4betting Preflop theorem

From Tannenj on 2+2 forum boards.
"This post was inspired by this thread, some brief AIM conversations with this guy, and a curiosity I’ve noticed I have about preflop play since I started datamining and dabbling with 400 NL -- a level known for a distinct spike in aggression, especially preflop -- a few weeks ago. The following discussion isn’t meant to be a be all, end all on preflop 4betting. If something like that could exist, poker wouldn’t be the fascinating game that has captivated the majority of this forum. Rather, I hope this post provides some food for thought about why to 4bet and helps some of SSNL with regard to how to think about villains’ ranges and how they should affect preflop decisions. If you read this post and think you got something out of it, you might like my Pooh-Bah Post.

Why 4bet?

It’s fairly simple, really. Two reasons (no order):

1. For value:

Not much explanation necessary. If A) you think you have the best hand and B) the villain has 3bet and you think it’s likely that he’ll call or shove over the top of your 4bet with a worse hand, the play is generally to 4bet for value. There are times when it makes sense to just call a 3bet with hands like AA and KK, but for the purpose of this article I’ll advocate an aggressive preflop strategy and assume that A) many villains 3bet light, B) to counter this, it’s helpful to have the light 4bet in your arsenal, and C) to balance light 4betting, you want to 4bet for value with AA and KK, and at times, AK, QQ, etc.

2. As a bluff:

As I touched on, this article assumes that villains 3bet light. In reality, at 200 NL, most villains don’t 3bet very light. I’ve found that some TAGs 3bet very light/frequently, especially if A) they suspect you’re raising light from late position or B) they have position on you. At 400 NL, most TAGs 3bet light. This is a personal observation, and my experience at the level is relatively limited. If you have played 400 NL, you might disagree. At the very least, however, there’s a significant increase in preflop aggression from the 1/2 games to the 2/4 games.

Say you’re sitting with a 26/22 LAGTAG and he’s 3betting the crap out of your open raises. You have a few options:

1) Leave the table. This is often a reasonable approach, especially if he has position on you. Too many of us play poker with our egos, and I’ve been guilty of doing so myself. We play to make money, and if the light 3bettor is competent postflop, it can be pretty tough to make money. However, we also play to have fun and to improve, so it often makes sense to stay at the table -- especially if there are one or two weak players sitting.
2) Open up your calling range. This can be useful, especially if you’re in position. Even out of position, it can make sense to mix in some speculative calls with the knowledge that A) you’ll hit sometimes and B) you can check-raise a bunch of flops profitably to throw the villain off guard.
3) Throw in some light 4bets. This is what I’ve been setting up for, and it’s a key component of this article. The situation follows: A) the villain knows you’re a TAG (or a LAG) and are opening with a pretty wide range, B) as a result, he’s 3betting with a wide range, and C) being 4bet with the bulk of the hands with which he’s 3betting is going to put the villain in some very tough spots.

If a villain is 3betting light, your 4bets should have a ton of fold equity (perhaps not after the villain realizes what’s happening, but at least for the first session or two). Against some villains, 4betting light is major +EV. With that said, much of the importance of the move is rooted in metagame. It’s very bad for your raises to be constantly 3bet, and in general, a villain will be less likely to 3bet after he notices that you’re willing to 4bet his 3bets.

Easy stuff. Show me some math:

When I decided to write this article, the purpose I had in mind was to explore the small 4bet. As I explained, there are two reasons for the preflop 4bet (or any bet or raise, really): for value, and as a bluff. The first one is easy; when betting and raising for value, it’s tough to run into trouble.

4bet bluffing preflop is trickier. The issue is that poker is a game of math. There’s this stuff called pot odds, and more often than not, it dictates your decisions in this game if your goal is to play soundly.

When you 4bet light, you’re hoping the other guy folds. Unfortunately, though, he won’t always fold. I suppose there will be times when he’ll call (especially if he’s bad -- but really, you shouldn’t be doing light 4betting against bad players). Much more often, however, he’ll shove. And when he shoves, things get fuzzy, because A) at this point, you’re being laid major odds, and B) obviously, you can’t know what hand he has (unless you’re Phil Hellmuth and have world class “reading ability”).

When things get fuzzy, it becomes more likely that you’ll make a mistake. When you make a mistake, you give up expected value. The more expected value you give up, the lower will be your win rate. I hope the following calculations illustrate some of the reasoning behind small preflop 4bets:

What follow are some assumptions I made for the purpose of my PokerStoving. At the risk of stating the (very) obvious, these things will rarely be exactly the case. This is a theory article.

