25nl: Turned top 2 after flop donk Axxxtt board

[The Bean Counter]

Get it in, r/f or flat call turn? What's the river plan if we flat or raise and get called?

Villain is 20/14 with no 3bets over ~40 hand sample. This is 2nd time villain has donked flop from 4 opportunities but otherwise fairly passive post-flop.


PokerStars No-Limit Hold'em, $0.25 BB (5 handed) - PokerStars Converter Tool from http://www.flopturnriver.com

SB ($27.92)
BB ($21.03)
UTG ($24.50)
MP ($28.48)
Hero (Button) ($31.53)

Preflop: Hero is Button with A, 8
2 folds, Hero bets $0.62, 1 fold, BB calls $0.37

Flop: ($1.34) 7, 2, A (2 players)
BB bets $1, Hero calls $1

Turn: ($3.34) 8 (2 players)BB bets $1.50

[MadMojoMonkey]

What's your range to ATS from BU, then call the donk-bet on this flop?

Of that, what's your range OTT to:
jam?
raise big?
raise small?
call?
fold?

[seven-deuce]

I'd prob raise for value, getting value from AT-AQ A2 A7 and to charge any FD's he may be betting.

[spoonitnow]

Thank you so very much for removing the results from this hand. That's such a big deal, and a lot of people don't realize that.

I'm pretty sure that a turn raise is the only play here. You can definitely expect to be the favorite against your opponent's calling range here. If you were going to slowplay a strong hand (big if), I think 88 (or slowplayed sets from the flop) would be better here because of Villain's reverse implied odds the times he has a draw.

For a raise size, I'd go for something in the range of $6.75 so. This sets up a situation where there is only a single bet left on the river, and that's what you want when you face a call on the turn. If a non-eight spade comes on the river and Villain shoves, then you can call on the river knowing that his turn was not profitable even if he always stacks you when a non-eight spade comes.

[The Bean Counter]

What's your range to ATS from BU, then call the donk-bet on this flop? {Ax without 2pr except AK, flush draws, 88-KK}. Raising 2pr and sets on this flop against this player

Of that, what's your range OTT to:
jam? raise big? raise small? Definitely getting it in with 88. Raising 87ss and prob T9ss strong. A8 is a raise but I'm not happy about getting it in if re-raised.
call? Other flush draws not in my raising range against this sizing
fold? 99-KK

.

[The Bean Counter]

I'd prob raise for value, getting value from AT-AQ A2 A7 and to charge any FD's he may be betting.

Indeed. But what's your plan for the hand? Are you getting it in on the turn if jammed on? What are you doing on what river cards if flat on the turn and then checked to, bet into again or facing a chk/jam?

[The Bean Counter]

Thank you so very much for removing the results from this hand. That's such a big deal, and a lot of people don't realize that.

I'm pretty sure that a turn raise is the only play here. You can definitely expect to be the favorite against your opponent's calling range here. If you were going to slowplay a strong hand (big if), I think 88 (or slowplayed sets from the flop) would be better here because of Villain's reverse implied odds the times he has a draw.

For a raise size, I'd go for something in the range of $6.75 so. This sets up a situation where there is only a single bet left on the river, and that's what you want when you face a call on the turn. If a non-eight spade comes on the river and Villain shoves, then you can call on the river knowing that his turn was not profitable even if he always stacks you when a non-eight spade comes.

Np on the results front spoon - thanks for taking the time out to reply.

What are your thoughts vs this villain's turn re-raising range? Get it in/call a jam?

Are you always stacking off if a non-8 spade rivers and you get jammed into?

[spoonitnow]

To clarify, I'm calling if he shoves the turn, and I'm calling if he shoves non-8 spade rivers. Work out the EV of his call with a bare flush draw on the turn assuming that he always stacks off on a non-8 spade river and you'll see what I mean.

[daviddem]

I'm calling if he shoves non-8 spade rivers

Not sure I get this. With what range do you think he flats a turn raise to $6.75 and donk-shoves a spade river, against which we can call a PSB profitably? You think villain is often turning his made hands into bluffs on spade rivers?

