Fold Equity (3) - Turn and River

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Fold Equity (3) – Turn and River

by MiiWiin

Up to now, you've read lots of the basic principles of fold equity and seen a detailed example of a continuation bet against a TAG range.

However, the hand doesn't have to end on the flop, you can also keep betting on the turn and the river (2nd and 3rd barrel) and take down the pot like that.

Multi-barrels

The mathematical example in part two was considered at a very simple level as far as results go. Your bet generates fold equity, if you can expect at least this fold equity against your opponent's range, then the bet is good.

FE (required) < FE (available)

However, this is only true if the flop is considered in isolation. It may easily be the case that the bet is good too, if you don't achieve enough fold equity. This is the case if you can generate significantly more fold equity with a further bet during the hand.

For example:

FE [Flop] (required) > FE [Flop] (available)

but

FE [Turn] (required) < FE [Turn] (available)

That's why the fold equity needs to be estimated precisely.

Genereally this isn't particularly easy against varied players, but the principle just mentioned can be illustrated in a further example.

You're up against an opponent who you know limp/calls every pocket pair preflop. However, on the flop he only continues to play with a set.

PartyPoker $25 NL Hold'em (6-handed)

Stacks & Stats
UTG ($25)
MP ($25)
CO ($25) limp/caller
Hero ($25)
SB ($25)
BB ($25)

Preflop: Hero is BU with
2 folds, CO calls $0.25, Hero raises $1.25, 2 folds, CO calls $1.25

Flop: ($2.85) (2 players)
CO checks, Hero bets $2.00, CO folds

This play results from the given information!

This play preflop might look a little strange as K2s doesn't have particularly brilliant equity and you estimate the fold equity to be about 0%.

Still, the raise is good because you know that you've got fold equity of over 80% on the flop (a pocket pair only hits every 7.5 times). With a clear +EV bet, you'll also win the pot regularly, which even includes the cutoff's 4 additional big blinds due your isolation raise preflop. 

Although in theory the preflop raise when viewed in isolation is -EV, the entire play can be seen as +EV.

The same goes for players who call too much on the flop, but give up on the turn or river.

Now take a look at the situation in part 2:

PartyPoker $25 NL Hold'em (6-handed)

Stacks & Stats
UTG ($25)
MP ($25)
CO ($25)
Hero ($25)
SB ($25) TAG
BB ($25)

Preflop: Hero is BU with
3 folds, Hero raises $0.75, SB calls $0.75, 1 fold

Flop: ($1.75) (2 players)
SB checks, Hero bets $1.20, SB calls $1.20

Turn: ($4.15) (2 players)
SB checks, Hero bets $2.75, SB...

Is the 2nd barrel good now or not?

A range has already been worked out. Villain could be holding the following on the turn here:

99, 50% proportion of TT => 6 combinations
50% of his T9s, 98s hands => 3 combinations
AJs-A9s, AQo-ATo => 39 combinations
75% of his Kx hands => 25 combinations
A7s/A2s => 6 combinations
33% proportion of 97s => 1 combinations

Making a total of 80 combinations.

Your 2nd barrel is in the region of a 2/3 pot size again, which generally comes with required fold equity of around 40%.

You bet $2.75 on the turn to win $4.15.

Required fold equity: 2.75 / (2.75 + 4.15) = 39.85%

How many combinations will have to fold according to this in order to make the 2nd barrel a +EV move?

0.4 * 80 = 32

What does Villain play on?

Now there's nothing else to do than check all of the combinations for playability.

The turn itself blanks with the . In your opponent's range this only helps the A2s combination to a two pair.

The following assumptions can be heavily disputed (as ever). However, we assume the following:

• The opponent plays solidly
• He doesn't try any tricky plays
• He has no reason to definitely take marginal hands to the showdown
• He knows the disadvantage of his position
• He'll rarely c/c made hands on the board on the turn just to give up on the river
• He believes Hero is a solid opponent

99, 50% proportion of TT => 6 combinations

How many of these six combinations can Villain continue to play? Ultimately, he's holding 3rd pair with TT, a bluff catcher. He's out of position and has to ask himself now how he's going to react to a potential 3rd barrel. The calldown was already right on the edge, maybe he'll play on with 1/3 of the TT combis again.

He certainly won't fold his sets with 99.

50% of his T9s, 98s hands => 3 combinations

A pair+flush draw might play on here, so is taken as a representative. He'll generally (have to) fold the other two.

