Poker is a zero sum game without rake. Who does our EV go to? One of the players calling with any two cards?

If you're talking about raising and likely playing HU vs. limping and likely being multi-way, I agree that the former is better, not because we're heads up, but because we're playing in a bigger pot where we have an edge.

The 'random range' thing was just to show how much our equity drops when we go mutliway. In reality against non random ranges it will drop even more.

As for EV, it increases for the player who has position.

Also, when you talk about playing a bigger pot where we have an edge, what edge are you referring to?

If our EV is going to the player in position, then are you saying it is correct for him to be calling with any two cards, or is it wrong only because the opener might have a hand stronger than AK sometimes?

If I have an edge I have a greater share of the pot than the money I'm contributing. EV increases as we maintain the same edge in a bigger pot. Edge will remain relatively constant (not exactly due to stack sizes and dead money) as ranges remain constant, which they are in the hypothetical.

Before we go any further, again you're speaking about an edge. Do you mean in terms of playing ability? If you are then we're getting a bit away from discussing theory as claiming to have an edge can be used to justify a lot of things that are theoretically bad. With 2 theoretically optimal players the one in position has an edge. Give them both the exact same range and the player in position will have the higher EV.

Maybe GTO preflop play, whatever that is, in a full ring range NLH game includes some sort of open limping strategy. Who knows. But practically for us humans it would be extremely difficult and complicated to implement, like you alluded to in your original post.

So yeah, thats why the majority of players choose to not have an open limping range. I think it's unlikely we will ever see "proof" (as in mathematically) that open limping + raising > only RFI in terms of EV.

The example of AK vs. n random hands is not two theoretically optimal players with the same range. It is one player with a hand that has a very large equity edge over the other ranges. The player in position is losing money by calling with atc, otherwise that would be a recommended play. That money is obviously not going to the first cold caller with atc. Some of it will go to the remaining players, but most of it will go to the out of position player with AK.

I am not talking about playing ability, although that could be a factor contributing to edge. Edge = EV/bet. If our edge is positive and constant, which it is roughly when ranges are insensitive to bet sizing, we would like the pot to be bigger.

I think we're getting side-tracked from the point. We are talking about the EV of raising relative to limping, not the EV of going 2 ways vs. 5 ways. We have little control over the latter. While it's true that some hands do better heads up than multi-way, we should hardly ever be raising to "thin the field." The primary reason to bet/raise is to win the pot (equity denial and bluffing), or to build a pot where we have an edge (value). For some hands, thinning the field is an additional benefit of raising, but it's not the primary reason we raise, and not the reason we raise AK specifically.

Since ranges are sensitive to bet sizing, a hand which is not profitable to raise because defending ranges are too strong may be profitable to limp. I think in some poker games limping may sometimes be a correct strategy, such as FE 7-card stud. I think NLHE is probably not one of those games.

But isn't "thinning the field" essentially the same thing as "denying equity"? Betting/raising to make people fold literally denies them the chance to realize equity cheaply.
FWIW, I think the vast majority of hands do better heads up than multiway. (One reason is that it's really hard to bluff much post-flop in multiway pots, and bluffing is a crucial part of winning). All of these factors partly explain why someone can open 85% of hands in a heads up game, but the optimal VPIP at 6-max will be 25% or less. Optimal play apparently doesn't feature much calling, except in the BB.
If thinning the field wasn't important, there would be no such thing as "isolation raises" or "light 3-bets", but clearly these are important strategic options, that are designed - in part - to prevent the hand going multiway to a flop.

Why is the BB allowed to call, if you think the other positions should not call?

Is it the better odds?

So what if it were an ante game where the other positions had better odds to call. Would calling show up then?

Better odds and the ability to close the action, often with the hand being played heads up, so that the BB only needs to win about 31% of the time to break even.
No other position can close the action, so 3-betting is often chosen as the best option, as it 'ruins' the pot odds of anyone else wanting to join in the fun. It's why hands like 65s are often 3-bets on the button, not calls. Forcing other people out of the pot means you get a bigger share of the dead money yourself.

I haven't studied ante games. I believe the play is looser, because there is more dead money to be won, but I'm not sure if it leads to more or less calling (except in BB, where there is LOADS of calling), let alone open-limping.

If you choose to open limp then you need to have a limp/fold, limp/call, and limp/raise range. Some players will only call or fold and this is a huge mistake. On top of this they open raise their strong hands which is even more of a mistake.

I've seen players implement a limping strategy that is well balanced. Basically they play as if their limp is a raise. Anyone who isolates the limper could be treated as a 3-bet. Players won't always adapt to this so easily. If there is an early limper and you see A9o on the Button you're probably going to raise to isolate the limper. Post flop you may end up giving the limper little respect for a hand and c-bet too often. I know I do this.

If you limp UTG you should only be calling a raise with something like KK-77, AQs-ATs, AKo. In the real world though, people don't do this. They limp call with all kinds of crap such as KTo.