[spoonitnow]

To clarify, I'm calling if he shoves the turn, and I'm calling if he shoves non-8 spade rivers. Work out the EV of his call with a bare flush draw on the turn assuming that he always stacks off on a non-8 spade river and you'll see what I mean.

I'm so damn good that I can answer questions even before they're asked.

I'm Villain, and I have a flush draw. You're Hero, and on the turn, you raise to $6.75. You will always call a river shove, and I will check/fold on any non-spade river. I also check/fold on the eight of spades just because I'm psychic like that.

Once you make your raise, there is $11.59 in the pot. If I make a river shove, it will be for $12.66. I'm calling $5.25 on the turn.

There are eight cards on the river that give me a flush, and there are 46 cards that I haven't seen yet. My chances of hitting the flush on the river are 8/46.

The EV of my call under these extremely favorable (for me) conditions is as follows:

(I hit my flush)($11.59+$12.66) + (I miss my flush)(-$5.25)
(8/46)($24.25) + (38/46)(-$5.25)
$4.22 - $4.34
-$0.12

So if our Villain is partially psychic, his call is still -EV if we always stack off against flushes on the river.

[daviddem]

yeah ok, I understand that the EV of the turn bet is > 0 even if we call a shove on non 8 spade rivers. But once we are on the non 8 spade river facing his shove, folding has an EV of 0 and calling has a negative EV. So why call??

As an aside, it looks to me that villain's turn bet is on the weak side, and he is a 20/14 so is he really going to call that big a turn raise with a naked FD?

[spoonitnow]

yeah ok, I understand that the EV of the turn bet is > 0 even if we call a shove on non 8 spade rivers. But once we are on the non 8 spade river facing his shove, folding has an EV of 0 and calling has a negative EV. So why call??

Because I don't negotiate with terrorists.

As an aside, it looks to me that villain's turn bet is on the weak side, and he is a 20/14 so is he really going to call that big a turn raise with a naked FD?

Do you want him to call if we raise much less?

[daviddem]

Do you want him to call if we raise much less?

Yes, if I decide firsthand that I will not spew chips to make a political statement on spade rivers facing a shove.

edit: is your strategy a good example of making a 0EV play for balance? (actually a negative EV play if you look only at the river)

[spoonitnow]

Which hands are you calling a shove with on the river if a spade comes in that doesn't pair the board? Can't call with sets unless you're in the habit of calling on wet flops when you get donked into against 25nl 80% stacks.

[The Bean Counter]

Sorry, I don't wish to divert your discussion as it's interesting, but..

I was interested in the plan for this hand when we raise the turn largely because I wasn't expecting this from villain. My turn raise is small as villain's turn continuation looked so weak (bad logic on my part), but this is the way fish butcher sets almost always, right?

Presumably this changes nothing and we still get it in?


SB ($27.92)
BB ($21.03)
UTG ($24.50)
MP ($28.48)
Hero (Button) ($31.53)

Preflop: Hero is Button with A, 8
2 folds, Hero bets $0.62, 1 fold, BB calls $0.37

Flop: ($1.34) 7, 2, A (2 players)
BB bets $1, Hero calls $1

Turn: ($3.34) 8 (2 players)
BB bets $1.50, Hero raises to $4.50, BB raises to $8.50,

[spoonitnow]

Yeah I'm still trying to get it in there, and I expect to be a slight favorite. We're 48% against 77, 22, A2s, A7s and two combinations of A7o. I expect his range to be weaker than this on average because of the weird turn bet and random pair + draw hands.

[daviddem]

Which hands are you calling a shove with on the river if a spade comes in that doesn't pair the board? Can't call with sets unless you're in the habit of calling on wet flops when you get donked into against 25nl 80% stacks.

