Since no one else has responded and I have been waiting all day for someone to, I will Take a shot at it. This will be just a guess. I could be way off base.

OK, so if we limp only our weak hands and raise our strong hands, it will incentivize BB to attack our limps aggressively and play fairly tight vs our raises. This would thus incentivize us to start limping our strong hands as well because now raising them is not as profitable as before and since BB is attacking our limps so aggressively the EV of limping AA should be greater than the EV or raising AA because BB is playing so tight vs our opens. Once we reach an equilibrium (of sorts) where BB knows we limp our entire range he should attack us less aggressively thus raising the EV of our weaker hands but reducing the EV of AA (since he is not raising as often) but the EV of limping AA should still be greater than the EV of open raising it because once we start open raising our strong hands and limping our weaker ones, we go back to where we started (BB playing tight and attacking our weak limps).

That's how I see it but I don't know if it is correct or not, and I have yet to develop this part of my game (I just open raise and do not limp). I just think it would be very hard to have a mixed strategy here and we should either limp our entire range or open our entire range. Again this is just my 2 cents and my 2 cents (after currency exchange) is not worth near as much as other posters.

edit: So since we have decided to have a limping range - limping w/ AA is the line that maximizes the EV of AA, doing anything else would allow villain to play better (or at least make our range more transparent and thus allow him to make easier decisions).

"When deciding how to play an individual combination, we always take the line which maximizes our EV."

Just change it to "we always take the line which we think that maximizes our EV. This is the principle of rationality. It is an assumption to every single solution of a game. 

the answer is shania.

Playing a pure strategy does not hide information, it gives away too much information, actually. Phil discussed it on his theory series, thinking out loud. 

To analyze EV in a proper manner we should make it from the whole hand and not from one street, which is kinda more complicated to explain without taking a lot of time. 

Playing a pure strategy does not hide information, it gives away too
much information, actually. 

As a logical statement, this is obviously false.  Take the example raising first in pre-flop. 

I think that what you're talking about is splitting your range.  It's certainly possible to split your range by playing a pure strategy, but it's not guaranteed (see the example above). 

themightyjim: the answer is shania.

I've heard this phrase before, but I don't know what it means.  When I did an internet search for "shania", I got like a million pages of Shania Twain. 

I just looked it up and found it on another poker site.  It seems like total horse-shit. 

The concept they're talking about is quite different from balance as we think of it today, and the first thing that Mathematics of Poker does is argue that "shania" does not exist.  Didn't Sauce make a video showing the concept of "Shania" as they refer to it is actually not possible? 

I haven't read the other replies yet but my short answer is that the assumptions you are making in your question may be wrong. For example, even in a vacuum I'm not sure if open raising AA from SB is higher EV than limping. Even if villain checks back pre with air that he'd fold to a raise, sometimes we end up making more money by either letting him catch up a little bit or bluff, plus we often get to play a bigger pot by limp/3betting vs some hands that would just call a raise but would raise/call vs a limp. Obviously there are arguments for raising as well, but there's a lot to consider. 

Edit: This plus the adjustment points that arizonabay mentioned in his first post. Also your specific example assumes that you want to open limp SB, which you may or may not want to do. Basically you haven't demonstrated the "contradiction" that you say you are confused about.

On the other hand, clearly there are sometimes specific 0 EV lines we should take over other 0 EV lines in order to maximize the EV of our overall strategy. If this is the case, then are there really no -.001bb EV lines which do better for our overall EV than a 0 EV line with the same hand? I'm not sure... 

I'm not sure that most of you guys know what GTO means.  It does not mean making a play that is most +EV in a vacuum given available information.  It is a strategy that would be (if actually solved which based on current computational power is essentially impossible) unexploitable.  That does not mean that it is the most profitable strategy, it just means that no matter what strategy you would choose against it the best you could do would be to break even.  And if you make any mistakes in range construction, essentially you do not play a GTO strategy in response, you will be giving up some EV vs a GTO playing opponent's range. 

You guys are confusing optimal exploitation vs GTO.  Assuming that your opponent does not play GTO (which is a safe assumption considering no one really knows what a GTO strategy would be) you should shoot for a strategy that maximally exploits your opponents tendencies.  The closer that opponent is to playing a GTO strategy the closer your maximally exploitative strategy will be to a GTO strategy.  It's a ying and yang give and take situation.  

edit: and if you don't really understand your opponents strategy (ie you don't have enough or have been unable to synthesize available information into a cogent exploitative counter strategy) the advantage of attempting to play a GTO strategy is that it protects you from exploitation in situations where the inherent rules of the game stack the odds against you (playing OOP like in the SB for example).

fwiw I don't see how exploitation can be part of a GTO strategy at least in the way that we commonly use the term when discussing poker strategies.  A GTO strategy would inherently ignore the available information that would in fact inform an exploitative strategy.  at least that is my understanding of the concept.