1) effective stacks are always 100 big blinds.
2) players won’t call your preflop 4bet; if they continue with the hand, they’ll 5bet all in.
3) Nit will 5bet all in with AA and KK and fold everything else (note: you might be thinking, “If this guy is a nit, why are you 4betting his 3bets light?” The response is, “Whatever, it’s theory. For the purpose of this article, ‘Nit’ doesn’t mean he doesn’t 3bet light, it’s just a name I’m giving him to differentiate his 5betting range. For the purpose of this article, don’t worry about 3bets, how they should affect these players’ 5betting ranges, etc. -- just go with the assumptions, even if some of them seem contrived.”).
4) Conservative TAG will 5bet all in with the above two hands plus AKs, AKo, and QQ (and fold all other hands).
5) Aggressive TAG will 5bet all in with the above hands plus AQs, JJ, and half the combinations of TT (and fold all other hands).
6) LAG will 5bet all in with the above hands plus AJs, half the combinations of ATs, the other two combinations of TT, 99, half the combinations of 88, and half the combinations of 87s (and fold all other hands).

I came up with these ranges in the span of a few minutes; if you think they seem “random,” you’re right. There’s little rhyme or reason to them, they’re just four ranges that start out tight and gradually increase in scope. Obviously, the number of ranges with which the following calculations can be performed is essentially infinite. Adding a broader range of ranges (ugh) to this analysis definitely has some merit, and hopefully someone will do this if it’s decided that the initial investigation is worthwhile.

First, let’s look at some scenarios in which you 4bet pot. Note that none of these 4bets are for value; each one is a bluff. The best hand you hold in the following scenarios is TT. Even TT is a dog against the widest of the villains’ 5betting ranges, so when you 4bet with this hand, you’ve decided to turn your hand into a bluff. TT does have solid equity against the range of LAG, but it’s important to realize what’s occurring. Turning TT into a bluff by 4betting isn’t necessarily a poor play; it is not, however, a value raise.

You’re UTG with 65s. You open to $7. Two folds to Nit on the button. Nit makes a pot 3bet to $24. You make a pot 4bet to $75. Nit 5bets all in.

Pot Odds:
You have to call $125 to win a pot of $278. You’re getting 278:125, or 2.22:1.

100/3.22 = 31.06.

Equity:

equity win tie pots won pots tied
Hand 0: 22.496% 22.29% 00.20% 18321336 168354.00 { 65s }
Hand 1: 77.504% 77.30% 00.20% 63532548 168354.00 { KK+ }

Are you priced in to call?

No. You need a little over 31% equity to call, and you only have about 22.5%. You can fold the hand without making a mistake.

You’re UTG with 65s. You open to $7. Two folds to Conservative TAG on the button. Conservative TAG makes a pot 3bet to $24. You make a pot 4bet to $75. Conservative TAG 5bets all in.

Pot Odds:
Same as above, $125 to win a pot of $278. You’re getting 278:125, or 2.22:1.

You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 31.144% 30.92% 00.22% 72010208 515570.00 { 65s }
Hand 1: 68.856% 68.63% 00.22% 159831996 515570.00 { QQ+, AKs, AKo }

Are you priced in to call?

Yes. You need a little over 31% equity to call, and you have it almost exactly. You’re priced in to call with only about 31% equity; you’ll get stacked more than twice as often as you’ll suck out, but calling is correct because folding would be a slight mathematical error.

Since you’re priced in to call all in after making a pot 4bet with 65s against the range of Conservative TAG, you’ll be priced in with the same hand after the same action against both Aggressive TAG and LAG as well. These guys have wider 5bet shoving ranges, so folding against them becomes a significantly bigger mistake. The good news is that your 65s has more equity against their ranges. The bad news is that they’ll 5bet shove on your 65s more often, and your hand is still a pretty big dog against the group of hands with which they’ll be pushing.

You’re UTG with TT. You open to $7. Two folds to Nit on the button. Nit makes a pot 3bet to $24. You make a pot 4bet to $75. Nit 5bets all in.

Pot Odds:
Still $125 to win a pot of $278. You’re getting 278:125, or 2.22:1.

