# Mathematically speaking, is open-limping that bad?

I do not limp, well in some rare situations I do but that's not the question. I would like to know why limping is bad mathematicaly speaking. There are many responses to the "limping is bad" question on the internet but most of the case, it's for metagame dependent reasons or to exploit weakness of players. I agree with it but I want to talk about GTO situations.

I tried to do the math but the polynomial side of the NLHold'em state-space blocked me in some way.

I have some intuition about it. Let's suppose I have AA on button, I have to raise because I'm sure my hand crush the opponent range and I want him to put more money in the pot. But what if I can get more money from him by limping and let him hit one paire of something he would fold because He would not have the odds to call ?

Please assume all players are trying to play optimally and we are in a GTO context.

EDIT : the thing is, as mentioned by Paparazzi, we are not in the case of "how many time should I bluff on the river to be break even". There are too many hypotheses to do the exact math on the limp case. For example, what amout of chips should we use to raise ? Should we consider in our equation if we loose money by autofolding any2 because we pay the blinds at some point ?

• From a non-mathematical viewpoint, if you open-limp some hands but you open-raise others, you allow yourself to become more readable. If you're going to open limp ever, you need to do it in a manner that balances your open action without regard for what the cards are. – mah Mar 29 '16 at 1:17
• Yes that's why I said I agree with most of the metagame reasons people often says ! – nicolas carrara Mar 29 '16 at 14:05

You never have aces on the button, but you always have a range on the button. If you limp with aces only, how can your opponent ever make a mistake? He will only put more money in the pot when he has you crushed.

We make money on the mistakes our opponents make.

You should try to construct a range to play OTB that gives you the highest EV. A percentage (bear in mind 0 is a percentage) of our range we will fold, a percentage we will limp and another percentage we will raise.

Let's say you assume you can raise the top 50% of hands and that will yield you 40BB/100 OTB. Awesome. Now you try to mix some limps on it and see how your overall EV goes. If you add AA there, your opponent never makes a mistake (as stated above) and your raising range also makes less money, since you've removed the strongest hand of it. And on it goes.

The problem here is not math per se, it is range construction.

I have analyzed limping versus raising over enormous samples (in the millions of hands) in poker tracking programs like HM and PT and found that from a profit standpoint there is no comparison. Raising is simply far more profitable than limping.

In the most general way, let's say that you assume you are an above average player (if you didn't, the most profitable thing to do is to not play poker!), and that whenever you voluntarily enter a hand you do so because you expect to make a profit from doing so (or else you would have folded!). Once you make a determination that playing a hand is profitable, then you should want the final pot to be as large as possible and a pre-flop raise amplifies the size of the pot on each successive street compared to a limp.

For example, if you limp in and everybody else folds, and then there is a pot sized bet and call on every street, the final pot will be 2.5*3*3*3 = 67.5 big blinds [the pot triples on each street because we have (starting pot) + (pot-size bet) + (call)]. If instead you had raised even the minimum to 2 big blinds, we end up with 4.5*3*3*3 = 121.5 big blinds. A larger raise amplifies that even more of course, but the point is that any addition to the pot early on causes the pot to expand in an exponential-ish way. With the assumption that you're playing a hand because you expect to profit for it, this is a good thing.

Limping can lead to big pots too because more players are likely to then enter to pot, but the tradeoff is that your chances of actually winning the hand go down. Another benefit to raising is that it help define your opponent's ranges which can help your decision-making on future streets. Also, don't forget that by raising, you can win the hand right there with no risk! Lastly, many if not most flops simply miss most pre-flop hands; being the pre-flop raiser gives you the impetus (or "benefit of the doubt") as to having the better hand, so that opponents may check to you (giving you more options than having to face a bet) or they'll give more credence to your bluffs (you can chase away a low pocket pair or even a hand that dominates yours when you have two face cards). You can win more pots and they'll also be larger than if you had limped preflop.

So what's the case for a good time to limp then? Since, as I mentioned above, you're only playing hands if you deem them to be profitable, the time to limp would be specifically when by limping you increase the amount you expect to profit on average. That means either increasing your chance of winning or increasing the size of the pot or some combination of that. Limping will never lead to fewer players, so you can rule out that your chances of actually winning the hand will ever go up, which leads to the option of increasing the size of the pot when you win. You could come up with some random scenarios where that may be possible, usually given specific table conditions. I mean, maybe it's possible that you're at a table full of fish who are very loose preflop and who regularly build up very large pots after the flop with weak holdings. Then you could limp in for cheap flops and still be assured at taking a large pot when you hit the board strongly (limping > raising because it's a reduced initial cost to speculate, but the same payoff). But it's rare to find such conditions that warrant limping.

