April 3, 2010 – 10:41 am | 7 Comments

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## Pre-flop Raise, then C-bet Strategy

Submitted by on March 29, 2010 – 12:28 am19 Comments

I was playing a live game on Friday night, and someone commented that I rely too much on the “preflop raise, then c-bet flop” strategy. It’s a strategy that I picked up from Daniel Negreanu’s books and videos (yes, I’m a huge fan-boy of Negreanu), and yes, it’s probably true that I use it a bit too often. But well, that got me thinking, how often does the strategy have to work?

A lot depends on the raise sizes. I currently play the microstakes SNGs online, where my standard raise size is between 2.5 to 3 big blinds (depending on how tight the table is), so let’s go with 3 big blinds.

For ease of calculation, let’s assume I’m in late position, it folds around to me, I raise to 3bb, and it folds around to the big blind, who calls. So that means there’s now 6.5 big blinds in the pot, when the flop comes.

The big blind is often going to check to me (the standard “check to the raiser”), and I’ll bet about half the pot here, so another 3.5 big blinds. That means I’ve placed a total of 6.5 big blinds in, and the pot is now 10 big blinds.

For a start, let’s assume that I can only win by my opponent folding. That means, if the probability of my opponent folding is x, then, for it to be a good play,

x * 10 – 6.5 > 0
x * 10 > 6.5
x > 6.5 / 10
x > 0.65 = 65%

Therefore, I need my opponent to fold at least 65% of the time on the flop, for this to be profitable. This is about equivalent to him folding every single time he does not pair the flop.

However, this also assumes that we never hit the flop ourselves. Let’s say I have a hand like AJ suited. I then have more than 13.75% chance of hitting TPTK or better, as discussed here. If we use that 13.75% (which, as one of the commentors pointed out, is in actual fact, a low estimate; it discounts the flops when you pair your Ace, and also discounts the times when you flop mid pair, or two overs, or draws) as the times we will win the hand, we then have the following.

x * 3.5 + (1-x) * (0.1375 * 13.5 – 6.5) > 0
x * 3.5 + 0.1375 * 13.5 – 6.5 – 0.1375x * 13.5 + 6.5x > 0
3.5x + 1.856 – 6.5 – 1.856x + 6.5x > 0
(3.5 – 1.856 + 6.5) x > 6.5 – 1.856
8.144x > 4.644
x > 4.644 / 8.144 = 0.57 = 57%

Meaning that your opponent would have to fold at least 57% of the time.

Let’s look at the co-relation of how often your opponent will have to fold (which we’ll represent with x), with how often you have to win if he calls (y). To break even, therefore:

x * 3.5 + (1-x) * (y * 13.5 – 6.5) = 0
3.5 x + 13.5y – 6.5 – xy + 6.5x = 0
10x + 13.5y – xy = 6.5
y(13.5 – x) = 6.5 – 10x
y = (6.5 – 10x) / (13.5 – x)

For a few values of x, we have the following table:

 How often Opp Folds (x) How often you need to win when called (y) 0% 48.15% 10% 41.04% 20% 33.83% 30% 26.52% 40% 19.08% 50% 11.54% 60% 3.88%

So, what does this show. Even if your opponent does not fold at all, you still need to win less than half the time to be profitable.

More realistically, though, you’ll win the pot about 10-25% of the time when called on the flop. I won’t go into the details of that here, there are too many possibilities to go through, but if you consider 13.45% the chance you flop TPTK with a J-high flop, or 2-pair or better, and add in the 9% of the time or so that you flop an A and no J, Q or K; as well as the semi-bluffs you may hit with if you flop and bet on a draw, I think 10-25% is not too much of a stretch.

Given that, that means you need your opponent to fold between about 30% to 50% of the flops. More specifically, 32% to 52% of the flops. If you average that to 42%, that means he needs to be “bluffing”/semi-bluffing the flop about 1/3 of the times he misses.

Whether this is profitable or not really depends on the player. But from the (admittedly limited players) I’ve encountered online, I think it’s not unrealistic to expect them to fold to about 35-40% of your c-bets, which would mean you need to win the hand about 23-27% of the time when they call. If you’re not playing too wide a range, that’s highly possible, I think (that’s about the chance of you hitting one of your cards on the flop).

