You suggested that your range was approximately JJ+, AhT+, AJ on the river with a board of:
7:heart: J:heart: 5:diamond: 3:heart: Q:heart:
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.
