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Folding Thresholds in 3-bet Pots: A Closer Look at your Opponents’ Strategy

Folding Thresholds in 3-bet Pots: A Closer Look at your Opponents’ Strategy

“If you know the enemy and know yourself you need not fear the results of a hundred battles.”
Sun Tzu, The Art of War

Our primary considerations when choosing a line of play in a poker hand are the particular exploitable tendencies of a given opponent. Both ‘math’ and ‘feel’ players alike will agree that if a player folds ‘too much’ on a given street we can bluff him more often. But exactly how much folding is too much?

Away from the table we can do some valuable analytical work to calculate folding thresholds beyond which a player is vulnerable to exploitative bluffing. In so doing we also discover how our own exploitative lines expose us to being exploited ourselves. Thus the calculation of folding thresholds helps us both to more accurately target imbalances in our opponent’s strategy and to recognize when they are adjusting correctly to our strategy.

3bet pots provide a veritable haven of diverse exploitative lines for the observant player. Given their bloated size relative to single-raised or limped pots they also account for a disproportionate amount of the money per hand which you win/lose at PLO. In this article, we consider the mathematics behind bet/calling, bet/folding and other decisions post-flop in 3-bet pots.

My writing in this post is not meant to be an exhaustive examination of 3bet pots so we shall not go into much detail regarding our opponents’ ranges. The purpose of this analysis is to discover the threshold frequencies at which bet/folding the flop leaves a player open to being exploited. A knowledge of these fundamental frequencies will improve our awareness in 3bet situations at the table.

For the duration of this article we shall assume starting stacks of 100BB, an open from a non-blind seat to 3BB and that a 3bet to 10BB from the SB is called by the player with positional advantage. The pot going to the flop is thus 21BB with 90BB behind.

The Flop Continuation Bet in a 3-bet Pot

If our opponent were to shove every time we continuation bet the flop we would have zero fold equity and need only consider our equity against his shoving range to decide whether or not to C-bet. In this limiting case, our required equity vs our opponent’s range to C-bet is 45%.

In practice, on most flop textures our opponent will fold the flop with a non-zero frequency and respond with a mixture of calls and shoves. So we need (much) less than 45% equity against his flop continuing range to justify a flop C-bet.

You (hopefully) understand that a flop continuation bet of 2/3 the pot size with zero realizable equity must take the pot down at least 40% of the time to be neutral EV. Fortunately we aren’t often dealt 2♠22♣2and so usually have at least some equity against our opponent’s continuing range.  We’ll investigate a situation where the quality of our realizable equity is very high:

Let us take some hands with only 4 nut outs going to the turn, such as J♠J♣9♠9 or A♠K♣85♠ on 7♠4♣2. Let us further assume that we only continue if we hit one of our 4 nut outs, and that we just take the pot down when we hit on the turn.

1

If our opponent were to shove 50% of his range on the flop, we have a profitable C-bet opportunity provided that he only calls the flop bet 11% of the time and folds 39% of the time. If we reduce his shoving frequency to 25% we can tolerate him folding only 35% of the time and still show a profit.

If our opponent chooses never to shove over our C-bet then we make 1.1bb per C-bet if he folds the flop 35% of the time. This result demonstrates that we should always C-bet hands with nut outs on boards that heavily favor us range versus range.

Indeed if our opponent insists on shoving only 15% of his range then we only need him to fold 30% of the time on the flop to have a +EV C-bet! In fact we can have as few as 3 nut outs (the same gut-shot on a two tone board) and still need only 32% folds to break even versus such a low shoving frequency. Even 2 nut outs only requires 34% folds on the flop if our opponent shoves as infrequently as 15% of the time. Thus bet/folding overpairs of AA/KK on boards where the A or King is live is usually profitable.

Take care to be betting nut outs; if you are relying on a small number of folds to make your line profitable then occasionally getting stacked with the second nut straight has a large negative impact on the EV of the flop C-bet. An 8+ out draw is usually a +EV C-bet, although there are circumstances under which check-raising the flop as the PFR will be more profitable.