Of course, any strategy that’s too obvious can be exploited, and any time you’re unable to adjust to your opponents, you’re losing out. But in general, I think it’s not a bad default strategy to fall back on, especially against unknowns. I think the required fold equities are highly possible, and with Poker Tracker, you can keep track of how often a player folds on the flop, and use that to make a decision as to whether this is a good strategy. I think, based on the above calculations, if he’s folding to at least 35-40% of flop c-bets, it’s a profitable strategy to use.

What do you think? How often do you c-bet your flops, and how much?

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• Titan says:

Nicely done. I haven't really thought about it in the mathematical sense, though I think I cbet about 80%. I mostly decide whether to cbet depending on my equity against an opponent's range and his fold equity, but most importantly, the board texture. I vary my bet sizing based on the board texture to keep my play unpredictable, and as long as you don't cbet 1/2 pot with top set on a superwet board multiway, I think bet sizing won't be that big of an issue.

• Derrick Kwa says:

Yeah, definitely. I think the board texture is slightly related to the other two factors, though? If the board is more “wet”, the opponent may be more likely to fold, for example; and your hand may have less equity vs your opponent's range. Havne't yet found a way to quantify that, but yeah, you're right.

• AW says:

When reading this post, I had a thought process similar to Titan. I think that board texture is a very important variable, and that it would be near impossible to quantify.

Not only do we have to figure out our fold equity against opponent's range, but we need to consider his perception of our range. You mention that we are raising in late position. A fish won't take any special notice, but a competent BB would recognize that we are raising in late position and possibly trying to steal with a much wider range. Therefore, we can't make any generalizations about his reactions on a wet or dry flop. He might think your wide range has matched up well with a coordinated flop and give it up. Or he could think you have raised with air and he might try to take the pot away from you with nothing. This would become even much more complicated if we had to factor in times we raise and c-bet in middle and early position.

All in all, very interesting post, I really enjoy your mathematical approach to these situations. It is far from my area of expertise! But you always definitely need to keep in mind how your opponents perceive you as a player. You mention your live opponents remarked on your strategy of c-betting almost all the time. This would definitely make me ease up on the missed c-bets as opposed to a situation where I've shown down some top pairs and other made hands.

• AW says:

I've been thinking about it a little more. You clearly understand well that we must avoid a strategy that's too predictable and must react to your opponents.

I think another important thing here is that opponents who do call the flop fall into very different categories. A calling station might be seeing showdown with any pair, while a more reasonable player might call with a draw but let it go to on the turn if facing more aggression. This is past the scope of the “flop c-bet” math, but maybe the next step to consider…

“How often you need to win when called” factors in villain's perception of our range on a particular board, our action on the turn and river, their player type, and much more.

Your equation is definitely a very good starting point vs. unknowns, but I'd be curious to hear more about how you'd proceed if called.

• Derrick Kwa says:

Definitely, you need to react to your opponents. You mentioned in your earlier post that you “need to keep in mind how your opponents perceive you as a player”. Yes, that's true, but I think I'd like to throw in a subtle point there. You need to keep in mind how your opponents *react* to you as a player. Your opponents seeing you in a certain way has no impact unless that affects how they play against you. You only need to adjust your strategy when your opponents are starting to play differently.

And that's where the variable of “how often your opponent folds” comes in. That's not constant, it's opponent and read specific. For example, the player who notices that I c-bet a lot, he isn't going to fold nearly as often. As such, my hand needs to win more often (ie my hand needs to be better) if I'm going to c-bet. So I feel like that's where the “reads” come in, in a sense. The variable of “how often your opponent folds” definitely factors in how your opponent plays. “How often you win when called”, yeah, factors in a lot of other variables as well, and in a sense, it's always an estimate – but for now I'll treat that as a general idea of hands trength.

As for how I'd proceed if called. Well, then I re-evaluate on the turn. Depending on the turn card, and the action on the turn, and the opponent. It becomes much much more complex, definitely, and the math isn't quite able to cover all of it. Ok, actually, let me rephrase. I haven't come up with a system to mathematically cover all the complexities. To quote Numb3rs (my favorite tv show), “if there's any limitation, it's got to be in the mathematician, not the math.” ;-). Definitely working on it, though. Any thoughts/suggestions from your end?

• Derrick Kwa says:

I generally tend to go the other way, actually. I c-bet about the same amount almost all the time (when I bet), so that it's harder to place me on a range. I think both approaches definitely have their advantages, though, and I feel that comes down more to style?

