the most interesting varianle for us, since we will be checking all our medium strength hands.

You didn't specify a betting or checking range for our opponent. I assume that he checks his entire range, which is quite reasonable since we have more straights and our opponent has more tp+ that he needs to protect. Also checking all his AJ works out well for his range.

The SB range has 11.2% "all hands", those must be missed flushdraws even though you said to bluff them all on the flop. That means a value to bluff ratio of 2.54. Hence we could bet 0.65 times the pot to make non AJ/Jx bluffcatchers indifferent between calling and folding.

After betting 0.65 times pot, our opponent can raise all-in for 1.78 times pot more. This gives us 2.29 to 1 pot odds or 30.4% equity needed. If our opponent raises with AJ and Jx only, he can have 60.8% Jx combos to make us indifferent between calling and folding with Jx. Hence he can have 1.55 Jx combos for every AJ combo, or 5.49%*1.55=8.51% range.

Note that us having a Jx does not have any significant card removal effect on the relative chance that our opponent holds AJ or Jx, since both players will have a J.

Now we can run game value estimation for this bet size with the following assumptions:

1.  Jx vs AJ/Jx has 25% card removal, all other card removal effects are ignored

2. Our opponent checks all hands

3. In order to use the game value estimation trick, we assume that our opponent folds all his bluffcatchers

4. In order to use the game value estimation trick, we assume that we fold all our Jx when our opponent shoves.

In order to use 1. first we must calculate the total range. This is not 100% * 100% but 100%*100% - 0.25*19.99%*28.5% = 98.58%

We calculate only our ex-showdown game value, that is the game value that we win purely by betting. This will be calculated as a fraction of the pot.

If we were to check down our 11.2%, it would lose against 70.5% of our opponents range, and on average chop against his 29.5% air (by assumption). Hence it wins 0.1475 times the pot, which attributes to a game value of

P(we have air)P(we win with air when checking)/P(total range) = 0.112*(0.1475)/0.9858 = 0.017

With our Jx combos, if we were to check, our game value would be:

P(we hold Jx)*[1*P(we win) + 0.5*P(we chop) + 0*P(we lose)] =

0.285*[1*0.800 + 0.75*0.145 + 0.75*0.0549]/0.9858 = 0.263

Hence combined our air and Jx have a game value of 0.280 (=0.017+0.263)when we check.

Now when we bet our 11.2% air for 0.65 times pot this gives a game value of:

P(we hold air)*[-0.65*P(we hold air and get called or raised) +1*P(we hold air and get a fold)]/P(total range)  =

0.112*[-0.65*0.200 + 1*0.800]/0.9858 = 0.076 <<where the 80% folds come from assumption 3.>>

Now when we bet our 28.5% straights this gives a game value of:

P(we hold Jx)*[-0.65*P(we get raised and we fold) + 0*P(we get called by a chop) + 1*P(our opponent folds)]/P(total range) =

0.285*[-0.65*0.75*(5.49%+8.51%)+ 0*0.75*(14.5%-8.51%) + 1*0.800]/0.9858 = 0.289

Hence combined our air and Jx have a game value of 0.365 (=0.076+0.289) when we check.

This gives us an ex-showdown game value of 0.365-0.280 = 0.085, or 8.5% of the pot that we win purely by our betting strategy.

Now the second betting example, where we bet all-in for 2.44 times the pot.

Again we must make bluffcatchers that have no card removal effects relative to our Jx and our air indifferent between calling and folding. Our opponent must call 2.44 times the pot to win 2.44+2.44+1 time the pot. Hence we must valuebet and bluff in a 3.44 to 2.44 ratio.

Hence, we bet 70.9%*28.5% of our worst combos as a bluff, a 20.2% range. We have 11.2% air thus, we take 9% out of our 19.2% one pair hands. For simplicity and to avoid card removal effects which our opponent can use, assume that we bet these 9% randomly out of our 19.2% range.

We calculate the game value of checking this 9% range:

P(we hold a hand from the 9% one pair range)*[1*P(we win) + 0.5*P(we chop) + 0*P(we lose)]/P(total range) = 

0.09*[1*0.295 + 0.5*0.0916 + 0*0.6134]/0.9858 = 0.031

We already did the game valaue calculations for bot our air and our Jx when we checked:

Game value for air = 0.017

Game value for Jx = 0.263

Hence, combined our air, 9% one pair bluffrange and Jx have a game value of 0.311 (=0.031+0.017+0.263) when we check.

