Game Plan (7): Barrelling Frequencies: River
Before going through this lesson you should already have read:
- Game Plan (1): Introduction
- Game Plan (6): Barrelling Frequencies
- Game Plan (5): Calldown Frequencies: River
If you are in position on the river, whether with a strong or a weak hand, it isn't always clear whether you should bet or check behind. In the lesson "Barrelling Frequencies" you have already got to know four factors that influence your betting frequency. However, factor 2: draw potential and factor 4: playability are not applicable on the river, and this simplification means you can define your betting frequency using factor 1: bet size and factor 3: favourable boards. In this lesson you will learn how to construct your river betting range using these two factors.
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If you are in position on the river, you can use Barrelling Frequencies: River (BFR) to determine which hands to bet with, or which to check behind. |
Your opponent's calling range is unknown
If you are on the river and your opponent checks to you, you can determine your betting range in two different ways:
- Can you estimate your opponent's calling range? Then calculate the EV of a bet and a check for every hand, then choose the more profitable option.
- You can't estimate your opponent's calling range? Then choose the BFR method.
You will sometimes find a situation where your opponent plays with a perfect calling frequency. If this is the case, your value bet and bluff limits are clearly defined. If your opponent calls more often than a perfect frequency would dictate, then both limits decrease towards weaker hands, meaning you value bet thinner and bluff less. Your VB-Ratio thus becomes very value heavy. If your opponent calls less often than a perfect fequency would dictate, then both limits increase towards stronger hands, meaning you value bet tighter and bluff more. Your VB-Ratio thus becomes bluff heavy.
If you can't determine what calling frequencies your opponent plays and with what probability, i.e. whether a value or bluff heavy VB-Ratio is better, then barrelling frequencies are your best option. You play the barrelling strategy that prevents hero calls and hero folds. This minimises the EV of your opponent's unknown calling range.
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You use BFR when... ...you have a potential bluff hand or thin value hand on the river ...you don't know your opponent's calling frequency. |
Seven steps to your betting range
Your betting range consists of all the hands where the EV of a bet is greater than the EV of a check. This is the case for your strong hands - value bets - and your weaker hands or bluffs. You then have a value limit and a bluff limit that are determined separately using the following seven steps:
Step 1: Determine your opponent's range on the river.
Step 2: Determine the EV of a check for every thin value hand.
Step 3: Determine your opponent's range against your bet (CFR villain).
Step 4: Determine the EV of a bet for every thin value hand.
Step 5: Determine your value betting range.
Step 6: Determine your bluff frequency.
Step 7: Determine your bluff hands.
These seven steps are shown in an example hand below.
Hero (BU): $1000 (100 bb)
SB: $1000 (100 bb)
BB: $1000 (100 bb)
Preflop: Hero is BU with 
Hero raises to $20, SB folds, BB calls $10
Flop: ($45) 
(2 players)
BB checks, Hero bets $22.50, BB calls $22.50
Turn: ($90) 
(2 players)
BB checks, Hero bets $60, BB calls $60
River: ($210) 
(2 players)
BB checks, Hero checks / Hero bets $140 ??
The requirements for BFR are met here.
- You have a thin value hand on the river and are in position.
- An estimate of how often your opponent calls here is not possible, because your opponent can use different game plans that lead to different calling frequencies on the river:
| Opponent's game plan |
Opponent's motivation | Calling frequency on the river |
| Loose call down: Call one - Call all |
All turn draws miss on the river. You will bluff far too much on the rive so a call is worth it with every hand. | Very high |
| Good call down: Call bluffcatcher with blocker |
A 3 barrel on an A-high board is made up of strong made hands and a few bluffs. A call with good bluff catchers is worth it. | Moderate |
| Tighter Calldown: Call Hands that beat Vbets |
Nobody bluffs enough on A-high boards on the river. It is only worth calling with two-pair plus. | Very low |
In this hand you don't know how your opponent plays so you can't attack his game plan, you can merely defend your own. You use the BFR method to decide which hands in your range to bet with and which ones to check.
First, you determine in three steps which is the best range your opponent can play against your bet.
Step 1: Determine your opponent's range on the river.