You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 19.057% 18.86% 00.20% 23252328 242334.00 { TT }
Hand 1: 80.943% 80.75% 00.20% 99548892 242334.00 { KK+ }

Are you priced in to call?

No. TT is generally a much better hand than 65s, but Nit’s range is so narrow that TT actually performs worse than 65s against the hands Nit could be holding. You need more than 31% equity to call and only have about 19%, so you can pitch it without making a mathematical error.

You’re UTG with TT. You open to $7. Two folds to Conservative TAG on the button. Conservative TAG makes a pot 3bet to $24. You make a pot 4bet to $75. Conservative TAG 5bets all in.

Pot Odds:
You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 36.413% 36.21% 00.20% 126502032 693396.00 { TT }
Hand 1: 63.587% 63.39% 00.20% 221421192 693396.00 { QQ+, AKs, AKo }

Are you priced in to call?

Yes, 36.4% is enough equity that you should be calling. Like in the first scenario, since you’re priced in against this guy, you’ll also have to call against the remaining two fictional players. It’s notable that even against LAG, your TT is a slight dog (about 45.5/55.5).

You’re UTG with KQo. You open to $7. Two folds to Nit on the button. Nit makes a pot 3bet to $24. You make a pot 4bet to $75. Nit 5bets all in.

Pot Odds:
You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 11.486% 11.15% 00.34% 20613120 628650.00 { KQo }
Hand 1: 88.514% 88.17% 00.34% 163058412 628650.00 { KK+ }

Are you priced in to call?

No.

You’re UTG with KQo. You open to $7. Two folds to Conservative TAG on the button. Conservative TAG makes a pot 3bet to $24. You make a pot 4bet to $75. Conservative TAG 5bets all in.

Pot Odds:
You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 20.751% 20.26% 00.49% 99910656 2423022.00 { KQo }
Hand 1: 79.249% 78.76% 00.49% 388386852 2423022.00 { QQ+, AKs, AKo }

Are you priced in to call?

No. You need a bit more than 31% equity to call, and you’re only getting about 21%.

You’re UTG with KQo. You open to $7. Two folds to Aggressive TAG on the button. Aggressive TAG makes a pot 3bet to $24. You make a pot 4bet to $75. Aggressive TAG 5bets all in.

Pot Odds:
You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 26.751% 26.33% 00.43% 194732364 3147627.00 { KQo }
Hand 1: 73.249% 72.82% 00.43% 538687710 3147627.00 { JJ+, TcTs, TdTs, ThTs, AQs+, AKo }

Are you priced in to call?

No, you’re still not getting enough equity against the range you’re facing.

You’re UTG with KQo. You open to $7. Two folds to LAG on the button. LAG makes a pot 3bet to $24. You make a pot 4bet to $75. LAG 5bets all in.

Pot Odds:
You need 31.06% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 32.027% 31.67% 00.35% 351420780 3937734.00 { KQo }
Hand 1: 67.973% 67.62% 00.35% 750276744 3937734.00 { 99+, 8c8s, 8d8s, 8h8s, AJs+, AhTh, AsTs, AKo }

Are you priced in to call?

Yes. This range is wide enough that you have to call, and when you do call, you’re a big dog.

Now, let’s look at some similar scenarios. Your hands will remain the same, as will the villains’ ranges. However, in the following hands, you make a small 4bet to 2.25 times the villain’s 3bet ($54) instead of 4betting pot. Such a 4bet risks less money while still denying villains the odds to set mine against you and to call with speculative hands. In addition, it makes calling a 5bet shove less attractive from a pot odds perspective, so you won’t be mathematically obligated to call shoves as often after making a light 4bet.

Important note: you might be thinking, “Fair enough, but a small 4bet doesn’t have the same fold equity as a 4bet to pot.” IN REALITY, THIS MAY OR MAY NOT BE THE CASE! For the purpose of this article, IT IS NOT THE CASE. If I were to assume that the villains’ 5betting ranges change when you decrease the size of your 4bet, the examples would become extremely lengthy and complicated. For the purpose of this article, assume that a small 4bet has the same fold equity as a big 4bet.

You’re UTG with 65s. You open to $7. Two folds to Nit on the button. Nit makes a pot 3bet to $24. You make a small 4bet to $54. Nit 5bets all in.

Pot Odds:
You have to call $146 to win a pot of $257. You’re getting 257:146, or 1.76:1.