Overall, I feel limping is an inferior strategy, but I doubt the EV difference is that great. It is much like open sizing. You can open 2.25x, 2.5x, 3x, 3.5x. The EV is different, but not that significant.

I do wonder if open/limping could be signifantly correct in late stage MTTs where the blind levels get to 20-15ish. I don't limp myself, but it wouldn't surprise me if we find out it is correct some day.

At fullring table at 100BBs deep, I highly doubt it is correct.

since I am a self-described expert in the GTO of pre flop, let me chime in:

there are hands that perform much better as a limp than a raise. an example might be A2s under the gun. This hand has zero ev. Go ahead and do a monte carlo simulation of it vs the entire deck. It doesn't make any money, doesn't lose any money. It's ev neutral. I take that to mean it's profitable to limp or should be profitable if you are a better player than the field.

That being said, i think open limping only works UTG or UTG+1. At other positions it's more profitable to raise, tho it probably isn't losing if you limp.

so there is a range of cards at UTG and UTG+1 that are more profitable and more theoretically sound to limp than to raise. probably includes low pairs like 33 and 22

then there are some hands that are profitable to (GASP!) overlimp but i guess the cool kids at 2p2 are not ready to discuss it

Overlimping with the same range that would be playable as an open-limp UTG is pretty standard. Iirc, none of us koolaiders said otherwise.

It's sound as long as it achieves a purpose. Betting should always achieve a purpose and not just be done "because it's the done thing"

If everyone is always calling preflop anyways then there's no point in open raising with anything but prime hands in order to get more money in the pot.

If someone after you always open-raises or rearises every hand then there's no point in open raising speculative hands...basically when you're just raising yourself out of the necessary odds to play profitably there's no point in open raising.

To me that says: if your opponents are very inelastic in how they behave the need for open raising sub-prime hands diminishes.

I was discussing 2 seperate things:
1) To show how much AK's raw equity drops even when we add a random range. I'd previously pointed this out though
2) The point about two optimal players with the same range was to explain that the player in position will have a higher EV (quite a bit in percentage terms)

Also, the player with AK won't necessarily have a very large equity edge over the other ranges. I think you're also assuming raw equity = best chance of winning the hand, which unless you're all in preflop isn't the case. EV, where it can be calculated, is a better indicator.

Even if we just use raw equity as an indicator of where the money is going, the majority of it is going to your opponents when it's multiway. It's possible you'll have a higher equity than them individually but not collectively.

I don't understand what you mean by much of this. Can you explain it in more detail?

Arty has covered much of this, equity denial and thinning the field are practially the same thing.
Our understanding of reasons for betting have evolved recently and Janda's explanation of betting to deny equity and/or make the pot bigger in case we win seem to be accepted by many good players these days.

You seem to be completely oblivious to the concept of pot odds. 50% equity getting 3:1 is significantly better than 65% equity getting 2:1. Obviously AK is going to have worse equity in a three-way pot as opposed to a heads up pot, unless the third person is drawing dead, but that hardly proves that AK would rather play heads up than three handed.

I'm going to approach this as someone whose specialty is live PLO, where I can make a much stronger case that always raising first in is clearly wrong.

The most obvious scenario where always raising first in in NLHE is when it mirrors the common live PLO situation, where you have zero preflop fold equity and are creating a bloated pot while playing OOP. Position has value. The worse your position, the less of your equity that you are going to realize. Having 24% equity in a five-way pot is more valuable OTB than in the BB. When I played more hold em, I tended to seek out the tables with 5+ players seeing the flop for no more than a single raise and avoided tables where the majority of flops were contested heads-up.

I do have the rare perspective of being a tight player whose raises were respected too much by bad players, so to get them to play pots with me, I had to limp more. Regardless, there are clear reasons for wanting to raise. I wouldn't be surprised if the GTO strategy is to always raise first in, but it should be easy to come up with table conditions where open-limping might be an exploitative strategy that has a higher EV than always open-raising. Those conditions might be the sort where there is no GTO strategy.

Again, as I've already explained, giving the equity % was just a very basic way of showing how quickly it drops when going multiway. In reality it's our EV that is important and in a limped multiway pot where we're going to be out of position against possibly multiple opponents a hand like AK will nosedive. I'm not even sure anyone is suggesting limping AK as that would pretty clearly be a poor play.

For anyone who advocates limping first it being theoretically sound (in any position other than SB) there are some questions:
1) In terms of theory, what conditions we are fulfilling taking this line?
2) Do any of the preflop solvers having a limping range? I have no idea other than knowing that Snowie doesn't
3) Give an example of a theoretically sound limping range

There seems to be a bit of a divide here with small stakes live players in general looking to defend the idea of a limping range but I'm not sure it's a thing anywhere else.

Your example is ineffective, though. Our absolute equity drops with any hand as we add players. Our pot share in dollars based on equity goes up with AK against random hands. Our true EV may or may not go down, but you showing the difference in equity is an argument that it should go up, not that it should go down. If you think the EV should go down, you should argue it a different way.

Of course all else equal, the IP player will have an edge when considering range vs. range and not specific hands. No one is arguing that position is not an advantage.