Step 2: Determine the EV of a check for every thin value hand.
Step 3: Determine your opponent's range against your bet (CFR Villain).
Step 1:
In order to know where your value limit is, you need to estimate which range your opponent reaches the river with.
In the example, your opponent reaches the river with this range:
66-22, ATs-A2s, K6s-K3s, Q5s-Q4s, 75s, 65s, 54s, 
, 
, 
, , 
, 
, 
, 
, , 
, , A9o-A2o, K6o (#161)
Step 2:
The EV of your hand is equal to the showdown value of your hand. This is the equity of your hand compared against the river range of your opponent, multiplied by the pot size, thus:
EV(Check) = (Equity vs Villains Range)(Pot)
In the example your has 73.6% equity against your opponent's range.
EV(Check AcKc) = 73.6% * $210 = $154.56
In the spoiler you will find the showdown value of your other, thinner value hands.
| Your hand |
Equity | Showdownvalue |
| A3 | 78.6% | $165.00 |
| 43 | 77.6% | $162.86 |
| AxKd |
73.8% | $155.00 |
| AxKs AxKc |
73.6% | $154.56 |
| AxKh | 72.5% | $152.25 |
| AxQ!h |
74.4% | $156.24 |
| AxQh | 74.0% | $155.44 |
| AxJ!h | 74.8% | $157.10 |
| AxJh | 74.2% | $155.86 |
| Ax9!h | 72.1% | $151.39 |
| Ax9h | 71.9% | $150.95 |
AxQ!h - AQ without the Qh
Step 3:
Use CFR from your opponent's view point against your bet. You need his folding frequency, calling range, and raising frequency.
In the above example your opponent raises #6.4, calls #89 and folds #65.6. His calling range is 66, 44-33, ATs-A9s, A7s-A2s, Ad8d, Ac8c, A9o, A7o-A2o.
In two steps you can determine the value limit for a chosen bet size.
Step 4: Determine the EV of a bet for every thin value hand.
Step 5: Determine your value betting range.
Step 4:
The EV of a bet is equal to your fold equity, plus the EV against both the raising and calling ranges of your opponent:
EV(Bet) = Fold equity + EV(Calls) + EV(Raise)
Your folding frequency is equal to the size of the pot multiplied by your opponent's folding frequency.
Your EV against the calling range is equal to the equity you have against his calling range, multiplied by the pot size: Pot + 2*Bet, minus your invested bet. Your EV against the calls is your EV against the calling range multiplied by the calling frequency of your opponent.
With a thin value hand you fold to a raise, therefore you lose your bet against such a raise. Your EV against the raising range is therefore (-bet). Your EV against a raise is your EV against the raising range multiplied by the raise frequency, giving:
EV(Bet) = (Fold%)(Pot) + (Call%)((Equity vs Call-Range)(Pot + 2*Bets) - Bet) + (Raise%)(-Bet)
In the example, has an equity of 53.2% against the calling range.
The raising, calling and folding frequencies of your opponent are always the number of raising, calling and folding combinations, divided by the total combinations. The exact number of combinations is often different for the different hands due to blocker effects. For this gives you
Total combinations of opponent's river range: #126
| Range Villain | #Combinations (Hero AcKc) | Frequency |
| All (#161) | #125 | 100% |
| Fold |
#56.6 | 45.3% |
| Call | #62 | 49.6% |
| Raise | #6.4 | 5.1% |
Using the above, you determine the EV of a value bet with to be:
EV(Bet AcKc) = (45.3%)($210) + (49.6%)((53.23%)*($210+$140+$140)) - $140) + (5.1%)(-$140) = $147.92
In the spoiler you will find the EV of a value bet with your other, thinner value hands.