100/2.76 = 36.23

Equity:

equity win tie pots won pots tied
Hand 0: 22.496% 22.29% 00.20% 18321336 168354.00 { 65s }
Hand 1: 77.504% 77.30% 00.20% 63532548 168354.00 { KK+ }

Are you priced in to call?

No. You need a little over 36% equity to call, and you only have about 22.5%. You can fold the hand without making a mistake.

You’re UTG with 65s. You open to $7. Two folds to Conservative TAG on the button. Conservative TAG makes a pot 3bet to $24. You make a small 4bet to $54. Conservative TAG 5bets all in.

Pot Odds:
Same as above, $146 to win a pot of $257. You’re getting 257:146, or 1.76:1.

You need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 31.144% 30.92% 00.22% 72010208 515570.00 { 65s }
Hand 1: 68.856% 68.63% 00.22% 159831996 515570.00 { QQ+, AKs, AKo }

Are you priced in to call?

No. You need a little over 36% equity to call, and you only have a bit more than 31%. Note that when you 4bet pot, you were mathematically forced to call off your stack after the 4bet bluff with this hand. When you make the small 4bet, though, you can fold to the shove without making a mistake.

You’re UTG with 65s. You open to $7. Two folds to Aggressive TAG on the button. Aggressive TAG makes a pot 3bet to $24. You make a small 4bet to $54. Conservative TAG 5bets all in.

Pot Odds:
Same as above, $146 to win a pot of $257. You’re getting 257:146, or 1.76:1.

You need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 30.045% 29.82% 00.22% 95998006 721187.00 { 65s }
Hand 1: 69.955% 69.73% 00.22% 224472772 721187.00 { JJ+, TcTs, TdTs, ThTs, AQs+, AKo }

Are you priced in to call?

No. You still don’t have enough equity to call. In fact, your equity has dropped slightly despite the fact that the range you’re facing has widened.

You’re UTG with 65s. You open to $7. Two folds to LAG on the button. LAG makes a pot 3bet to $24. You make a small 4bet to $54. LAG 5bets all in.

Pot Odds:
Same as above, $146 to win a pot of $257. You’re getting 257:146, or 1.76:1.

You need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 29.368% 29.06% 00.31% 133367872 1400403.00 { 65s }
Hand 1: 70.632% 70.33% 00.31% 322728794 1400403.00 { 99+, 8c8s, 8d8s, 8h8s, AJs+, AhTh, AsTs, 8h7h, 8s7s, AKo }

Are you priced in to call?

No. Even against LAG’s range, you can fold 65s to the shove after 4betting.

You’re UTG with TT. You open to $7. Two folds to Nit on the button. Nit makes a pot 3bet to $24. You make a small 4bet to $54. Nit 5bets all in.

Pot Odds:
You’re getting 1.76:1 and need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 19.057% 18.86% 00.20% 23252328 242334.00 { TT }
Hand 1: 80.943% 80.75% 00.20% 99548892 242334.00 { KK+ }

Are you priced in to call?

No. You weren’t priced in to call against this guy even after making a pot 4bet, so clearly you’re not priced in now that your pot odds have become less favorable.

You’re UTG with TT. You open to $7. Two folds to Conservative TAG on the button. Conservative TAG makes a pot 3bet to $24. You make a small 4bet to $54. Conservative TAG 5bets all in.

Pot Odds:
You’re getting1.76:1 and need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 36.413% 36.21% 00.20% 126502032 693396.00 { TT }
Hand 1: 63.587% 63.39% 00.20% 221421192 693396.00 { QQ+, AKs, AKo }

Are you priced in to call?

Yes, barely. Note that TT also became a call against this guy when you were 4betting pot, but that then, the decision wasn’t a close one. Math dictates that with TT, the four villains’ 5bets should be dealt with the same way despite the difference in 4bet size (since you’re priced in against Conservative TAG, you’ll be priced in against Aggressive TAG and LAG too). However, folding would only be a very minor mistake because the small 4bet leads to much less favorable pot odds on a call.

You’re UTG with KQo. You open to $7. Two folds to Nit on the button. Nit makes a pot 3bet to $24. You make a small 4bet to $54. Nit 5bets all in.

Pot Odds:
You’re getting 1.76:1 and need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 11.486% 11.15% 00.34% 20613120 628650.00 { KQo }
Hand 1: 88.514% 88.17% 00.34% 163058412 628650.00 { KK+ }

Are you priced in to call?