• I realized that with this answer, the question could arise: "Then why ever make a call before the flop--why not always raise?". One factor is that there's always some finite amount of chips available for the hand, so "amplifying" the size of the pot is only necessary to a point. But the main thing is that when the positive value of your hand relies on implied odds or being able to play post-flop in position, there's a point where paying too much pre-flop becomes negative. Additionally, there's the ever-looming threat that you'll be re-raised for even more. – Dr.DrfbagIII Mar 25 '16 at 19:03
• I am not see the GTO context here. – paparazzo Mar 25 '16 at 19:47
• I guess the overall point of view here is" there is a tradeoff : if I raise,the pot can be bigger but i'll scare players. If I limp, many player'll be in the pot, so it's a big pot, but my chance of winning are reduced". I'm ok with it and I have the intuition (and the experience) that it is better to raise but this is supposition, I need exact math ! – nicolas carrara Mar 29 '16 at 14:05
• @nicolascarrara I'd say that no matter how you simplify or complicate a theoretical preflop setup, limping will lead to an additional option for the player in the big blind to check and essentially get "free" equity--an option that's never available if you instead raised. The only way you could show limping to be better is by it affecting post-flop play in such a way that it counteracts the benefit of raising preflop. – Dr.DrfbagIII Mar 29 '16 at 15:28
• If it helps at all, here is a link to some stuff about Perseus, the computer program that "solved" limit heads-up holdem: preposterousuniverse.com/blog/2015/01/09/… It's solely game theory based and never open limps from the small blind. – Dr.DrfbagIII Mar 29 '16 at 15:29

Extract value by limping a monster on the button preflop is more EV than a true GTO. River against a single opponent is a better place to start with GTO. But limp on the button GTO is the stated question and this is the GTO math.
GTO vs Exploitative Poker Strategy | Ask SplitSuit
GTO - Game Theoretical Optimum

On GTO you want your opponent to have to have zero EV to fold versus call. BB does not fold to a limp preflop - ever.

To make the math easy let's assume BB is 2 and SB is 1. Let's assume the SB can assume the BB will just check. SB assume he will win versus BB 1/2. Will win to your bluff 100% and will lose 100% if you are not bluffing.

SB is on a mid hand and needs to decide if you are bluffing (do you have nothing or a monster).
br = you bluff rate
EV fold = 0
EV call = -1 + (br)1/2(6)
even if it is always a bluff (br = 1) then still need to get past BB
set EV to call = 0 = -1 + 3br
1 = 3br
br = 1/3
you should bluff 1/3 the time from a GTO perspective
so lets say 3 hands and you bluff 1
sb fold all 3 then EV = 0
sb call all 3 then = -1 -1 -1 + 3 = 0
your br has given sb zero incentive to call versus fold

my EV
sb fold all 3 then EV = 9
sb call all 3 then = -1 -1 -1 + +6 +6 = 9
Let say I just bet the nuts twice and they fold cause they know I only bet the nuts
EV = 6

lets say you push it and bluff 1/2 the time for two hands
sb fold all 2 then EV = 0
sb call all 2 then EV = -1 -1 +3 = 1
if you bluff too much then sb should look you up

lets say you bluff too little 1/4 for 4 hands
sb fold all 4 then EV = 0
sb call all 4 then EV = -1 -1 -1 -1 +3 = -1
if you bluff too little the sb should always fold

let's say you bet 3 bb - that changes the GTO
EV call = -2 + (br)1/2(9)
2 = 9/2 br
br = 4/9
yes you you should actually bluff more more with a bigger sized bluff

Yes in real life they are going to call with good hands and fold with weak hands. When they have a medium hand that will beat a bluff you need them to have to make the tuff decision (if they are capable of calling with a bluff catcher).

GTO is more about getting action when you have a hand. Getting action on a small pot just is not that appealing. You are better off getting optimal action on a bigger pot. If you want to play deceptively then go with a standard button raise of like 3 bb and vary your buff frequency. Pros will reverse bluff monsters but they are also playing pros.

• EV call = -1 + (br)1/2(6) is OK if BBvsSBvsBU <=> 50% vs 50% vs 0% if BU bluff in your hypotheses. Am I correct ? – nicolas carrara Mar 29 '16 at 14:00
• @nicolascarrara This is the math to the stated question. I had to make some assumptions as the question failed to make necessary assumptions. As I stated this is not a good "GTO context" question. – paparazzo Mar 29 '16 at 15:02