| Your hand | EV Valuebet |
| Ac3c | $167.21 |
| 43 | $190.62 |
| AcKc | $147.92 |
| AhKh | $146.42 |
| AcQc | $154.71 |
| AcJc | $155.55 |
EV(Bet) = 1/(#All) * ((#Fold)*(Pot) + (#Call)*((Equity vs Call-Range)(Pot + 2*Bets) - Bet) + (#Raise)(-Bet))
EV(Bet Ac3c) = 1/126*(61.6*$210 + 58*(60.34%*($490)-$140) + 6.4*(-$140)) = $167.21
EV(Bet 43) = 1/147*(65.6*$210 + 89*(63.29%*($490)-$140) + 6.4*(-$140)) = $190.62
EV(Bet AhKh) = 1/120*(50.6*$210 + 63*(53.97%*($490)-$140) + 6.4*(-$140)) = $146.42
EV(Bet AcQc) = 1/129*(60.6*$210 + 62*(53.23%*($490)-$140) + 6.4*(-$140)) = $154.71
EV(Bet AcJc) = 1/131*(62.6*$210 + 62*(53.23%*($490)-$140) + 6.4*(-$140)) = $155.55
Step 5:
For every thin value hand you compare the EV(check) with the EV(bet) and choose the play with the higher EV.
In the example, your has an EV of $154.56 as a check and an EV of $147.92 as a value bet. You thus choose to check behind, as a value bet is too thin.
In the spoiler you will find the EV of your other, thinner value hands.
| Hand | EV(Check) | EV(Bet) | Action |
| 75s | $207.86 | $315.10 | Bet |
| Ac3c | $165.00 | $167.21 | Bet |
| 43 | $162.86 | $190.62 | Bet |
| AcKc | $154.56 | $147.92 | Check |
| AhKh | $152.25 | $146.42 | Check |
| AcQc | $156.24 | $154.71 | Check |
| AcJc | $157.10 | $155.55 | Check |
AJ-AK are close checks. If your opponent has less two pair and sets on the river than you assumed in your CFR analysis, for example because he raised the flop or turn, then AJ-AK become a value bet.
Alternatively you could choose a smaller river betsize to be able to value bet AJ-AK. Against a smaller bet your opponent has to call more often. If you do additionally value bet AJ-AK however, your opponent can then raise his sets for value. This would mean he raises more often.
You value bet the river with two pair+, #66 combinations: AA, 66, 44-33, A6s, A4s-A3s, 75s, 64s-63s, 52s, 43s, A6o, A4o-A3o, 75o.
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Shortcut (slightly optimistic approximation) If your equity against your opponent's call range minus his raise frequency is bigger than 50%, then a value bet is better than a check. |
As soon as you get money into the pot with more than 50% equity your EV increases - you therefore have a value bet.
A raise from your opponent against a thin value bet has the same EV as a call with a better hand - you lose the bet because you fold to the raise.
If your equity against his calls minus the raise frequency is more than 50%, you can value bet.
Your in the example has 53% equity, but, your opponent's calling range is raised in 5% of cases and since 53% minus 5% is obviously less than 50%, the approximation straight away tells you that you can't value bet.
A value bet generates additional value compared to a check for every call of
(Equity vs Call)(2*Bet) – Bet
A value bet loses EV compared to a check for every raise of
(Equity vs Raise-Hand)(Pot) + Bet
The additionally generated value exceeds the lost EV exactly then when,
(Call%) * ((Equity vs Call)(2*Bet) – Bet) > (Raise%)*((Equity vs Raise-Hand)(Pot) + Bet)
The equity against the raise range is close to zero, since we only win with a check against a bluff raise. This is however only the smaller part of the raise range.
(Call%)((Bet)(2*(Equity vs Call) – 1)) > (Raise%)(Bet)
Cancel out the (Bet) and divide by (Call%).
(2*(Equity vs Call)) - 1 > (Raise%)/(Call%)
Divide by two and sort
(Equity vs Call) – ½ > (Raise%)/2(Call%)
(Equity vs Call) – (Raise%)/2(Call%) > ½
(Call%) generally lies between 33% and 66%. In approximation 2(Call%) is around 100%.
(Equity vs Call) – (Raise%) > 50%
And this is the rule of thumb. Please note
(Equity vs Call-Range) – (Raise%) > 50% ⇒ EV(Bet) > EV(Check
The exact formula is
Valuebet > Check when
(Equity vs Call) – ((Raise%)(1 + (Equity vs Raise)(Pot/Bet)))/(2(Call%)) > 50%
This step mirrors factor 3: favourable boards. If you have the stronger range you value bet more often.
