A Note from Two Plus Two: the following is an excerpt from our upcoming book, Analytical No-Limit Hold ’em by Thomas Bakker. We’re hoping to publish it sometime in November, but it’s still being worked on and the actual publication date could be later. Also, we’re interested in any comments anyone might have on this material, so please feel free to post away.
Perceived Ranges
We constantly ask ourselves what the ranges of our opponents are, but our opponents will be doing the exact same thing! So in order to effectively plan ahead, we will need to analyze what our opponents think our range is, which to them is our “perceived range,” and how they will respond to it. If it’s strong, our opponents will be cautious; if it’s weak, our opponents will be confident about their hands.
Let’s analyze how our villain perceives our range in our main hand.
TABLE LAYOUT:
UTG: $895
Hero (CO): $1,752
BTN: $191
SB: $2,755
BB: $1,103
Blinds: $5/$10
Pot: $15
Hero’s hand: Q
J
Action: The first player folds. Hero raises to $35.
We are the cutoff and make a pot-sized raise. Since our villains think we are good and aggressive, they will assume we have a wide range here. So a range of approximately 36 percent of the hands would not be far off:
22+, A2s+, K3s+, Q6s+, J7s+, T7s+, 97s+, 87s, A4o+, K8o+, Q9o+, J9o+, and T9o
Action: The button folds and both blinds call the $35 raise. The pot is now $105.
Flop: T
9
5
Action: The small blind checks.
At this point, the small blind will check with most of his hands as discussed in “Continuation Betting” starting on page XXX. But how does this flop hit our perceived range? From quickly looking at our range on paper, it seems like our range hits the flop with many hands, but a lot of the weaker hands, such as ace-rag and king-rag have completely missed. And if we analyze the exact numbers, the results are:
Hand Type |
Probability |
|
No Pair or Draw |
47.1% |
|
Any Pair |
43.1% |
|
Top Pair |
12.7% |
|
Second Pair |
12.7% |
|
Open-Ended Straight Draw |
5.6% |
|
Set / Two Pair |
4.2% |
|
Overcards |
22.6% |
Recall that we earlier calculated that the probability the small blind had a hand of absolutely no value was 8.8 percent. That is, over 90 percent of his hands made something.
So the small blind’s range is much stronger than our perceived range. This means that, on average, the small blind will be somewhat confident about the strength of his hand.
Now that the small blind has realized all of this, plus he’s probably thinking about his perceived range as well, realizing that it’s quite strong, and it’s his turn to act.
Action: (After the small blind checks) the big blind also checks. Hero bets $60.
Our continuation-bet is relatively small. The small blind will notice this and try to interpret its meaning. Why do we bet $60 here? With what hands would we bet more?
He most likely understands that we realize that many of his and the big blinds hands contain a straight draw of some sort. So if we were trying to protect our hand against their draws, we would probably have bet more, trying to make them fold. Thus the fact that we did not bet big gives the small blind a clue (but in no way a guarantee) that our hand is either a complete bluff, a draw (such as our actual QJ), or an extremely strong hand which here would be a set, or a pair of jacks or queens which both block many straight draws.
The fact that we did bet even though we knew our opponents’ ranges were strong, could tell the small blind that our hand has equity. Recall the formula we used to decide if continuation betting would be good showed that the expectation of a $60 bet was $170.71.
$170.71 = (0.559)(0.541)($105) + [1 – (0.559)(0.541)][(0.384)*$675) - $60]
Most of this information, except for our equity when called (and the probabilities that they fold are a bit different to him since he does not know we have queen-jack, making hands that contain a queen or a jack less likely in the small blind’s and big blind’s ranges), is also available to the small blind. So in theory he could solve the above formula for an unknown range and calculate the equity that makes a $60 bet more profitable than a bigger bet. This would give him an idea what the strength of our hand is because we decided that $60 was the most profitable bet.
Of course, nobody will ever make these profitability calculations at the table, let alone make them for someone else to find out what their equity is, but it’s interesting to realize that this is possible.
With or without these calculations, the small blind probably deduces that we have at least some kind of hand on this flop, so he can further narrow our range to something like the following:
55, 99-AA, A5o, A9o+, K9o+, Q9o+, J9o+, T9o,
A5s, A9s+, K5s, K9s+, Q8s+, J7s+, T7s+, 97s+, and 87s
Manipulating Your Perceived Range
All of the above is correct when everyone plays in an honest, straight-forward way. However, sometimes you will want to represent a different range than you actually have. This is essentially the idea behind one type of bluffing. But there is also another type of bluffing, and the two types are listed below:
1. Misrepresenting your range
2. Overplaying a hand that is in your perceived range
For example, consider the following board:
T T 5 A 9
On this river, our opponent checks and we make a bet. Now, if we have thus far played our hand extremely aggressively, we are representing a range with a lot of tens and maybe a pair of fives, but no weaker hands. If we do not actually have a ten but instead are betting ace-king, we will have misrepresented our range. And even though we are “bluffing” (by representing a hand that is stronger than our actual hand), we are hoping for a call by an opponent who does not believe we have the hand we are representing. He might call us with a weaker ace or even a pair of kings, queens, or jacks. This is a Type 1 Bluff.
Now, consider the following board:
T 5 4 A 9
If we have been playing our hand as either an ace or a flush-draw, and in fact do have an ace, our bet would be a bluff if we were strongly betting this river. However, our bluffing hands are in our perceived range, so it’s easy for our opponent to realize that we might be bluffing. In the hand above, since we are representing a completely different range than our actual hand, it’s much harder for our opponent to guess our hand. This is a Type 2 Bluff.
Now, let’s look at this board:
5 4 3 J A
If we have been slow-playing a flopped straight, either by check/calling or in some other way, our perceived range will be weaker than our actual hand, allowing us to value-bet and get called lighter. This is a Type 1 Bluff; by slow-playing, we pretended to have a weaker range than we actually had.