AJs-A9s, AQo-ATo => 39 combinations

Things are very difficult here now. A9s is holding two pair, there's no need to discuss that. But what about the other Ax hands? Does he fold any kind of top pair?

We could philosophise here, however, the opponent is defending his small blind against a button raise, so you can't assume that a top pair folds.

That's why he'll call all the combinations again.

75% of his Kx hands => 25 combinations

We are compromising here. If he's always willing to call with Ax hands, he is more likely to fold Kx hands.

It's reasonable to assume that a third of this range will stay in the hand.

A7s/A2s => 6 combinations

A2s remains in the hand, however as it's assumed that he'll call every Ax above, he may part from A7.

33% proportion of 97s => 1 combinations

He can comfortably give up these combinations on the turn.

Still in the hand:

99, 33% proportion of TT => 4 combinations
=> 1 combination
AJs-A9s, AQo-ATo => 39 combinations
33% of his Kx hands => 8 combinations
A2s => 3 combinations

Total: 55 combinations

check/fold it:

66% proportion of TT => 2 combinations
T9s, 98s hands => 2 combinations
66% of his Kx hands => 17 combinations
A7s: 3 combinations
97s => 1 combinations

Total: 25 combinations

We need: FE (required) < FE (available)
FE (required) = 39.85%
FE (available) = 25 combinations / (25+55) combinations = 31.25%
FE (required) < FE (available) = 39.85% < 31.25% (not fulfilled)

This means the 2nd barrel wasn't good.

The 3rd barrel

PartyPoker $25 NL Hold'em (6-handed)

Stacks & Stats
UTG ($25)
MP ($25)
CO ($25)
Hero ($25)
SB ($25) TAG
BB ($25)

Preflop: Hero is BU with
3 folds, Hero raises $0.75, SB calls $0.75, 1 fold

Flop: ($1.75) (2 players)
SB checks, Hero bets $1.20, SB calls $1.20

Turn: ($4.15) (2 players)
SB checks, Hero bets $2.75, SB calls $2.75

River: ($9.65) (2 players)
SB checks, Hero...

After the previous analysis, you don't need to count any more combinations on the river. A very large portion of Villain's range is made up from Ax hands, where the assumption was made earlier that he'll rarely check/call the turn just to fold on the river.

The only thing that helps here is the saying "if you call twice out of position, you won't be folding on the river..."

Whether your opponent is in a position to fold hands such as ATo or not is pure speculation. You shouldn't assume that the 3rd barrel can't help you any more either.

Extras

With semi-bluffs plays such as these are hard to calculate, because as well as fold equity you have to also take your equity into consideration.

Put simply: theoretically you need to consider with which hands Villain calls your semi-bluff bet, but stays in the hand if there's a potential hit. 

There are innumerable cases to determine between here as if your hand improves on the turn or the river it would be ideal for you not to attain any more fold equity.

Apart from a very complicated calculation, so many assumptions also have to be made that every result should be seriously challenged.

There's already content eavailable on the theory:

A Mathematical Reflection on Draws
Semi-bluffs for Advanced Players

One main factor changes out of position: your opponent will bet on floats more often and therefore call on the flop more frequently.

In general he'll call more, as a good opponent knows the advantage of position and knows that he can call there a bit more than he can out of position.

In multi-way pots plays are usually more honest. Even though you won't catch bluff raises as often there, you should be aware that lots of opposing hands may also have hit something.

Therefore, you need to assume less fold equity.

Example: even if every opponent messes up 70% of his range, you can still expect (0.7 * 0.7 * 0.7 = ) 34.3% fold equity against 3 players. 

You've seen that you achieve around 40% fold equity with a 2/3 pot size bet and that's within a reasonable range. Experience shows that you won't make any massive mistakes with this betsize. Deviations are still possible.

If you place a higher bet, the fold equity will increase a little, however it has to work more often. If you bet less, it doesn't need to work as often, but the actual fold equity will also be smaller.

Example: if you bet just one big blind, it only has to work every 12th time. However, you can assume that no TAG will ever fold to a 1BB contibet.

Nonethelss, you can experiment a bit here, the 2/3 isn't set in stone, it should fall between 0.5 and 1.0 x pot size.

LINKS
Related articles: Fold Equity (1) - General Fold Equity (2) - Continuation Bet