I won't claim to truly know the answer to this but I believe this is wrong, a GTO strategy only ignores the other strategy after it is solved. But in order to find the GTO solution it is very much concerned w/ exploitation. Again, I have been under the impression that a GTO solution is a maximally exploitative strategy only it is exploiting a nemesis villain who is also maximally exploiting us. Both players are exploiting each other so much that they both lose incentive to deviate thus once this point is reached we are at the N.E. - once we have found this, we could always use it and never be exploited by another opponent. This strategy might not always make the most money (almost certainly not unless our opponent was also using this strategy) but it would prevent us from being exploited accidentally by an unknown opponent and the best that unknown opponent could do vs us was to play the GTO strategy himself. 

Skimmed over the responses, and don´t feel like it´s necessary to involve into the general "GTO"-discussion, most has been said already. :)

So, back to OP´s question:

"For example, AA is always worth at least + 1.5 BB's to shove AA
pre-flop, and so anything we do with AA pre-flop must be greater than or
equal to 1.5 BB's in EV.  This should imply that we never limp AA
pre-flop, since Villain can simply improve their expected value against
our Aces by checking any two cards."

You always have to think about it as strategy vs. strategy, not hand vs. hand. Take your example. We shove AA preflop (and say, we´re only shoving AA and nothing else) and Villain correctly folds, as we always have the nuts. This strategy "guarantees" us 1.5bb / hand with AA.

Now, as you correctly say, any other strategy, limping / raising smaller included have to yield a bigger profit than 1.5bb. Now, assume, for the sake of whatever I implemented the following strategy: Besides raising some of my top-range (excluded AA), I limp with some crap hands (say 76s-54s) that aren´t good enough to raise, but for getting 3:1, I still pay the price. To not let you roll over me, I limp with AA as well. If you raise (say to 5 bb), I´ll 3bet my AA and fold all my junk. When I 3-bet you, you fold your entire range (knowing I have AA).

Now, obviously this is no particular intelligent strategy (for neither player), but as you can see, the EV of my AA now is +5bb instead of +1.5bb (and your overall EV is 0), so in a vacuum, I found a way to make limping way more profitable to raise with my AA.

That demonstrates, how strategies aren´t made of iron but always can go back and forth ... once Villain found a counterstrategy to combat our "Limp-AA"-strategy, we might be "forced" to always raise AA again, because EV of raising is higher than limping. And maybe (who knows?!) eventually we will find a Nash Equilibrium, where limping has the same EV as raising with our Aces and then - and ONLY then - we would start to play the so-called "mixed strategy" and sometimes raise and sometimes limp with our Aces - and get the "deception" you where talking about.

Got clear?

  I'm focusing on the concept of GTO in an actual poker game being
played currently as opposed to an MES using available information from
that game.  You're discussing the development of a GTO strategy via a
process of MES against nemesis opponent(s). 

By definition, a game theory optimal strategy (pair) is two strategies which are maximally exploiting each other.  You converge upon the GTO strategy (pair) through maximal exploitation. 

For example, even in a vacuum I'm not sure if open raising AA from SB is
higher EV than limping.

When you open-limp pre-flop in SB vs BB, the Big Blind can do the following with every hand they would have folded to a raise:

1)  Check with every hand that they would have folded to a raise.

2)  Fold every flop on which they do not have the nuts.

It should be very clear that this entitles them to a larger percentage of the pre-flop pot than if you shove AA, or even if you raise to any amount at which they would have folded for that matter (in which case they would have folded, and get zero percentage of the pre-flop pot).  This is the same reason why people believe open-limping from any other position is dominated by the strategy of raising first in. 

Even if villain checks back pre with air that
he'd fold to a raise, sometimes we end up making more money by either
letting him catch up a little bit or bluff

This seems to be a by-product of people not understanding the above.  For the formation of Small Blind vs Big Blind, there shouldn't be anything inherent about this in structure of the game.  I don't need to bluff you post-flop if I get a free-play preflop.  I've still increased my expectation with every hand by checking (if I know you have AA).  It doesn't matter if you know I always have the nuts when I bet. Then you just fold some small % of the time that you limp, and you've won less than if you would have raised pre-flop (with AA). 

stevejpa

By the way, just want to say thank you to everybody for contributing to the thread. 

mightyjim:

IE there are some hands that as UTG raises in 6max NLHE are basically
breakeven. But having them as part of our range adds equity to our other
hands by allowing us to value bet and bluff certain board textures that
would otherwise be excluded by our range.  More importantly they allow
us to meet minimum defense frequencies so we aren't just exploitatively
playing too tight.