And yeah, definitely board texture plays a part. But if you think about it, board texture is an underlying factor that leads to the first two variables (equity against an opponent's range, and his fold equity). If you're bluffing at a wet board, you're assuming your opponent is more likely to fold because of the board texture, etc, doesn't it?

I dunno, I could be wrong, but I feel like equity against opponent's range, and opponent's fold equity, are the two most important *end* variables. But how you arrive at the values of those variables is where the subtleties comes in – what range you put them on, what range they put you on (which affects how often they'll fold), etc etc. And I feel like board texture is one of those subtleties. In a way it's like a chain. Board texture (and reads, etc) impact the two variables I mentioned, which impact whether you should c-bet. I've only looked at the surface of the chain, though.

So yeah, thanks for bringing up board texture. =). It's a really important point, definitely. I'm actually trying to figure out a way to quantify that, so that I can start looking further down the chain. Any ideas?

• AaronBrown says:

I think this analysis is too narrow. The standard advice is the preflop aggressor should c-bet if she hits the flop, or if it's likely no one hit the flop. This is not a deception bet, the other players can see clearly which one you're doing.

Therefore, you have to consider three types of flops, the ones you hit, the ones you miss but have danger cards, and the ones it's likely no-one hit.

One major benefit to c-betting that you left out is it protects your strong hands. When you do hit, you can raise and get called. Of course, the other benefit is you get a lot of folds when everyone misses.

On the other side, you left out the major risk of a c-bet, when it is reraised. The other player doesn't have to call and then check to showdown.

I don't think it's wrong to always c-bet, because you're not trying to deceive. Whether you want to mix things up or not, either by varying the amount or by sometimes not c-betting, depends on the circumstances and your style. But you don't have to do it.

• derrickkwa says:

Yeah, definitely, there are many more variables, especially future betting. I did admit that in an earlier comment. Haven’t quite found a way to analyze that as yet, though.

And of course, you don’t have to do it. I’m just trying to figure out what conditions make it favorable to do so.

Oh,and again, the two variables I’ve used in my analysis are quite ‘high-level’, in terms of hierarchy in the thought process. Whether you hit the flop, how likely it is that your opponents hit the flop – those are all things that lead to the two variables I used.

• Andrew Walsh says:

I think that future betting is going to be next to impossible to quantify. It is very opponent dependent, includes our table image and the exact turn and river cards.

Now, I do think it is worthwhile to try to calculate our chances for the opponent to fold right away to our c-bet on the flop, not to a double or triple barrel.

But then “How often you need to win when called” is problematic because it gets right back into future action.

I’m not sure what to do with this, but I feel that the second variable is always going to be a pretty rough estimate.

• AaronBrown says:

I wasn’t clear. I don’t mean you don’t have to c-bet, I meant you don’t have to vary c-betting. That is, you can always do it in the specified circumstances, and always at the same amount (half pot is a good general amount). It’s hard to imagine a sound strategy that never made a c-bet.

The obvious way to quantify future betting is to put a number on it, and see how varying that number changes the decision. When you are the preflop aggressor, there are three possible flops, ones you hit, ones it’s likely no one hit and ones you miss that have danger cards.

There are various plausible strategies that you want to compare. Say you want to consider c-betting half the pot in the first two cases, and c-betting half the pot half the time in the last one. For each of these cases, there’s a chance the other players will reraise, call or fold. Suppose you reraise a reraise if you hit the flop, and fold otherwise.

Let P be the pre-flop pot and all profits are stated relative to your stack just before the flop is dealt. Now you’ve got seven possible scenarios:

Hit flop – C-bet – players fold – Win P
Hit flop – C-bet – players call – Expected win 0.5P
Hit flop – C-bet – multiple raises – Expected win 0.25P
Miss flop – C-bet – players fold – Win P
Miss flop – C-bet – players call – Expected win -0.25P
Miss flop – C-bet – players reraise – Win -0.5P
Miss flop – don’t C-bet – Expected win 0.5P

In three of them you know your outcome, in the other four you have to guess (I stuck in some sample values). You can compute the probability of hitting or not hitting the flop, and making a c-bet or not; the other players’ reactions and your expected win in the four cases in which play continues have to be estimated either by analysis or sampling.

• Sean Kaur says:

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• Gel Pen says:

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• Enrique Turtura says:

Numb3rs is a great tv series since it combines mathematics to solve crimes. ..

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