AK still has a very large equity edge against typical cold-calling ranges in most situations.

We're not playing against two hands collectively when each player bets independently of each other. Using raw equity to calculate EV, our EV goes up with AK as players are added with a range of any two cards.

Ignoring any dead money such as blinds:

When a second player cold calls, our EV and the first cold caller's EV go up. The third player's EV is being distributed to the others because he's making a losing call. Based on equity and ignoring factors such as post-flop playability and position, AK is happy to see the third player call.

Quote:
I am not talking about playing ability, although that could be a factor contributing to edge. Edge = EV/bet. If our edge is positive and constant, which it is roughly when ranges are insensitive to bet sizing, we would like the pot to be bigger.

I can try. Maybe you're not used to considering edge as a quantifiable term. Edge is the fraction of our bet we expect to gain/lose by having placed the bet. For example, for every dollar bet at single zero roullete, our expected value is -$.027, and our edge is -.027. If we bet $2, our EV is -$.054, and our edge is still -.027. Having a positive edge is the same as being +EV, but EV describes an amount, and edge describes a ratio.

Edge remains constant as we scale our bet sizing in roulette. Edge does not scale this way in poker, because an opponent still has the ability to make a decision after us. However, if you pre-determine that our opponent will call with his entire range. Our edge will be roughly insensitive to bet sizing.

In case it's not clear. I'm not arguing for limping AK or any other hand. I think it has no place in GTO play outside of the SB (although some place in exploitative play). I'm just arguing that thinning the field is not a very good reason to raise AK.

I do not consider thinning the field as the same as equity denial. Thinning the field implies we have a hand that would not like to go multiway. Equity denial is a different concept. Arty is right that a lot of hands don't like to go multi-way, but I'd counter that by mentioning that most hands prefer to go zero-way . Certainly our very best hands want to go multi-way. AA should absolutely be happy to get as many callers as possible. AKs probably is happy to get as many callers as possible. AKo probably wants to be heads up or win the blinds against optimal opponents. Against atc fish I'm happy to get everyone to call.

So, here's a LLSNL scenario that I actually have encountered. I'm known in my room as a good player who doesn't do stupid stuff and still tries to play good when playing smaller. Against certain lineups, if I limp, most of the table will limp with 50-70% of hands and only raise a range of QQ+, maybe TT+/AK. If I raise, they will respect my raises. Some of them will fold hands like AJs against my UTG raise and they usually won't 3bet KK. (This will change if I start raising every hand, but I never do that.)

Against this lineup, I think I should definitely have an exploitative limping range. I'm not sure if AK should be in that range, but I end up sometimes raising and sometimes limping with that hand.

Um, this is standard and exactly what they should be doing 9 handed. Limping and having a bunch of callers covering the whole board with position on you for no extra charge would be the nut worst imaginable position to put yourself in with the 3rd/4th strongest starting hand.

It's not really a problem, so long as you don't spew when you flop TPTK. I find it a lot easier than playing AK OOP in a 3bet HU pot.

Rather than go through everything here I'm going to drill down to what I see as the main issues and there are 2 in particular. I don't need to harp on about certain things given you admit this.

Sorry but saying in poker edge = ev/bet is just wrong and using roulette as an example to try to explain this shows it doesn't apply in poker. In poker your edge, outwith any skill advantage, comes from position and range advantage.

You may not consider thinning the field the same as equity denial but there's no way of getting away from the fact that it is a consequence of it. Denying equity means you're not letting your opponent or opponents see the next street.

Also I'm pretty amazed that you say you want AA and AKs to get as many callers as possible. Their winrates will be higher HU than they will be mutliway. AKs can at least make the nut flush byt AA is in general a one pair hand at showdown. Having a quick look at my database (664,000 hands) confirms that for me.

I think most people, when playing AA multiway, make mistakes that they are unaware of and stack off too easily. You have to identify the players that are playing fit or fold with pocket pairs and which ones are stacking off TPTK and other over pairs.

On some boards with certain players you may even want to check/fold AA. An obvious example would be you raise from EP and get called by 3 regs in position who are squeaky tight with their calling. Flop comes JT9. You check. Next player bets out 2/3 pot. The other two fold. And you fold too since he only does this with JJ, TT, KQs and maybe QQ.

You seem to be acknowledging that "edge" can have more than one meaning, but you're rejecting that one of those meanings can apply to poker. Edge can be used in any game with bets and payouts. It absolutely can be applied to poker the same way it can be applied to roulette. This is an objective truth. I suppose whether it is a useful measure is arguable, though.

I bet you're going to get a lot of disagreement here if you ask people other than me.

Your database is very weak evidence. A lot of the heads up pots are likely against the BB where we have position, or 3-bet+ pots. Most of the multi-way pots we'll be OOP. You have to control for positional advantage and raise size. Also, 664,000 hands may seem like a lot, but it's not. You are only expected to get AA about 3000 times, and even fewer times that satisfy the needed filters. The difference would have to be pretty glaring for this sample to be statistically significant.