This seems more important as we get deeper, as the example of 10 BB starting stacks would show.  I used to agree with the pre-flop "minimum defense" thing as applied to opening from a non-blind position, but Sauce has kind of convinced of the opposite.  We open all hands that are +EV and fold all hands which aren't.  I guess the idea behind "Shania" is that it's impossible to consider the EV of any specific holding (except for very specific instances like where we have the nuts on the river)? 

Also, it seems like we don't consider "EV", but rather we try to find "the minimum EV" of every hand by looking at dominating strategies, if that makes sense.  In other words, we don't say "XX is worth this much", we say that "XX should be worth at least this much, or it's likely we're dominated."

I want to say "of course it does", with regards to your first sentence.  Suppose you form a strategy pair through some other method besides alternating maximal exploitation.  Then one of two things has happened:

1)  You are at equilibrium, in which case you have two strategies which are maximally exploiting each other.  You have just achieved a GTO strategy pair by alternating maximal exploitation by alternating zero times.

2)  You are not at equilibrium.  Then, by definition, somebody has incentive to change.  They will never change to a strategy which lowers their expectation, so whichever strategy they change to will be closer to the maximal exploitative strategy than the previous one.  In this case, it is clear that by iterating this process, we should converge to equilibrium (eventually). 

Because these are the only two cases (by the law of the excluded middle), then every way of converging to "GTO" should be equivalent to the process of alternating maximal exploitation. 

I don't want to get into a big discussion about semantics but the way I
understand it is that technically a single GTO strategy is the MES vs.
whatever strategy you're playing against. It's the optimal strategy
because it makes you the most money, and exploitation is often involved.

The term "GTO strategy is just referring to a (single) strategy which is part of the equilibrium.  You're talking about something different:

the MES vs.
whatever strategy you're playing against.

This is arguably more important, and it actually has it's own name.  It's called the "nemesis" strategy, and it's distinct (but related) to "GTO" strategy (pairs).

You may be confused because, as others have alluded to, you are not
considering your whole range and how one aspect of your opponent's
strategy is that she/he never knows exactly what hand you have.

Essentially, this whole thing is about how to construct a range. 

There seem to be two approaches. 

The first approach, as often endorsed by Matthew Janda, is that the range which we end up in any given situation is a by-product of taking the most +EV line with every combination.  We do not "balance for the sake of balance". 

However, I have also heard him say numerous times that he thinks a "GTO" strategy would involve several mixings (bet sizing, range construction, etc.)  He may have reversed his stance, on the first point, but I don't think he has. 

The second approach is "shania".  That is constructing ranges for the sake of balance.

There certainly seem to be aspects of each of these approaches which merge. 

I'm trying to understand when the two approaches merge and when they don't. 

For example, in the first approach, when you can argue that the EV of two strategic options are the same, you can play a mixed strategy.  Otherwise, your strategy is clearly dominated.  

The problem that I have with constructing ranges using "shania" is that it basically runs the risk of protecting a bad strategy.  You see all this time, if you think of somebody who goes "well...I want to value bet [insert hand they shouldn't value bet], so I'll obviously need to be bluffing more." 

You still need the first approach before you use "shania".  It seems that "shania" is a nice litmus test of your strategy, but it seems that it's a very weak approach.  (I mean "weak" in the logical sense here).

The first approach, as often endorsed by Matthew Janda, is that the range which we end up in any given situation is a by-product of taking the most +EV line with every combination.  We do not "balance for the sake of balance".  

However, I have also heard him say numerous times that he thinks a "GTO" strategy would involve several mixings (bet sizing, range construction, etc.)  He may have reversed his stance, on the first point, but I don't think he has.

These aren't necessarily contradictory stances. There are a lot of situations where hands are indifferent to multiple actions at equilibrium* and in those cases you will have the same hand mixed among those actions. 
*This is at least true for some slightly simplified flop/turn/river games as shown in Will Tipton's 2nd book.

Is the idea that they are somehow forced to raise a certain percentage 
when we limp, or else we have improved the expected value of our weakest
holdings by limping?  

