Here, I discussed preflop 4bet sizing. The thread is referenced occasionally to this day.
I do a lot of light 4betting these days. Mid-stakes games are very aggressive and tend to feature LAGTAGs who -- perhaps due to some level of distaste for postflop action -- are quite 3bet-happy (read: many guys 3bet eight or more percent of the time the opportunity arises), especially when facing opens from the cutoff and the button. Such players are begging to be 4bet.
You'll notice that in the original article, I advocated 4betting to 27 percent of the effective stack size (27 big blinds, assuming an effective stack size of 100 big blinds). These days, I go even smaller, in general; 23 to 25 big blinds is my sweet spot. I 4bet quite a bit more often these days than I did at the time the original article was written, and my current sizing provides me with increased "wiggling room," fosters more of a "leveling" battle, and my estimate is that it causes my opponents to make more mistakes -- essentially, I believe that the new sizing is an improvement over the old sizing for the same reasons that the old sizing was an improvement over 4betting to pot (admittedly, the improvement is far from as great).
Given this logic, you might be wondering, "Why not take this idea to the extreme and 4bet the minimum (to 20-21.5 big blinds or so, in general)?" I don't think this is such a bad idea, actually, but I'd rather not have villains calling my 4bets with almost all of the range with which they 3bet and choose not to 5bet all in (I'll avoid commenting on the extent to which playing this way would be a mistake).
Anyway, the reason for this post is that I'd been finding myself in lots of spots in which I opened from late position preflop with some ****ty ace, got 3bet by a guy who 3bets either somewhat often or very often, and was tempted to make a 4bet.
mdm13 mentioned in one of his
videos that such spots are great for 4betting (his videos are excellent, by the way; if you haven't checked them out, I strongly recommend doing so).
The concept in play here is card removal. Put simply, the 3bettor is quite a bit less likely to have several of the hands with which he can continue (specifically, AA, AK and AQ) due to the fact that we have an ace in our holding. How much less likely? Well, our holding the
ace of spades eliminates three of the six combos of AA, three of the 12 combos of AKo, one of the four combos of AKs, three of the 12 combos of AQo, and one of the four combos of AQs. That's a total of 11 of the 38 combos of those five hands; the answer to the posed question is, "About 29 percent less likely."
I had
4bet all in with some frequency in the spot in question, but until recently, I hadn't 4bet small because I was unsure about the math. In other words, I didn't know if a small 4bet would cause me to be priced in. So, clearly, I decided to go ahead and perform this math. For what it's worth, I found the results of doing so to be surprising.
In general, as touched on, villains shove over small 4bets with a higher frequency than they do over big 4bets. Furthermore, villains shove over small 4bets with a higher frequency than the frequency with which they call 4bets all in. The reason is obvious -- villains understand that they have a relatively small amount of fold equity when faced with a big 4bet (such as a 4bet to pot when effective stacks are 100 big blinds) and zero fold equity, of course, when faced with a 4bet all in. I'll refrain from commenting on the way in which villains "should" react to small 4bets. Perhaps I'll attempt something like this in the future, but it's something that's very complicated (and almost everyone's 4betting strategy is wack, anyway, in my opinion, so it's not an issue about which I'm overly concerned).
In light of the content in the above paragraph, I went ahead and performed my first calculation by assuming that the villain has a very wide 5betting range -- 88+, AQo+, AJs+, KQs, and A4s:
equity win tie pots won pots tied
Hand 0: 29.132% 27.37% 01.76% 34209861 2204338.50 { As6h }
Hand 1: 70.868% 69.10% 01.76% 86379654 2204338.50 { 88+, AJs+, A4s, KQs, AQo+ }
This is a range against which A6o is a ~2.43:1 dog.
When we 4bet to 24 big blinds from the cutoff over a 3bet from the button, we're getting ~1.65:1 when the button shoves (again, I'm assuming that effective stacks are 100 big blinds). The pot odds on a shove are slightly less generous when we're in the cutoff/button facing a 5bet all in from one of the blinds (think ~1.64:1) and slightly less generous than that when we're the small blind in a blinds battle (think ~1.63:1). In each of these scenarios, though, we need approximately 38 percent equity to break even on a call.