No way. You need more than three times the equity you have before calling becomes the right play.

You’re UTG with KQo. You open to $7. Two folds to Conservative TAG on the button. Conservative TAG makes a pot 3bet to $24. You make a small 4bet to $54. Conservative TAG 5bets all in.

Pot Odds:
You’re getting 1.76:1 and need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 20.751% 20.26% 00.49% 99910656 2423022.00 { KQo }
Hand 1: 79.249% 78.76% 00.49% 388386852 2423022.00 { QQ+, AKs, AKo }

Are you priced in to call?

No, and it’s still not close.

You’re UTG with KQo. You open to $7. Two folds to Aggressive TAG on the button. Aggressive TAG makes a pot 3bet to $24. You make a small 4bet to $54. Aggressive TAG 5bets all in.

Pot Odds:
You’re getting 1.76:1 and need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 26.751% 26.33% 00.43% 194732364 3147627.00 { KQo }
Hand 1: 73.249% 72.82% 00.43% 538687710 3147627.00 { JJ+, TcTs, TdTs, ThTs, AQs+, AKo }

Are you priced in to call?

Nope. 26.8 < 36.2.

You’re UTG with KQo. You open to $7. Two folds to LAG on the button. LAG makes a pot 3bet to $24. You make a small 4bet to $54. LAG 5bets all in.

Pot Odds:
You’re getting 1.76:1 and need 36.23% equity or more to call.

Equity:

equity win tie pots won pots tied
Hand 0: 32.865% 32.51% 00.36% 360688620 3972930.00 { KQo }
Hand 1: 67.135% 66.78% 00.36% 740938512 3972930.00 { 99+, 8c8s, 8d8s, 8h8s, AJs+, 8h7h, 8s7s, AKo }

Are you priced in to call?

No. Even against this guy you can fold and avoid getting all in preflop as a 67/33 dog.

Analysis

When you 4bet pot with 65s, you were able to fold to a shove from Nit, but you were priced in to call against the other three players. When you 4bet bluffed small with 65s, meanwhile, you were able to fold against all four players’ shoves without making a mathematical mistake.

When you 4bet pot with pocket tens, you were able to fold to Nit’s shove, but you had to call against the other three. The same was the case when you decreased the size of your 4bet with the tens: folding was correct against Nit, but you had to call against the other three (though against Conservative TAG -- the player with the next narrowest range -- the decision to call in this spot was extremely close).

When you 4bet pot with KQo, you were able to fold to shoves from the first three players, but you were priced in to call against LAG. However, when you made your 4bet bluff smaller with this hand, you were able to correctly fold against all four players.

When holding TT, decreasing the size of your 4bet changed nothing with regard to the correctness of calling a 5bet shove against the given ranges. However, when holding the other two hands, the smaller 4bet provided you with increased leverage; with 65s and KQo, decreasing the size of your 4bet made it possible to correctly fold to 5bet shoves in spots where you would’ve had to call if your 4bet were bigger.

The results were especially drastic with 65s, the small suited connector. With this hand, you were priced in to call three times out of four when making a 4bet to pot. When you made a small 4bet, though, you were priced in zero times out of four.

Being mathematically forced to call off most of your stack is bad when you know you are a significant dog in the pot. Being able to 4bet bluff with the knowledge that you can fold to a 5bet all in is a nice luxury. Decreasing the size of a 4bet makes this process possible, perhaps at the expense of some fold equity. How much fold equity (if any)? Tough to say.

Won’t villains adjust to my small 4bets?

Probably not, especially at 200 NL. But yes, if you’re pulling this crap often with junk, smart villains should notice eventually and begin to shove over your 4bets with a wider range. If they adjust, you adjust. 4betting small with premium hands is a good adjustment and is important for balance.

So, according to this article, does it never make sense to 4bet pot?

It never makes sense to 4bet pot according to the assumptions in this article. If I had some way of knowing that light 3betters react the same way to small 4bets as to pot 4bets, I’d eliminate the pot 4bet from my arsenal because if this were the case, 4betting pot would be inferior to both 4betting small and to 4bet shoving, in my opinion.

However, it’s impossible to prove that fold equity is the same regardless of 4betting size.

With the information I have, I’ll probably begin to replace pot 4bets with a combination of small 4bets and 4bet shoves. Note that it’s important to make both small 4bets and 4bet pushes with both hands that are marginal and hands that are powerful."