In these two steps you determine your bluff limit for any given bet size.
Step 6: Determine your bluff frequency.
Step 7: Choose your bluff hands.
Step 6:
Your bluff frequency is determined by your river bet size and the pot size.
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The bluff frequency of your betting range is equal to b/(2b+p) where b = bet size p = pot size |
Your bluffing range is b/(b+p) times as big as your value range.
In the example you bluff 28.6%. This matches your opponent's pot odds so your bluffing range is 0.4 times as big your value range.
b = $140 - your betsize is equal to $140
p = $210 - the pot size is equal to $210
Bluff frequency = b/(2b+p) = $140/(2*$140+$210) = 14/49 = 28.6%
Bluff multiplier = b/(b+p) = $140/($140+$210) = 2/5 = 40%
In the example you value bet #66 combinations on the river: AA, 66, 44-33, A6s, A4s-A3s, 75s, 64s-63s, 52s, 43s, A6o, A4o-A3o, 75o.
Bluff combinations = #66*40% = #26.4
This step mirrors factor 1: bet size . With a smaller bet you bluff less often than with a big bet.
Step 7:
Choose those hands with the least showdown value from your range and add these to your bluffing range, until you reach your bluff frequency.
Hands where the showdown value (EV of a check) is lower, profit more from a bluff than hands whose EV(check) is higher.
In the example you construct your bluffing range from hands with minimal showdown value until you reach #26.4 combinations. is the worst hand you can hold on the river and has $0 showdown value; this then is your first bluff combination. Your other bluff combinations are 8-high with 85 and 87s as well as busted heart draws with 95, 97, J2 (40%), J5 and J7.
| Hand | EV(Check) | #Combinations |
| 7h2h 8h5x 8x5h 8h7h 9h5h 9h8h Jh5h |
$0 | #12 |
| 85s | $0.71 | #9 |
| 87s | $1.37 | #3 |
| Jh7h | $1.37 | #1 |
| 9h7h | $1.39 | #1 |
| Jh2h | $1.41 | #1 |
Your river bluff range is then: 87s, 85s, 9h8h, Jh7h (40%), 9h7h, Jh5h, 9h5h, Jh2h, 7h2h, 85o (#26.4).
In the spoiler you will find the calculation of the bluff EV for every bluff hand compared to the check EV
Hands with identical showdown value, ie the same EV when checking, only differ in bluff EV due to blocker effects. If you are holding hands with the same EV(Check) of which you can only bluff bet a part to reach your bluff frequency, you sort by two direct blocker effects:
- Block of the opponent's raise and call range
- Block of the opponent's fold range
If like in the example you know the exact raise-, call- and fold range of your opponent, then you can calculate the bluff EV of every hand taking blockers into consideration.
You need to pay attention however. If you bluff too much due to blocker effects, or too little, or with hands that have more showdown value than hands you are checking behind, your opponent can exploit this by adjusting his range. Your showdown value never changes, therefore using the showdown value as the criterion to choose bluff hands with is stable and adequate as a heuristic.
An exact bluff range construction takes into consideration showdown value and blocker effects of both players, this is ignored here. If there are no blocker effects then hands below your opponent's showdown value limit can always bluff profitably.
Against the blocked raise-, call- and fold ranges of your opponent you get the following bluff-EVs.