The game value for our 20.2% bluffing range when we bet 2.44 times the pot, again using the calculation trick that all blfufcatchers fold:

P(we hold a bluff hand)*[-2.44*P(our opponent calls) + 1*P(our opponent folds)]/P(total range) = 

0.202*[-2.44*0.200 + 1*0.800]/0.9858 = 0.064

And the game value for our Jx now becomes:

P(we hold Jx)*[-2.44*P(opponent has AJ) + 0*P(oppnent has a chop) +1*P(opponent folds)]/P(total range) =

0.285*[-2.44*0.75*0.0549 + 0*0.75*0.145 + 1*0.800]/0.9858 = 0.202

Hence combined our bluffs and Jx have a game value of 0.266 (=0.064+0.202) when we bet all-in for 2.44 times the pot.

This gives a NEGATIVE ex-showdown value of 0.266-0.311 = -0.045.

That is, purely by betting all-in with a balanced range, we lose 4.5% as compared to checking.

In particular, since betting 0.65 times the pot gave us a POSITIVE ex-showdown value of 8.5%, betting 2.44 times the pot instead of 0.65 times the pot costs us 13% of the pot!

thoroughly enjoying the entire discussion.  have reread the thread once, and figure I will do it several more times this week.  was out of town the whole weekend so it's a real treat to come back and find awesome posters providing outstanding analysis.  thanks to everybody.

Hey guys, imfromsweden here from the 2p2 thread! Really nice discussion here, and thought I'd give my input on what I think of ranges here at least. I think me calling Q8/AT etc on the turn seems very very unlikely. My range is mostly going to be Jx/FDs/turned 2pairs/perhaps a hand such as AQ that has a good pair + the nut gutshot, but it's not going to be wider than that. I also don't think he's going to bet top pair or the better on the turn (a lot of 2pairs and top pairs will reason that they are rarely getting called by worse on the turn and have a better chance of extracting value by check/calling turn), so I think his range is mostly polarized with air, Jx/AJ, and maybe a few hands such as TTT or similar. Given those assumptions, mr GameTheory seems correct in his calculations that we should bet small. I'm not sure about the assumption that he's going to check/shove Jx a relatively high % of the time though, I think he check/calls Jx a vast majority of the time. So not sure how many combos of Jx we should be calling to a CRAI tbh. 

Hi imfromsweden, Mr. GameTheory here!

I think me calling Q8/AT etc on the turn seems very very unlikely. My range is mostly going to be Jx/FDs/turned 2pairs/perhaps a hand such as AQ that has a good pair + the nut gutshot, but it's not going to be wider than that

I gave you this range distribution, feel free to make (slight) adjustments if needed:

This range distribution has you on 108 combos, of which 15 are total air by the river. And 60 are straights, that means that your range is very straight heavy. Thus your opponent cannot bluff with a big bet size, unless he is trying to make you fold a straight.

Here is your opponents range:

I gave him all A*s without a pair and 87s as bluff range. I assume that he calls most Jx combos preflop to your minraise. Maybe you want to add more combos to his range, for instance Kc*c or medium pairs etc. Or discount some of them to give a better representation of his actions on previous streets.

I'm not sure about the assumption that he's going to check/shove Jx a relatively high % of the time though, I think he check/calls Jx a vast majority of the time. So not sure how many combos of Jx we should be calling to a CRAI tbh. 

This was a result from the EmptyPromises assumptions that we never hold AJ, and hence our opponent should be check raising us with AJ and some Jx or air combos as a bluff.

Do you think both you and your opponent have (almost) full AJ combos in our range, or how heavily do you want to discount those from both ranges?

I think he check/calls Jx a vast majority of the time.

Then why did you call the check raise, if you don't expect him to let you fold a J? Also, if he puts you on air, there is no need to raise that big for him.

I know this topic is probably a bit old now, but I was playing around with some excel numbers.  If he's always calling his Jx, which on the range you gave above (ignore for the moment that it may or may not be accurate, it's a good a guess as anyone's), he'll call an overbet shove and you'll split 44.44% of the time.

That would mean that for the EV of a shove to be 0, he would need AJ 14.11% of the time, and he's folding 41.44% of the time.

Can someone check those numbers for me?  It's a pretty easy calc (i.e. when you shove and split EV = ~0, when you shove and get called by AJ EV = -375.92, and when you shove and he folds EV=128) - ignoring rake effects.

Which would mean that if he has AJ 12/81 times, i.e. 14.81%, he needs to fold 43.51% to make EV 0.