Calling with K, Q, maybe J high flushes because I can't think of any non flush hand with which villain flats a turn raise, then proceeds to open shove the river.

edit: OP sorry for hijack. For your last question, personally I wouldn't mind folding because he does not seem to be the kind who would do that with just top pair (unless maybe you have a very bluffy image). When he 3b your raise, he means business. I doubt he does that with a hand as vulnerable as A2. That is not a bluff/semi-bluff, because he cannot expect you to fold after your raise (again, unless you have a very bluffy image). Looks to me that his small turn open was more a suck in bet than weakness in the end.

edit2: and OP please don't include "xxx" in the titles of your post, my proxy filter at work doesn't like it

[bikes]

wow he took the only line i would consider folding

[rong]

Because I don't negotiate with terrorists.

Hahaha

[spoonitnow]

Calling with K, Q, maybe J high flushes because I can't think of any non flush hand with which villain flats a turn raise, then proceeds to open shove the river.

Fixed the last word in your post, but how are there any K/Q/J-high flushes in your range? Do you call the flop with flush draws when you're getting crappy odds and then raise the turn when you're getting amazing odds? A8 is probably the best hand in our range.

[daviddem]

OK you have a point even though I might take this line sometimes facing a turn bet that looks weak. Also call flop/raise turn is a much stronger line so hopefully our increased fold equity compensates somewhat for the worse odds (maybe wishful thinking).

This being said, even if I don't have flushes in my range and A8 is the top of my range, why in the world should I call a shove from a range that crushes me? Now you can say "if you're never calling then you are exploitable". Well I may be, and villain may know it and consider calling the turn raise with any crappy hand comforted by the fact that he will be able to bluff me out whenever a spade hits. Point is, his spade outs, bluffing or real, will only hit 16% of the time. So yeah he will win the hand 16% of the time (a little more considering he could hit better two pairs or trips) but in the long run I still win because he called a turn bet that vastly denies him the odds to win the hand 16 or 20% of the time, doesn't matter if he bluffs or bets for value OTR.

Now yes, if it gets to the point that I am loosing too much value because I am getting bluffed too much, it will be time to adjust and make hero calls to teach'em bastards that I am not negotiating, even if it costs me chips now to loose less later.

But for now, all I am interested in is whether a call on a spade river facing a shove is +EV, and the answer is no.

[spoonitnow]

Point is, his spade outs, bluffing or real, will only hit 16% of the time. So yeah he will win the hand 16% of the time (a little more considering he could hit better two pairs or trips) but in the long run I still win because he called a turn bet that vastly denies him the odds to win the hand 16 or 20% of the time, doesn't matter if he bluffs or bets for value OTR.

But why would you fold to worse hands on the river at all!!!111oneoneonecos(0) That wouldn't be +EV!

But for now, all I am interested in is whether a call on a spade river facing a shove is +EV, and the answer is no.

This is the type of thinking that's a major problem for a lot of players. The point isn't to try to be completely unexploitable; it's to avoid being exceptionally exploitable.

[daviddem]

But why would you fold to worse hands on the river at all!!!111oneoneonecos(0) That wouldn't be +EV!

Sorry wut? I am not folding to a hand he might or might not have, I am folding to a range that crushes me.

"Exceptionally exploitable" would mean something close to folding the non nuts every time we are raised. Seems to me our two pairs, even if they are the top of our range, is very far from the nuts and way behind the range of villain when he shoves spade river. Surely, there are better spots to make balance plays.

[spoonitnow]

"Exceptionally exploitable" would mean something close to folding the non nuts every time we are raised.

Wat.

[daviddem]

I mean folding the 2nd nuts because we assume that villain has the nuts every time he raises is exceptionally exploitable.

Seems to me that all the balance thing is worth it 1) when the range of our opponent and ours are merged or at least significantly overlapping and 2) against an opponent who is going to exploit us if we do not play close enough to balance.

By 1) I mean how can we have a value range or a c/c range when we are facing a range that crushes us (for example against a nit who never bluffs and shoves into our strong line)? Why would we want to be balanced against such a player when we can exploit him to death by folding?