Given that A6o is a 2.43:1 dog, the math here isn't even close. Calling the 5bet is quite -EV; in fact, we would have to 4bet to about 40 percent (!) of the size of effective stacks before we could even begin to consider taking the hand to the felt. That's $40 at 100 NL, $80 at 200 NL, $160 at $2/$4, $240 at $3/$6, and $400 at $5/$10.
I continued my experiment by tightening the 5betting range in question. I did this for the hell of it; I figured that it would be very unlikely for it to make sense to get all in with A6o against a tight range when it was -EV to do so against a loose range, but surely it wouldn't hurt to learn how drastically our equity changes against a more conservative villain:
equity win tie pots won pots tied
Hand 0: 24.836% 23.59% 01.24% 12118655 639228.50 { As6h }
Hand 1: 75.164% 73.92% 01.24% 37972008 639228.50 { QQ+, AQs+, AKo }
We drop from a 2.43:1 dog to a 3.03:1 dog, a change of almost 25 percent. 3.03 is even further from 1.65 than 2.43 is; against the narrower range, calling all in with an A-rag hand becomes even more of a mistake.
I decided to do a similar set of calculations with a low pocket pair due to a suggestion from
carnivalhobo (THE carnivalhobo). I understand that I did a bunch of calculations like these in the original article, but 1. I failed to do the math on a situation in which our hero holds a low pocket pair and 2. the original article assumes that we 4bet "small" to 27 big blinds (rather than to 24). When I pitted 66 against the loose 5bet range seen above, PokerStove spit out the following:
equity win tie pots won pots tied
Hand 0: 37.323% 37.10% 00.23% 327781056 1990068.00 { 66 }
Hand 1: 62.677% 62.45% 00.23% 551787672 1990068.00 { 88+, AJs+, A4s, KQs, AQo+ }
66 is a 1.68:1 dog against this range. That's better equity than A6o has (recall that it's a 2.43:1 dog). Still, we need about 38 percent equity to call all in, here, and we're not quite getting it.
equity win tie pots won pots tied
Hand 0: 37.583% 37.39% 00.19% 145991964 735246.00 { 66 }
Hand 1: 62.417% 62.23% 00.19% 242942856 735246.00 { QQ+, AQs+, AKo }
Against the tighter range, interestingly, 66's equity improves by about a fourth of a percentage point. This is an anomaly to some extent and is the case because of the absence of 88-JJ in the tighter range; these hands don't exist within the confines of a tight player's 5betting range, but they each have a heavy equity advantage over low pocket pairs.
66 is a 1.66:1 dog here. Again, recall that we need about 38 percent equity to break even when calling a 5bet all in. 66 has about 37.6 percent equity against the range designated above. 38 and 37.6 are very close -- too close for comfort, in my opinion; like George Bush, I don't enjoy when math gets fuzzy (of course, even fourth grade math is fuzzy when you're Mr. Bush).
Keep in mind that we're doing math based on fabricated ranges. There's no chance that A6o has enough equity to call a 5bet all in against even the craziest LAGTAG; 29 percent equity is too far from 38 percent equity for calling to be a reasonable consideration. 37.6 percent equity is quite close to 38 percent equity, though; consequently, it's very possible that we'll make a mistake by folding a hand like 66 to a 5bet.
Conclusion: It's very clear that we can make a small 4bet and fold to a 5bet all in 100 big blinds deep when holding ****ty Ax hands. Given this, it will be +EV for me to increase the frequency with which I make 4bets when in possession of these holdings.
Small pocket pairs are a different story, however. The math here when effective stacks are 100 big blinds is close; I'll not mess around with this trick unless effective stack size is larger and I can therefore reasonably 4bet to something like 18 percent of effective stacks.
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