Introduction/Disclaimer

When I first thought about doing a COTW, I had a number of ideas but at the time I was experimenting with pre flop bet sizing, and hence thought it would be a great idea… … … but then, when I actually started the research I realized that preflop bet sizing is probably one of the best analyzed topics on 2p2. So, me being me (not who you might call a pro), I had quite a task in front of me. Hence I ask you not to judge too strictly.

“4BB+1BB Per Limper” Mantra

I think that every modern player must have heard or knows of the 4BB+1BB mantra, and indeed there is a lot of topics about it, for instance a very good discussion back in the archives:

http://archives1.twoplustwo.com/showflat.php?Cat=0&Board=ssplnlpoker&Number=616675 7&fpart=1&PHPSESSID=

However, in my experience a lot of people still argue that 3BB+1 or any other lower (Than 4BB+1BB) alternative is feasible as a standard bet. This, is where I disagree and one of the main points of this COTW is to make it very clear why 4BB+1BB per limper is “correct” (I know, I know… it still depends, and we shall address this later).

To get to this point, let us first look at the following example with the computer hand or what one may consider the average NLHE hand:


Poker Stars $0.10/$0.25 No Limit Hold'em - 9 players
The Official 2+2 Hand Converter Powered By DeucesCracked.com

UTG+2: $10.00
MP1: $9.65
MP2: $25.70
CO: $83.70
BTN: $4.55
SB: $25.00
Hero (BB): $52.35
UTG: $25.00
UTG+1: $26.00

UTG+2 posts a big blind ($0.25)

Pre Flop: ($0.60) Hero is BB with Q 7
2 folds, UTG+2 checks, MP1 raises to $0.75, 4 folds Hero ???


Now then, how do we go about a decision to call? First of all we need to know what are our odds to hit on the flop and then to evaluate our pot odds and estimate how we are doing against the villains range. For example our odds to hit are:

Chance of getting 4-of-a-kind on the flop: 0.01%
Chance of getting a Full House on the flop: 0.09%
Chance of getting 3-of-a-kind on the flop: 1.57%
Chance of getting 2 Pair on the flop: 4.04%
Chance of getting 1 Pair on the flop: 26.94%
Chance of getting a Four Flush on the flop: 2.24%

(Courtesy of http://www.texasholdem-poker.com/calculator)

All in all, we have a chance of around 34.89% (Odds of 1.86 to 1) of the time. Now then, the villain only raised to 3BB and there was a limper in the hand giving us pot odds of 27.027% (Odds of 2.7 to 1). So we are clearly getting the correct odds to call. Moreover the equity without any further betting against any other non-pair hand is 31.89% (Odds of 2.14 to 1).


equity win tie pots won pots tied
Hand 0: 68.109% 67.90% 00.21% 223217964 700158.00 { AKs, AKo }
Hand 1: 31.891% 31.68% 00.21% 104144088 700158.00 { Q7o }

Thus, we can see that if we make our raise sizing too small then we are allowing villains to call with too wide a range, profitably and without making a mistake.

Let us generalize this scenario. With any non-paired hand the odds of hitting at least one pair on the flop is 32.4% or 2.1 to 1. (Source) If we add a little bit of % for flops where we hit something better than a pair, or something playable… then it is clear that one needs to offer pot odds that are worse for the big blind (as he has already put the BB into the pot and thus has the best pot odds amongst the players) than 2.0 to 1 namely, I am going to suggest this number to be at least 1.9 to 1. There are some details here which can argue for both offering slightly worse or slightly better odds, I think they cancel out but feel free to discuss and argue this point!

By doing a little bit of math with a help of my friend (Who for now shall remain known as PoorLeno), who doesn’t play poker but is rather good at math, we came up with the following formula:



This gave us the following result (In big blinds):


As you can see the 4BB+1 mantra holds fairly well. However with a number of limpers increasing, one may consider increasing the raise amount slightly. For example with three limpers, I might suggest that it may be advantageous to bet 7.5 big blinds. (Also note that by raising 3.5BB+1BB will always offer 2 to 1 odds)

The effect of a different betting structure.

Not all games run under the assumption that one big blind is equal to two small blinds. In some games such as 25NL, 1 BB = 2.5 SB. Let us take a look at how that will change the situation:


The difference, as you can see is pretty miniscule and could generally be disregarded. However what I want you to take from this is the following: with increasing ratio of the structure in regards to SB to BB ratio, you will need to bet LESS pre-flop.

RAKE?

Now then, while we were at it, we thought to see how does rake affect this scenario. Given that rake formulas are different for different sites and even between stakes, we decided to greatly generalize and just did a calculation for 4% rake. This analysis assumes that there will be no further betting of course.