So now in your example if I understand correctly you are worried that villain can exploit us if we do not call his river shove sometimes on spade rivers. He can bluff shove profitably if we fold 43% of the time or more to his bluffs (bet/(bet+pot)). So obviously if we always fold to a shove he can very much bluff shove profitably. So to be balanced and zero his EV we should call his bluffs exactly 43% of the time.

When he shoves we are being offered 30% pot odds. The EV of our play will be zero if we win 30% of the time when we call. So for his range to be balanced he needs to have 30% bluffs in it and 70% value hands (this is assuming that his value range always wins against our calling range, which it does if we go by your assumptions that A8 is the top of our range).

So now we know he has 30% bluffs in his balanced shoving range, so we should call his shoves 0.43*0.3 = 12.9% of the time for a balanced strategy.

So keeping this balanced situation in mind, when I said above "I don't think that calling a river shove is +EV", this translates into "I think his range is not balanced and it is skewed towards value (more than 70% value and less than 30% bluffs). So I will exploit this imbalance in his play by folding more. In the extreme case, if his range is 100% value, my optimal exploitative strategy is to always fold. However playing this optimal exploitative strategy also makes me exploitable if villain adjusts: his optimal exploitative strategy is to move all the hands in his c/f range to his bluffing range.

So with this in mind, we should play a balanced strategy against players who are both 1) good enough that we cannot exploit them properly and 2) so good that if we even attempt to exploit them, they will successfully exploit us (or similarly: we cannot exploit them as good as they can exploit us once we stray from balance to try and play an exploitative strategy).

Now let's say that this is the case of our villain in this hand. And let's say we accept that A8 is the top of our range. So when he shoves a spade river, we have in hand the part of our range that we should call the shove with to be balanced. I agree with that. We made a balanced play vs a super-opponent who would otherwise exploit us.

However where your logic seems to break down is the turn: you don't want to pay the price of this river balanced play, not even the few times a spade will hit on the river and he will really have a flush. So you make a large turn raise to compensate for this. So in the end this turn raise is a glaringly obvious attempt at exploitation. Damn, we just opened our flank, and if villain is any good as assumed, he is never going to call or raise such a fat value bet with worse.

Another thing: you say that A8 is the top of our range when we raise the turn. But from what you said, it's also the bottom, since you said we are not allowed to have flush draws or other two pairs (would raise other two pairs and flopped sets OTF, would not flat OTF with flush draws and raise turn with them later etc). So now that we have raised the turn, we have practically turned our hand over on the table and allowed him to play perfectly against us. Not a very balanced strategy, uh? Looks like we could add a few bluffs in there to compensate? Maybe a few K, Q, J high flush draws?

Also if A8 is the only hand in my range, for a river balanced strategy I should fold A8 to a shove on a river spade 87.1% of the time, and only call with A8 12.9% of the time.

So in conclusion, I don't think I am being exceptionally exploited if I can get him to call even a 1/2 PSR OTT with a worse hand and I fold 100% on spade rivers facing a shove.

edit: actually, if you ask me, a much more interesting scenario is where Hero makes a 1/2 or 2/3 PSR OTT, villain flats and open shoves a non spade river that does not pair the board. Now it's much more interesting because his range is very possibly polarized with bluffs (busted flush draws, for example) and sets. Now there really are serious balance considerations at play.

[MadMojoMonkey]

gold

[rong]

Looking forward to a response.

[spoonitnow]

So now in your example if I understand correctly you are worried that villain can exploit us if we do not call his river shove sometimes on spade rivers. He can bluff shove profitably if we fold 43% of the time or more to his bluffs (bet/(bet+pot)). So obviously if we always fold to a shove he can very much bluff shove profitably. So to be balanced and zero his EV we should call his bluffs exactly 43% of the time.

So now we know he has 30% bluffs in his balanced shoving range, so we should call his shoves 0.43*0.3 = 12.9% of the time for a balanced strategy.

