April 3, 2010 – 10:41 am | 7 Comments

I’ve compiled a short (just 7-pages) e-book, an introduction to the mathematics of poker. It’s basically covers how to calculate your expected value in a certain spot – starting with explaining what EV is, all …

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## Small Ball Strategy – Pre Flop

Submitted by on December 12, 2009 – 3:00 am4 Comments

I just came across a post from the folks at True Gamble, saying the the “Small Ball” strategy (hugely publicized by Daniel Negreanu), is not good for beginners.

Personally, I like the idea of the Small Ball strategy (mainly because I’m a Negreanu fanboy, to be honest, but the logic seems sound). But either way, let’s break down the example they’ve used, and try to come to a standard conclusion. I’ll break it down into a couple of posts, and today, we’ll look at the preflop play.

Blinds are 100/200, John has A J and is using the ‘Small Ball‘. He minimum raises from a latter position and Dave calls it.

So, first off. John gets dealt A J. For the sake of simplicity, we’ll assume that they are offsuit. Let’s look at the odds an A J offsuit has against a few different ranges of hands.

• vs any hand: 63.6% of winning
• vs any pocket pair, any broadway, A7+ (21.6% of hands): 54.5%
• vs any pocket pair, any suited cards, any broadway (38.5% of hands): 57.9%
• vs any pocket pair, any Ace, any suited connectors (30.0% of hands): 58.0%

I could go through a number of other ranges, but I think we can safely say that John has about 55-65% chance of winning the hand, against whatever range his opponent may have.

So John min. raises to 400. His opponent, Dave, now has to call 200 into a pot of 700. That gives him odds of 28.6%. If he assumes that John will make the raise with any two cards, the call makes sense no matter what two cards Dave has. If he puts John on a range of any pocket pair, any broadway or A7+ (like what we assumed in our preflop analyses), he should still call with any hand other than T2o, 92o, 82o, 72o, 32o. With implied odds considerations he can probably make a case to call with these hands as well.

Should John have raised higher, then? Let’s say John raises to 500 (which is the 2.5 big blind raise that Daniel Negreanu recommends). If Dave thinks that John would raise with any two cards, the call is still valid for any cards. However, a larger raise would indicate a slightly better range of hands, so assuming the range of any pocket pair, any broadway or A7+, you’ll see that the calling range for Dave is narrowed.

Now, Dave should fold with a lot of his hands. He should call with any Ace, any suited King, any Broadway, and any suited connectors, but that’s about it (the exact range is 22+,A2s+,K2s+,Q8s+,J9s+,T9s,98s,87s,A2o+,K9o+,QTo+). It narrows Daves range to 31.8% of all possible preflop hands . Against these 31.8% of hands, John is a 58.2% favorite.

As such, John’s expected value from raising to 500 (ignoring action on future streets) is

EV = 0.682 * 800 + 0.318 * 0.582 * 1000 – 500
EV = 230.676

Compare that to the expected value of 0.636 * 900 – 400 = 172.5 (making the reasonable assumption that Dave would call the min raise with any two hands), and you can see that the raise to 2.5 times the big blind is the better play.

What if John had raised even higher? If Dave puts John on that same range (and folds the appropriate hands), a raise to 3 times the big blind would give John an even higher expected value of 262.145; a raise to 4 times the big blind gives 268.415

The increase in expected value, however, seems to get more and more minimal. I haven’t yet figured out how to do the analysis on that (if you have any suggestions, let me know), but it seems like the increase in expected value (from the narrowing of John’s range) is small enough to not warrant the raise to 4 big blinds (maybe 3 is acceptable), as compared to the 2.5 big blinds raise.

Because of the much narrower range that Dave would have if he called a raise of 3 big blinds, John is likely to be behind. As such, John stands to lose more money on later streets, where a minimum of one bet (200) is already many times more than the slight gain he gets from the larger preflop raise.

Of course, that’s just my intuition, but from this example, it would seem that a pre-flop raise of 2.5 to 3 times the big blind is about the optimal range. 2 times is too little, and 4 times seems a bit too much.

What do you guys think? What is your standard pre-flop raise, and why?

• stevebrogan says:

When ever I hear a discussion of a hand using math to evaluate it, my eyes start to glaze over and I go into a zen like trance. Math is my weak point and I am going to have to embrace it and use it if I intend to be profitable at this game.

In answer to your question, I come into an un-raised pot with a bet of the pot size as my bet. In a .25/.50 game with .75 already in the pot, my post size bet amount to a raise of \$1.75 3.5 times the size of the big blind. This appears to be a standard practice at the tables I play at. And I am not sure if just betting \$1.50 would be as effective. Your point is well made.

I like your blog design, by the way.

• Derrick Kwa says:

Hi, Thanks for dropping by. =). I'm kind of the opposite of you, I guess. My weak spot is in observing the players at the table, gaining reads and adjusting to it. And I definitely need to find a way to improve on that. Maybe we can help each other out? 😉

As for the raise, I think the analysis demonstrates that 2.5 to 3 times is a good bet size if you're the first in the pot. With more players, it might be a good idea to raise slightly more.

I haven't exactly done the math on this, but essentially, the probability of you winning the hand decreases as more players are in the hand, and so the expected value from playing the flop decreases. Also, the pot odds you are presenting to your opponents would be higher (because of the larger pot size from earlier callers), and as such there will be less folds. Intuitively, I would think that a larger bet size would cancel out those factors.

Oh, and the design is actually a free WordPress theme. Got it from http://www.poker-themes.com/poker-style/ There are some errors in the code, though; I might have to either fix them or change it soon.

• Derrick Kwa says:

Hi, Thanks for dropping by. =). I'm kind of the opposite of you, I guess. My weak spot is in observing the players at the table, gaining reads and adjusting to it. And I definitely need to find a way to improve on that. Maybe we can help each other out? 😉

As for the raise, I think the analysis demonstrates that 2.5 to 3 times is a good bet size if you're the first in the pot. With more players, it might be a good idea to raise slightly more.

I haven't exactly done the math on this, but essentially, the probability of you winning the hand decreases as more players are in the hand, and so the expected value from playing the flop decreases. Also, the pot odds you are presenting to your opponents would be higher (because of the larger pot size from earlier callers), and as such there will be less folds. Intuitively, I would think that a larger bet size would cancel out those factors.

Oh, and the design is actually a free WordPress theme. Got it from http://www.poker-themes.com/poker-style/ There are some errors in the code, though; I might have to either fix them or change it soon.

• […] a previous post, I looked at this example from TrueGamble, and studied the preflop action; concluding that the […]