The result can be displayed in a graph:

X-Axis -> Required bet amount. (In small Blinds) Y-Axis -> Pot Odds, with the yellow line being the needed 1.9 to 1.

The red line is the needed bet amount without rake and the blue line is with rake. As you can see the with rake taken into effect, the needed bet amount is slightly less. However the difference is very small indeed, but would be greater with higher rake %.

ANTE Games

With the introduction of ANTE games at PS, I feel like this section might be of most use to players. Lets jump right into it. We have analyzed two scenarios, first where ante is 25% of the BB and then where ante is 20% of the BB, as this is typically the range for ante, you can guesstimate for higher and lower ante values from the analysis. The results are:

Ante at 25%



Ante at 20%



From these graphs you can come up with your own betting system, but personally I would suggest 6.5BB+1BB per limper for 25% antes and 5BB+1BB per limper for 20% antes. One thing to note here would be that with increasing number of limpers, one may need to add more than 1BB per limper to correctly adjust.

Conclusion:

All of that being said, don’t forget to think about the table, specific players and all the other variables when choosing your bet sizing. This post is intended just as a guideline to help you start thinking about how the pre flop betting size can be analyzed under different game structures. This can be taken further and extended to pre-flop betting depending on position, adjusting to calling stations and nits. For instance one may choose to raise 5bb+1bb per limper on loose tables running 25VPIP+ while make it 3bb+1bb per limper on 8VPIP tables. Moreover there are strong implications for further streets. One other thing I’d like you to think about and maybe discuss is how bet sizing affects stealing situations, and how you can use the combined VPIP of the SB and BB in order to capitalize on stealing opportunities. One other thing you might want to think about is the fact that the small mistakes that you can allow yourself to make pre-flop can be too small to count at later streets with competent opponents. This COTW is mostly intended as a starting point with a strong discussion to follow, and hopefully I will be able to make a few replies with further analysis.

Aknowledgements:

Along with my 1000th Post, this is one of the first serious strat posts that I have made, its been a nice journey from a complete beginner a year or so again, to becoming a reg. Although I have only played around 150k hands meanwhile, I feel like I’ve gained a lot of understanding in poker and thanks to poker (Which I talk a lot about in my 1000th post). For this, I would like to thank everybody who has been helpful in my poker education. Special thanks to PokerRon for helping me greatly in my journey, to JLBorloo for taking a look at my draft and giving me suggestions. Also, I would like to express my gratitude to my best friend, PoorLeno for helping me with the math behind this post. PoorLeno asked me to forward this to you: “If you would like us to tackle any interesting poker and math related research, please do let us know and if we have the time we shall try to deliver.” (PM me if you have good ideas) Thanks to all previous COTW writers as well.

On a side note: Yesterday when playing one of the SCOOP tournaments, I forgot to follow my own advice and while being a short stack, raised to 2.5BB with 66, the BB called. I pushed on a A27 flop and he called with T7o 

Resources that have covered Pre-Flop aspects previously:

Basic Theory Discussion -

http://archives1.twoplustwo.com/show...fpart=all&vc=1

Playing PP’s –

http://archives1.twoplustwo.com/show...rt=&PHPSESSID=

http://archives1.twoplustwo.com/show...fpart=all&vc=1

Suited Connectors –

http://forumserver.twoplustwo.com/78...ectors-705585/

http://archives1.twoplustwo.com/show...1&fpart=2&vc=1

Positional Awareness –

http://forumserver.twoplustwo.com/78...p Re-Raising –

http://forumserver.twoplustwo.com/78...sizing-574867/

Stealing –

http://archives1.twoplustwo.com/show...Number=5348855

ISO –

http://archives1.twoplustwo.com/show...Number=6073737

Playing from the blinds -

http://archives1.twoplustwo.com/show...=0#Post7459990

These are just some of the resources, there is plenty more out there, also if any link does not work for some reason, do let me know.

Nice!

almost first!!!!!!

Someone writing a lot of good and long stuff recently.

Calling a raise with Q7o in your example is going to lose money whether it is 3bb+1 or 4bb+1, there is more than your expressed odds preflop to consider. Naturally it loses more in the latter case but if the first is more likely to be called then that should be what you raise.

fancy maths. Nice post.

Not to be an *** but doesn't that mean that it would be better to raise 5x+1bb per limper? What about 6x? Why not more?

Yeah there are other considerations that we need to address like how big the resulting pot will be relative to stacks, and what is the impact of our open raise size when we get 3 bet?

Nice post, ty.

Stop the madness! There are enough preflop donk shovers already.

nice stuff dude.