Logic fail @ the two bolded parts. You can't simultaneously call 43 percent of the time and 12.9 percent of the time.

Seems to me that all the balance thing is worth it 1) when the range of our opponent and ours are merged or at least significantly overlapping and 2) against an opponent who is going to exploit us if we do not play close enough to balance.

Point #1 here excludes times when either player has a polarized range, and that makes this argument invalid on its own. Point #2 assumes that your opponent in any given scenario always plays exactly like you believe he does which is questionable at best.

By 1) I mean how can we have a value range or a c/c range when we are facing a range that crushes us (for example against a nit who never bluffs and shoves into our strong line)? Why would we want to be balanced against such a player when we can exploit him to death by folding?

To the bold, if someone is shoving into your strongest line, and you are absolutely crushed, then you have balance issues. To the underlined portion, you could always exploit him by folding if you are clairvoyant. Since you're not, you don't know if you're the one getting exploited or not. When you are playing in an optimal exploitative manner (like the strategies you recommend here and elsewhere), your opponent only has to change his or her strategy a small amount to drastically change your EV.

So keeping this balanced situation in mind, when I said above "I don't think that calling a river shove is +EV", this translates into "I think his range is not balanced and it is skewed toward value (more than 70% value and less than 30% bluffs). So I will exploit this imbalance in his play by folding more. In the extreme case, if his range is 100% value, my optimal exploitative strategy is to always fold.

The bold here simultaneously shows a lack of understanding of what the optimal exploitative strategy is and proves my point. If his range is 100% value, your optimal exploitative strategy is not to fold your entire range. As regards this specific spot, a call with A8 can be part of the optimal exploitative strategy if Villain has a range of 100% value bets.

So with this in mind, we should play a balanced strategy against players who are both 1) good enough that we cannot exploit them properly and 2) so good that if we even attempt to exploit them, they will successfully exploit us

Both of these points are correct. However, neither of them have anything to do with calling A8 here because I am not advocating a balanced strategy in this river spot. Don't be misled by the above two points: They are not the only conditions under which playing with a balanced strategy can be favorable.

Another thing: you say that A8 is the top of our range when we raise the turn. But from what you said, it's also the bottom, since you said we are not allowed to have flush draws or other two pairs (would raise other two pairs and flopped sets OTF, would not flat OTF with flush draws and raise turn with them later etc). So now that we have raised the turn, we have practically turned our hand over on the table and allowed him to play perfectly against us. Not a very balanced strategy, uh?

The underlined combined with the bold shows that you misunderstand what balanced strategies are. Whether our opponent knows our range (even if it's a single hand) is irrelevant to having a balanced strategy. It also doesn't allow our opponent to play perfectly against us unless he wants to play balanced as well. If we did have a single hand in our range, and if our opponent did know that, then that would be a subcase of playing against a polarized range like I referenced earlier. In that case according to your quote a few blocks above, you don't believe that playing against a polarized range can have a balanced strategy (which is absurd).

Also if A8 is the only hand in my range, for a river balanced strategy I should fold A8 to a shove on a river spade 87.1% of the time, and only call with A8 12.9% of the time.

You immediately go from saying you couldn't have a balanced strategy to claiming to know what the balanced strategy is. The strategy that you propose also contradicts what you said earlier about needing to fold 43 percent of the time. This shows again that you don't understand what a balanced strategy would look like for Hero on a river -- the same strategy that you are arguing against using and that I never advocated that Hero should be using.

Overall, you don't seem to have a solid grasp of what balanced strategies or optimal exploitative strategies are or how to find them in given scenarios. You seem to have a problem with using exploitative bet sizing on the turn in the hand while simultaneously battling against any non-optimal exploitative strategy on the river. The result of all of this is that you don't seem to understand that there are strategies that lie between a balanced strategy and the optimal exploitative strategy. Put another way, you make a mistake about poker thinking that a lot of people do, and that mistake is that the correct way to play is to take every +EV spot you can and avoid every -EV spot you can.