Yes. If bb raises our sb limps less than an equilibrium strat does then he probably reduces the EV of our strong hands but increases the EV of our weak hands more than enough to compensate. We know it's at least enough to compensate (assuming there's some SB limping at GTO), because if it wasn't then the GTO strat wouldn't be the GTO strat!

I can't read this thread cause its insane, but the answer to your question in short is...

Is the idea that they are somehow forced to raise a certain percentage when we limp, or else we have improved the expected value of our weakest holdings by limping? 

Yes. They aren't "forced" but they'll raise their EV signficantly by raising a wide range preflop. In which than you can obviously 3-bet AA... which possibly might make AA higher EV as a limp than raise preflop. 

Yes. They aren't "forced" but they'll raise their EV signficantly by
raising a wide range preflop. In which than you can obviously 3-bet
AA... which possibly might make AA higher EV as a limp than raise
preflop. 

Great, thanks.  That's what I thought. 

"Forced" = They must do this to maximize their expectation in the game.  

I use "forced" to reflect the idea that if they do not do this, then the expectation they are sacrificing should show up somewhere else, and there-by raise the expectation of everybody else at the table.  In this case, it would raise our own EV of calling with hands that we would otherwise have folded.

Is that correct?

These aren't necessarily contradictory stances. There are a lot of
situations where hands are indifferent to multiple actions at
equilibrium* and in those cases you will have the same hand mixed among
those actions. 
*This is at least true for some slightly simplified flop/turn/river games as shown in Will Tipton's 2nd book.

I don't think they're contradictory at all.  What I was suggesting is that one approach seems stronger than the other. 

Santaur: they'll raise their EV signficantly by raising a
wide range preflop. In which than you can obviously 3-bet AA... which
possibly might make AA higher EV as a limp than raise preflop. 

This would imply that at the equilibrium, we could at least conclude that AA is higher EV to limp-shove than open-shove. 

Is it then valid to conclude that it also must be higher to limp than to raise for normal sizing?

The argument would be a kind of transitive thing between our various strategic options, using the "fact" that the EV of open raising (to 3BB, for example) must be greater than the EV of open-shoving, so we would have:

EV of raising to 3BB < EV of open shoving

EV of open shoving < EV of limping 

==> EV of raising to 3BB < EV of limping

Edit:  As stevejpa pointed out, this equation is invalid.  

"Your first line should say EV of raising to 3BB > EV of open shoving
given your previous statement and so you can conclude nothing."

For example, AA is always worth at least + 1.5 BB's to shove AA pre-flop, and so anything we do with AA pre-flop must be greater than or equal to 1.5 BB's in EV.  This should imply that we never limp AA pre-flop, since Villain can simply improve their expected value against our Aces by checking any two cards.

basically +1 to santaur.  i think you're confusing yourself by thinking that if he exploits AA with a subsection of his range (his PF checks), it means AA is getting exploited on the whole in the limp line.  but he could still be raising enough pre to make the EV of AA > or = to the RFI line

basically +1 to santaur.  i think you're confusing yourself by thinking
that if he exploits AA with a subsection of his range (his PF checks),
it means AA is getting exploited on the whole in the limp line.  but he
could still be raising enough pre to make the EV of AA > or = to the
RFI line

Well, that was kind of the whole question--

He certainly can exploit AA with a subsection of his range, but the question was can that subsection actually be his entire range?

The answer seems to be "no" if Villain is trying to maximize their expectation. 

You're right, though.

I've been playing so much Fast-Fold poker without reviewing my hands, due to this promotion on the site I'm playing on, that it's definitely messed with my head. 

I think you're right. I think it's actually kind of easy to argue (mathematically) that it dominates raising.

It does not seem easy or intuitive to me...want to walk me through the argument? 

To address Nick's point, if BB is putting in substantially more money vs a limp than a raise then of course we should limp our strong hands, but we should probably also raise our weak but still strong enough to not fold hands, which would incentivize bb to adjust. It's possible that limping big hands is a good exploitative adjustment in current games but I don't quite follow the reasoning why it must be better than raising vs good opponents. (edit: I'm not saying it's not best to limp, I'm saying I have no idea but I don't think anyone's presented a conclusive argument either way)

I've thought about where I was making errors in my thinking when I wrote the original post, and I think that's I was confusing the idea of game theory dictating that we "always play the highest EV strategy" with the statement that "we always play each hand in the most +EV manner", where the second statement should actually be quite meaningless on it's own, except for a few situations.