Yeah, I don't have much time for an analysis of pre-flop play that uses pot odds because the sums of money involved pale in comparison with those involved in the implied odds. Q7o is a terrible hand to call oop with against a pre-flop raiser.

I really feel like that Q7o AK thing is flawed. The obvious point is if you are facing a taggy player open from MP to 3bb and you are sitting in the bb do you call with Q7o. Of coarse not.

There are a few reasons for this.

First you are not playing against just AK you you are playing against a range and you are doing worse against his range than against Ak because of all of the pairs and dominations.

Also flopping a pair or better doesn't mean you win. It just means you are oop with a pair and a bad kicker with no initiative. Doesn't sound that profitable.

There are situations where 4+1bb opens are best. They are generally loose passive games where there is a lot of c/f on the flop. I play 6max so I am not sure how much this comes into play in FR but you really only need one semi competent light 3better to make your life hell if you are opening that big.

You also are not taking into account position or skill advantage. when I am playing I usually assume at least some (if not a great) skill advantage at my limit until proven other wise and by raising smaller ip I get to play with a bigger SPR which benefits me as the better player. I also get a bigger SPR ip which means more money could potentially go in with me making more informed decisions

Baluga Whale does a good job explaining this in the new Coaching Kristy videos when he talks about the fact that if you have about 50/50 equity say AQo v 77 the player ip will win in the long run by a pretty big margin. He goes on to say that you can probably get away with even as much as a 35-40/60-65 disadvantage as long as you are ip.

Not trying to flame the OP or to say I know better. I am just trying to put forth at least some counter argument to the premise of the post

Thanks for getting us started OP

nice post thanks for the thoughts

math is sound, but it assumes that post flop play is non-conditional.

Great post, but I disagree with the 4xbb+1bb mantra due to several reasons.

I was trying to think of a way to respond to this COTW in a way that wasnt harsh, but explained what I was feeling. This response...is what I would have said

Props to OP for putting this together. Clearly a decent amount of thought + work put into this COTW. However, I personally am not really taking anything useful from this. The general consensus is 4bb+1 per limper in EP, however if you are doing this in MP on with aggressive 3bettors in the BTN, CO and/or blinds ur asking for it. 3bb from MP and 2bb in LP work really well so long as your range widens a lot as you move towards the button. People can call with their Q7o from the blinds vs my 2bb open all they want, but in the long run I'm going to rape them because I'm raising Q8o+ from the hijack on given ideal table dynamics and will have position on them. I'll also have the initiative and will make them fold Q high on the flop which will often be the best hand or have decent equity when we both miss.

Sometimes it may be correct to raise larger, say 7bb+ but it will almost always be the right play to do so due to the table dynamic, or for setting up smaller SPRs on the flop when we are semi deep, and not the preflop pot odds we are offering to limpers or the blinds.

I also find myself isolating short stacked limpers (20-30bb) to only 3bb. I usually do so knowing that they will call close to 100% but will also call a 4 or 5bb iso raise. Some hands that I iso with will favor a slightly larger SPR.

The logic behind defending steals with such weak hands is quite flawed. You seem to forget all about reverse implied odds which is a huge factor, which is why I prefer to have a hand that plays particularly well vs the stealers range in order to flat. I'd much rather 3bet or muck. Even in LHE where you can only ever minraise, I'm pretty sure that Q7o is a fold, and if its a call, it should be damn near the bottom of your range.

WRT the Q7o hand, this is very, very wrong:

Only a fool (or a fish) would flat an open with Q7o in the blinds. Many people might flat this citing 'pot odds', but they would be wrong to do so. If this hand were played with 0.75 stacks you might be correct to do so, because the pot odds you're being offered are the equivalent to your chances to win the hand. But with money left to play postflop, pot odds are not the whole story. Indeed, I might go so far as to say that considering all the considerations for playing Q7o vs an MP open in the blinds, pot odds are close to the least important. Other considerations are (in no particular order):

1) Your absolute position. You will be first to act on every street. This will make it difficult for you to control the size of the pot, leaving you in a position where you will get less money than you should with your good hands, and lose too much with your bad & 2nd-best hands. It also leaves you with a compromised ability to show down a marginal hand cheaply, not get bluffed off the best hand, and not yourself end up bluffing with the best hand or betting in to a better hand.

2) Your relative position. Which, in the case of a HU pot is the same as your absolute position.

3) Your implied odds. If you can't get paid enough when you do hit a winner, you will not make up for those times when you hit a loser. Being out of position limits your implied odds. So does the effective stack size.