[daviddem]

No what I mean is that we have to call 43% of the time when he bluffs. And since he bluffs only 30% of the time, and the entire other 70% of the time his hand beats us, then we have to call a shove altogether 43%*30%=12.9% to zero the EV of his bluffs.

[spoonitnow]

No what I mean is that we have to call 43% of the time when he bluffs. And since he bluffs only 30% of the time, and the entire other 70% of the time his hand beats us, then we have to call a shove altogether 43%*30%=12.9% to zero the EV of his bluffs.

Maybe you didn't get the point, but this is wrong. Betting 12.9% will not give his bluffs an EV of zero. This is part of what I meant in the above (long ass edited post) that you don't understand what the strategies are.

I have already spent too much time on this thread, but if you'd give what his bet size would be on the river, then I can come back at some point and show you why your math is wrong.

[daviddem]

OK if I don't get it then work with me here. Villain open shoves b into pot p OTR. His range is a polarized range of 1) bluffs which Hero's entire range always beat and 2) value hands that beat Hero's entire range. Graphically:
<-Villain's bluffs-><-Hero's range-><-Villain's value hands->

Basically it's like an AKQ game, where Hero has K in position and Villain has either A or Q. Hero always checks behind if checked to (simplifying assumption). The pot size is 1 and the bet size is b.

Calculate the balanced strategies of both players (that is what I tried to do above and apparently failed to do).

If you don't want to bother solving it, I'll try my hand at it tomorrow. I know I can do it, I should have started with the AKQ situation and referred to your 5000th post...

As a sidenote, in my mind when I was talking about "overlapping" ranges, the above ranges ARE "overlapping" even though they have no intersection. On the other hand ranges like <-Hero->.....<-Villain-> are not overlapping in the sense that I meant it. Maybe I should have used another term.

[spoonitnow]

OK if I don't get it then work with me here. Villain open shoves b into pot p OTR. His range is a polarized range of 1) bluffs which Hero's entire range always beat and 2) value hands that beat Hero's entire range. Graphically:
<-Villain's bluffs-><-Hero's range-><-Villain's value hands->

Basically it's like an AKQ game, where Hero has K in position and Villain has either A or Q. Hero always checks behind if checked to (simplifying assumption). The pot size is 1 and the bet size is b.

Calculate the balanced strategies of both players (that is what I tried to do above and apparently failed to do).

If you don't want to bother solving it, I'll try my hand at it tomorrow. I know I can do it, I should have started with the AKQ situation and referred to your 5000th post...

As a sidenote, in my mind when I was talking about "overlapping" ranges, the above ranges ARE "overlapping" even though they have no intersection. On the other hand ranges like <-Hero->.....<-Villain-> are not overlapping in the sense that I meant it. Maybe I should have used another term.

The scenario that you've described above is too different from the situation in the hand. It's also more complicated than the situation in the hand because of all of the attempts to simplify it.

Two things overlapping implies an intersection. If there is no intersection, then they are not overlapping.

What bet size were you suggesting that Villain would be making on the river in this hand?

The link in my signature explains this situation fwiw. I'm surprised you didn't pick up on that.

[Fnord]

BB bets $1.50, Hero raises to $4.50, BB raises to $8.50,

I'm all-in. If he can beat top 2, good for him I hope the River is an Ace.

[Icanhastreebet]

all!!!111oneoneonecos(0)

nh wp.

[daviddem]

OK I will solve that AKQ game first, then I will try to show why I think the situation is similar in your scenario for this hand. Then you can prove me wrong.

Hero is IP with a K, villain is OOP with an A or a Q. Pot size is 1, bet size (all-in) is b. Villain can check or open shove b. Hero always checks behind when checked to and can either call b or fold when shoved into. It makes sense for Hero to always check behind when checked to, because Villain always calls with an A and always folds a Q, so there is no sense in betting. Now this game may seem unfair because Hero always has a K and Villain knows it. In real poker that would correspond to a Villain who is good at putting Hero on a range and the action on previous streets tells Villain that Hero has neither a nutty type of hand, nor complete air. Hero is not bad himself and he knows that Villain knows that, and he also figures that the range of Villain when he bets is made up of nut hands and bluffs.