| Your hand | Showdownvalue |
EV Bluff | Action |
| 7h2h | $0 | $0.95 | Bluff |
| 8!h5!h | $0.71 | -$1.90 | Check |
| 8!h5h | $0 | -$5.24 | Check |
| 8h5!h | $0 | -$5.79 | Check |
| 8h7h | $0 | -$1.85 | Check |
| 8h7!h | $0 | -$0.46 | Check |
| 8!h7!h | $1.37 | $4.55 | Bluff |
| 8!h7h | $1.37 | $3.20 | Bluff |
| 9h5h | $0 | -$1.92 | Check |
| 9h8h | $0 | -$2.76 | Check |
| Jh2h | $1.41 | -$5.60 | Check |
| Jh5h | $0 | -$6.15 | Check |
| Jh7h | $1.37 | $0.92 | Check |
| Kc2c | $11.97 | -$7.60 | Check |
| 5c3c | $30.62 | $3.40 | Check |
EV(Bet 7h2h) = 1/148 * (59.6*$210 - 83*$140 - 5.4*$140) = $0.95
EV(Bet 8!h5!h) = 1/147 * (58*$210 - 85*$140 - 4*$140) = -$1.90
EV(Bet 8h5!h) = 1/145 * (56*$210 - 86*$140 - 4*$140) = -$5.79
EV(Bet 8!h5h) = 1/147 * (56.6*$210 - 85*$140 - 5.4*$140) = -$5.24
EV(Bet 8h7h) = 1/151 * (59.6*$210 - 86*$140 - 5.4*$140) = -$1.85
EV(Bet 8h7!h) = 1/152 * (60.6*$210 - 86*$140 - 5.4*$140) = -$0.46
EV(Bet 8!h7!h) = 1/154 * (63.6*$210 - 85*$140 - 5.4*$140) = $4.55
EV(Bet 8!h7h) = 1/153 * (62.6*$210 - 85*$140 - 5.4*$140) = $3.20
EV(Bet 9h5h) = 1/146 * (57.6*$210 - 83*$140 - 5.4*$140) = -$1.92
EV(Bet 9h7h) = 1/152 * (63.6*$210 - 83*$140 - 5.4*$140) = $6.48
EV(Bet 9h8h) = 1/152 * (59.6*$210 - 86*$140 - 6.4*$140) = -$2.76
EV(Bet Jh2h) = 1/150 * (57.6*$210 - 86*$140 - 6.4*$140) = -$5.60
EV(Bet Jh5h) = 1/148 * (56.6*$210 - 86*$140 - 5.4*$140) = -$6.15
EV(Bet Jh7h) = 1/153 * (61.6*$210 - 86*$140 - 5.4*$140) = $0.92
EV(Bet Kc2c) = 1/149 * (56.6*$210 - 86*$140 - 6.4*$140) = -$7.60
EV(Bet 5c3c) = 1/144 * (59*$210 - 81*$140 - 4*$140) = $3.40
You bluff on the river against the exact range of your opponent with 7h2h and #12 8!h7. All other hands are better as a check. (Be careful! You therefore bluff only #13, too little.)
This step again mirrors factor 3: favourable boards. If you have the stronger range, you bluff more often.
Summarised, your river ranges in the example look like the following:
You value bet #66 (AA, 66, 44-33, A6s, A4s-A3s, 75s, 64s-63s, 52s, 43s, A6o, A4o-A3o, 75o)
You bluff bet #26.4 (87s, 85s, 9h8h, Jh7h, 9h7h, Jh5h, 9h5h, Jh2h (40%), 7h2h, 85o)
You call all-in #32 (AA, 66, 44-33, 75s, 52s, 75o)
When not to use BFR
BFR can be used for most river bet or river raise situations. In the following situations however, you should refrain from this method:
- Multiway pots
- Protected pots
- Boards with a high chance of a split pot
- When you have made strong adjustments to your range based on your opponent. Your knowledge about your opponent also needs to go into your river decision.
- Based on blocker affected betting ranges, BFR suggests betting ranges that can lead to strong changes in your opponent's range.
Summary
BFR divides your river range into value, bluff and check ranges and you use the method against unknown ranges of your opponent.
The seven steps to your call range:
Determine your opponent's river range:
Determine which hands your opponent reaches the river with.
Determine your showdown value:
Determine the EV of a check for every thin value hand.
Determine your opponent's ranges against your bet(CFR):
Determine the raising, calling and folding ranges of your opponent.
Determine your value bet EV:
Determine the EV of a bet for every thin value bet.
Determine your value range:
Determine your value hands for the EV(Bet) > EV(Check).
Determine your bluff frequency:
Determine your bluff multiplier using your bet size.
Determine your bluff hands:
Determine the hands with the least showdown value and bluff these until you reach your bluff frequency - be mindful of blockers.