4) Your absolute & relative hand strength. Sure, you can hit TP. But if you hit TP with Q7o on a board like Q82r and all the money gets in when you're both 100BB+ deep, you are rarely going to win. On the other hand, when you bet your TP on that board, you are very limited in your ability to get a 2nd-best hand to pay you off.

Although my original plan was to withstand the discussion for a while in order to see how it develops However I think I should make a point. (I shall make another one later, but I think the discussion is focusing too heavily on the example hand at the moment)

What I ask of you is to please read the post carefully, at no point does COTW state that we should call in the example hand. The example hand was create to demonstrate how some people might think about pot odds, and why it is then important not to give them the needed pot odds.

However, I really like that controversy and discussion is starting to take place and I encourage it, in fact the more wrong I am, the closer to the truth we can get.

If anyone wants to call oop with marginal holdings citing pot odds or "lucky hand" what ever they like I welcome that. The point I am (and I think the others) are trying to make is that denying the blinds the express odds to call is not enough to justify 4+1bb being the best open size.

There are defiantly times when it is (probably almost every table 10NL and below) but as you get into the higher micros you have to consider other things. Like what kind of players are in the blinds

OK, fine. I concede the point. But you did say,

...and this is factually incorrect because villain would not be correct to call. There are more variables to the correctness of villain's action than simple pot odds, as I state above. I suppose you could think of this as a math problem -- personally I don't. I think it's valid to approach this as a math problem, but if you do the equation is far, far more complex than simply (villan's thoughtstream here):

It is closer to:

So in short you are advising against raising 3x in this case because villain would be correct to call. I say this is the opposite of correct. In fact I think raising to 3x or even less can be better than raising more because it would encourage Villain to call, and by calling he is making a mistake regardless of pot odds.

you said that if a mp raised to 3x and villain was in the bb it would not be a mistake to call. If it is not a mistake it is correct. So you kinda implicitly said you should call.

poker could theoretically be broken down into math and I respect your attempt to do that here but there are express as well as implicit odds. Your post only seems to deal with express odds so is flawed from the start

I think the original 2+2 mantra of 4x+1 per limper was developed for when the games were alot more passive then they are now, and no longer applies beyond ~Nl10.

At 25nl+ and especially at 50nl+ how many winning regs acctually still use this size?

Even the bad regs in todays game understand three betting a the option of three betting light to a certain extent.

If we open to 4x+1...not only do we lose more when we get 3-balled, we lose our abilty to four bet light to counter their strategy as we can no longer 4 bet fold.

The reason:
Hero raises to 4x. villian 3 bets to 12-15. Hero can four bet light, but its effectively a shove b/c he will have to commit more then 1/3 of his stack, so he will have to be right way more often, or his four bet will basically be a min raise.

Hero raises to 3x villian 3 bet 9-11, Hero 4 bet/folds to 23-25bbs, ez game.

Postflop:
In games where we're not simply bet bet shoving tptk for value, some room to play post flop gives the skilled player more of an advantage. Since the pot grows exponentially, 4x+1 leaves very little room to make any play OTT or OTR, that will get alot of fold equity.

Lastly, in a game where everyone else is opening to 3x+1 do you want to be the only one opening to 4+1. Who is the fish more likely to call with trash? (Its like split said in one of his articles, betsizing in poker is like marketing)

You or Split owe the forum an explanation. To tell the OP, "You're wrong, but I won't explain why," is unfair to him. At least a list of, "You haven't considered these factors."

I've posted and Split has posted a lot about this in the past, its just sprinkled in.


First off (not the most important, just the first one I am addressing), its success % factors in steal attempts, if we calculate how much more successful a steal attempt needs to be just by 1 bb, and a cbet..then we start to see.


The next has to do with the game itself. We win poker with
1: Card Strength
2: Position
3: Skill edge.

As we play deeper (post flop), 2 and 3 raises in importance, as we play shallower 1 becomes the most important. For recent examples of this, look at a good explanation of short-stack strategy and SPR concepts.

So as we play more hands against weaker opponent IP, we want to play deep, this allows us to abuse position against them. I think a lot of us do not really understand positional advantage given the limitations that a 4bb+1bb in a 100bb game does.

Since we have a skill and positional advantage, we will want to play more hands, but one thing to note even though we WANT to play more hands the combinations of AK+, TT+ stays the same, so we add hands. We had SPECULATIVE hands, which to be very profitable we want to play deep, with large IO.

Finally, as we really explore playing deep post flop with position and skill advantage, we will awaken to this awesome thing tagged backside aggression, and how we can utilize it to make pretty graphs with nice red and green lines.

Q regarding the bolded text above: What exactly does that mean? I've never heard that phrase before and just want to make sure I'm not missing anything. I otherwise understand and agree with your whole post.