Villain's strategy: always bet an A and bet a Q a fraction x of the time
Hero's strategy: fold a fraction y of the time

Preliminary remark: since villain bets all his A and a fraction x of his Q, it means that, whenever villain bets, he holds an A a fraction 1/(1+x) of the time and he holds a Q a fraction x/(1+x) of the time.

If both players play unexploitably ("at balance"), Villain's EV of betting a Q is zero (betting or checking a Q has the same expectation for him), and Hero's EV of calling is zero (calling or folding has the same expectation for him).

Villain's EV of betting a Q:
EV = y*1 - (1-y)*b
EV = y - b + yb
EV = y*(1+b) - b

Hero's EV of calling:
EV = (x/(1+x))*(1+b) - (1/(1+x))*b
EV = (1/(1+x)) * ( x*(1+b) - b )

When we zero both EV's, we find:
x = b / (1+b)
y = b / (1+b)

Which is the familiar bet / (pot+bet) ratio.

So now with your scenario for the hand above, the pot OTR after your $6.75 turn bet is $16.84 and the bet size OTR is $12.66. That gives us a bet/pot ratio b of 0.7518.

To be unexploitable, villain should bluff x=42.9% of the time with the bottom part of his range and check the rest of that bottom part. It also means that when Villain plays unexploitably and he bets, he has almost exactly 30% bluffs and 70% value hands in his range (a ratio of 0.429:1, just like in Spoon's signature's post).

To be unexploitable, Hero should fold y=42.9% of the time, and so he should call a seemingly massive 57.1% of the time (that's where I completely messed up when I said above that Hero should only call 12.9%).

As an aside, it intuitively seems like an awful lot of calling when we are crushed 70% of the time, but this is exactly made up for by the 30% times we win his bet AND all the dead money in the pot. Some reasons it may feel weird to call so much is 1) we are used to try and make fat +EV plays rather than 0 or near 0 EV plays (especially at microstakes), which is another way of saying that we tend to try to play closer to an optimal exploitative strategy than close to balance and 2) we are not used to play against Villains who have very balanced ranges themselves.

Before I go further and try to show why I think this game is a decent model of the river situation you describe with this hand, please let me know if the above is correct.

[spoonitnow]

OK I will solve that AKQ game first, then I will try to show why I think the situation is similar in your scenario for this hand. Then you can prove me wrong.

Hero is IP with a K, villain is OOP with an A or a Q. Pot size is 1, bet size (all-in) is b. Villain can check or open shove b. Hero always checks behind when checked to and can either call b or fold when shoved into. It makes sense for Hero to always check behind when checked to, because Villain always calls with an A and always folds a Q, so there is no sense in betting. Now this game may seem unfair because Hero always has a K and Villain knows it. In real poker that would correspond to a Villain who is good at putting Hero on a range and the action on previous streets tells Villain that Hero has neither a nutty type of hand, nor complete air. Hero is not bad himself and he knows that Villain knows that, and he also figures that the range of Villain when he bets is made up of nut hands and bluffs.

Villain's strategy: always bet an A and bet a Q a fraction x of the time
Hero's strategy: fold a fraction y of the time

Preliminary remark: since villain bets all his A and a fraction x of his Q, it means that, whenever villain bets, he holds an A a fraction 1/(1+x) of the time and he holds a Q a fraction x/(1+x) of the time.

Good so far. This is a good way to model facing a polarized range.

If both players play unexploitably ("at balance"), Villain's EV of betting a Q is zero (betting or checking a Q has the same expectation for him), and Hero's EV of calling is zero (calling or folding has the same expectation for him).

