You suggested that your
range was approximately
JJ+, AhT+, AJ on the
river with a board of:
7

J

5

3
Q
Here's how your
range breaks down in order of hand strength from lowest to highest with the number of hand combinations in parentheses (assuming you play these hands the same way 100 percent of the time for the sake of example).
AxJx(9), KxKx(3), AxAx(3),
JJ(3),
QQ(3), KhKx(3), AhAx(3), AhTx(4), AhQx(3), AhKx(4), AhJx(3)
9 -
Pair of jacks
3 -
Pair of kings
3 -
Pair of aces
3 -
Three of a kind, jacks
3 -
Three of a kind, queens
3 - King-high
flush
17 -
Ace-high flush
If you
shove, your opponent is going to have to make a decision about how to play hands that beat your
three of a kind in queens and lose to your king-high
flush. Hands that are worse than this are almost always going to
fold, and hands that are better than this are almost always going to
call. We're targeting the
range of hands that he actually makes a decision with.
Without taking out the
rake for the sake of simplicity, your opponent will need to be winning at least 1.07/4.01 = 27% percent of the time to make a
call with (for example) a ten-high
flush if you
shove on the
river.
Suppose you're betting F combinations of your flushes. To be bluffing 27 percent of the time, you'll need to be betting B combinations of the bottom of your
range so that B/(B+F) = 27 percent. A quick calculation gives us a useful formula.
B/(B+F) = 0.27
B = 0.27(B+F)
B = 0.27B + 0.27F
0.73B = 0.27F
B = 0.27F/0.73
B = 0.37F
As an aside, guess what else 0.37 is? It's bet/(bet+pot) when you make the
river shove. That number shows up a lot in poker. This formula tells us that the number of bluffing hands in our
range will need to be 37% of the number of value hands for us to have a balanced
range. In general, if you're in a
shove/
check situation where you'll be betting a
polarized range, a quick shortcut to find a balanced
bluff frequency is that your number of bluffs should be bet/(bet+pot) percentage of your number of value bets.
Anyway, if we decided to
shove all 20 combinations of our flushes and 7.8 combinations (on average) of our single pairs of jacks, for example, then our opponent will not have a correct way to play when he has a ten-high
flush (or any of the other in-between hands). We can scale
back these numbers appropriately to put nut hands in our checking
range while remaining balanced if we wish to do so.
Note: It's worth pointing out that we could also expand our value betting
range to include sets of queens if we
felt the need to. We could then scale up the number of bluffs that we had accordingly. We're not going to do that, though.
This leaves us with several suitable hands that we could use as
bluff catchers in our checking
range if we wanted. We can combine these with some percentage of nut hands if we want to make life tough on
Villain.
-- The Checking
Range --
If we
check and our opponent shoves, then he'll need us to
fold 37 percent of the time. If we really want to piss him off, then we could
set up our ranges so that we were folding exactly 37 percent of the time. That would really rustle his jimmies.
But what about his value bets with something like a Ten-high
flush? If we only checked with
bluff catchers, then he could bet all of his flushes and get paid off 37 percent of the time. We can't allow him to print money like that.
Villain needs to be winning at least 50% of the time when he value bets with
marginal flush hands for it to be the correct decision when compared to checking. If we really wanted to piss him off, then we could make it so that one-half of our
calling range would beat those
marginal flush hands.
This means that our checking
range really just breaks down into three parts like this:
|---- c/f (X) ----|---- c/c
bluff catchers (Y) ----|---- c/c
nuts (Z) ----|
We know that we're only going to be folding 37 percent of the time, so that X is going to be 37 percent of our checking
range. We also know that Y and Z have to be the same number of combinations to make his bets with
marginal flushes have about a 50%
equity against our
calling range. A quick calculation shows that Y and Z are each 31.5 percent of the checking
range. To sum things up so far:
B + F + X + Y + Z = 100% of the
range
B =
bluff shove
F = value
shove
X =
check/
fold
Y =
check/
call (
bluff catcher)
Z =
check/
call (strong hands)
We need to do some work on the equation B + F + X + Y + Z = 1 to figure out the right proportions for our ranges to
balance both the shoving
range and the checking
range at the same time.
B + F + X + Y + Z = 1
X + Y + Z = 1 - B - F
Note that X/(X+Y+Z) = 0.37 since X is 37 percent of the checking
range (X+Y+Z).
X/(X + Y + Z) = X/(1 - B - F)
0.37 = X/(1 - B - F)
0.37(1 - B - F) = X
We found earlier that B = 0.37F when the betting
range is balanced.
0.37(1 - 0.37F - F) = X
0.37(1 - 1.37F) = X
0.37 - 0.51F = X
So we put in F and we get out X.
Remember that X/Y = 0.37/0.315 based on the ratios we figured out earlier.
X/Y = 0.37/0.315
X/Y = 1.175
1/Y = 1.175/X
Y = X/1.175
Now we can put in F and get out Y. Since Y = Z, this also gives us Z. Long story short, we can put in F and get the rest of our
range now (B, X, Y and Z).
-- Fleshing out the ranges --
What all of this means is that we can pick how many combinations we want to
value bet, and our formulas above will spit out how to play the rest of our
range to keep our betting and checking ranges balanced simultaneously. It's important to note that there are several different ways to play a
range that's balanced, and we'd need to evaluate the EV of each one to determine the balanced strategy that performed the best.
Instead of doing that (because it will take fucking forever), we're going to operate on the assumption that we always wanted to
check/
call with a
flush and never anything worse. This will narrow down the possibilities. Since 20 of our combinations were flushes, that's 20/41 = 48.8 percent of our
range. We would require that F + Z <= 0.488 so that we aren't shoving or
check/calling with anything other than flushes. Some quick work on a spreadsheet shows that we'll have to
shove with at least 13 combinations of flushes to avoid ever
check/folding a
flush (when we
shove 13 flushes, we
check/
call with about 7).
For a reminder of what our
range looked like:
9 -
Pair of jacks
3 -
Pair of kings
3 -
Pair of aces
3 -
Three of a kind, jacks
3 -
Three of a kind, queens
3 - King-high
flush
17 -
Ace-high flush
Here's an example of a balanced strategy where we never
check a
flush: bet 20 hands for value,
bluff about 7 hands,
check/
fold about 5 hands,
check/
call about 9 hands total. Here's how the example
range would play out:
Bet 7 pairs of jacks
Check/
fold 2 pairs of jacks, 3 pairs of kings
Check/
call 3 pairs of aces, 3 sets of jacks, 3 sets of queens
Bet 3 king-high flushes, 17
ace-high flushes
This strategy will probably seem somewhat unreasonable to a lot of people.