Villain's EV of betting a Q:
EV = y*1 - (1-y)*b
EV = y - b + yb
EV = y*(1+b) - b

Hero's EV of calling:
EV = (x/(1+x))*(1+b) - (1/(1+x))*b
EV = (1/(1+x)) * ( x*(1+b) - b )

When we zero both EV's, we find:
x = b / (1+b)
y = b / (1+b)

Which is the familiar bet / (pot+bet) ratio.

You were doing well until you got to this part. Nobody knows what x and y are at this point. I'm going to assume that y is how often Hero folds? If so, you have Villain's EV of a bluff being:

<Villain, Bluff> = (Hero folds)(win pot) + (Hero calls)(lose bet)

And that's right. You defined x as the percentage of the time that Villain bluffs with a Q, so you have Hero's EV of calling:

<Hero, Call> = (Villain's bluffing)(win bet+pot) + (Villain's value betting)(lose bet)

And this is right. You set both EVs to zero, and you get.

0 = y*(1+b) - b
0 = (1/(1+x)) * ( x*(1+b) - b )

Note that he's "pulled out" 1/(1+x) in the second equation if you're trying to follow along. If you solve these equations, you find that x = y = b/(1+b) like he's said. This is correct. So far, so good.

Shortcut: Villain's balanced bluffing range is always an alpha:1 ratio of bluffs to value bets in river all-in spots (or y:1 in the above). Hero's balanced folding range is to always fold the bottom alpha % of hands (x in the above). This would have saved you a lot of time and chances for mistakes.

So now with your scenario for the hand above, the pot OTR after your $6.75 turn bet is $16.84 and the bet size OTR is $12.66. That gives us a bet/pot ratio b of 0.7518.

To be unexploitable, villain should bluff x=42.9% of the time with the bottom part of his range and check the rest of that bottom part.

The bet size is $12.66, and the pot size is $16.84. The alpha value is 12.66/(12.66+16.84) = 0.429 or 42.9%. Villain will bluff with a ratio of 0.429:1 to value bets when balanced here. Along similar lines, Hero will fold 42.9% when balanced.

It also means that when Villain plays unexploitably and he bets, he has almost exactly 30% bluffs and 70% value hands in his range (a ratio of 0.429:1, just like in Spoon's signature's post).

Shortcut: You can do bet/(2*bet + pot) to get what percentage of Villain's balanced betting range will be bluffs. Here it's 30.0%.

To be unexploitable, Hero should fold y=42.9% of the time, and so he should call a seemingly massive 57.1% of the time (that's where I completely messed up when I said above that Hero should only call 12.9%).

Yep. That was your big mistake, and it makes an absolutely huge difference. Note that now Hero can still call like 15-20% of the time and be exploiting Villain to a large degree without over-exposing himself to be exploited in return. In effect, you're giving up a small amount of EV currently to protect the larger part of your EV in the long run.

As an aside, it intuitively seems like an awful lot of calling when we are crushed 70% of the time, but this is exactly made up for by the 30% times we win his bet AND all the dead money in the pot. Some reasons it may feel weird to call so much is 1) we are used to try and make fat +EV plays rather than 0 or near 0 EV plays (especially at microstakes), which is another way of saying that we tend to try to play closer to an optimal exploitative strategy than close to balance and 2) we are not used to play against Villains who have very balanced ranges themselves.

It's more about not understanding how a small degree of balance will help your EV more than it hurts it. Poker players who start studying and trying to get better get geared to think in terms of making all of the plays that they can that are +EV in a vacuum and avoiding all of the plays they they believe are -EV in a vacuum. It's hard to see beyond that since it's so easy to question it and stay in the comfort zone of trying to play optimal exploitative poker.

More to the point: I'm not advocating a balanced strategy in this hand on the river. I'm advocating a not-quite-optimal exploitative strategy.

[daviddem]

You were doing well until you got to this part. Nobody knows what x and y are at this point. I'm going to assume that y is how often Hero folds? If so, you have Villain's EV of a bluff being:

Villain's strategy: always bet an A and bet a Q a fraction x of the time
Hero's strategy: fold a fraction y of the time

I said what x and y are but I should have highlighted them