In this analysis we have focused on hands with high quality realizable equity. It is a lot more complex to evaluate the EV of C-betting hands with no nut outs. For that we would need to consider the quality of our equity on various turn cards. We shall save that exercise for another article (or perhaps a book!).

We now turn our attention to the mathematics that guides our decisions when we face a raise of our flop continuation bet.

Facing a Raise on the Flop in a 3-bet Pot

Raises of flop continuation bets fall into two distinct categories, depending on whether or not they pass the ‘stack-commitment’ threshold. We’ll consider both types of raises below and in so doing notice that committing raises from an opponent make it far easier for us to plan a continuation betting strategy.

Committing raises
If we bet 14BB into 21BB and our opponent shoves we need 38% equity to call off the remainder of our stack. Since a common line as the pre-flop 3bettor is to bet/fold the flop, we need to calculate how wide our opponent can shove the flop if we bet/fold with a given frequency. The table below shows how often our opponent needs us to fold on the disconnected rainbow board J♠6♣3 once we bet and he shoves a low equity hand.

Opponent Equity when called Threshold f(Bet/Fold)
0% 72%
20% (Naked Gutter versus 2P+) 59%
23% (QQ + BDFD versus 2P+) 56%
31% (Bare OESD versus 2P+) 44%

 

An opponent who is shoving any open-ended straight draw will show a profit against us unless we stack off at least 56% of the time when we C-bet. Thus even if we only C-bet 50% of the time we would need to bet/call 28% of our total range assuming we had enough hands worth 2 pair or better.

Generally speaking, when you are 3betting a strong range you will not have difficulty finding hands to stack off with on boards with two cards nine or above. However, on innocuous boards such as {7♠6♣2/8♠5♣2/9♠7♣3} the 3-bettor has virtually no two pairs or better in his range. Hence bet/folding AA unimproved will leave you heavily exploitable.

Once your 3-betting range widens (as it will in HU games or in the blinds against an aggressive BT) even boards with a lone high card can make unimproved AA a terrible hand to bet/fold. When I examined my own 3bet range versus a MP open on a K62r board I discovered that I had at most 20% of hands which were 2 pair or better, the majority of which were KKxx hands which I would not always 3bet.

Given the lack of draws here and the high frequency with which I would C-bet this board, if I am bet/folding AAxx here and my opponent is smart enough to kamikaze shove any crappy King here (whose main value is serving as a blocker to my sets since my calling range is so nutted that he only has 13% equity) then he only needs me to bet/fold 65% of the time. His King blocker reduces my calling range to around 10% and so if my Cbet is over 30% (and it is by a lot) then he banks an easy profit.

Fortunately for the brave shover, once I start calling AA he has 31% equity against my calling range and so only requires 44% folds to break even. He will be losing around 28BB per call of his shove but picks up 35BB per fold. If I call 60% of the time he loses 2.8BB on the play. Since this is the same loss as simply folding to a 3bet pre-flop it is fair to say that the “suicide shove” is not as bad as it would first appear2.

Non-committing raises
Against thinking players (where thinking is very broadly extended to those who are not simply clicking buttons) we will tend to encounter smaller flop raises on dry boards only if they have either a pair on them or a made straight.

On a paired flop, a raise is polarized to trips/air on a disconnected board, although we can encounter some straight draws when the cards are closer in rank. Assessing your opponent’s range in this spot involves detailed analysis and I shall not address it today. Suffice to say, your opponent’s tendency to raise/call with “vulnerable boats” and non-nut trips such as 76♠5♣4 on 7♠7♣Q is extremely important in determining how weighted they are towards a bluff.
If your opponent raises your flop continuation bet of 14bb to 31bb, he has risked 31bb to win 35bb immediately. We must thus continue with more than 53% of our range on the flop (different bet sizes will affect this value, we have chosen these sizes for thematic consistency). If we call, we go to the turn facing a pot size of 83bb with 59bb behind.

We need to bluff-catch the turn with 59% of our turn range to prevent our opponent having a +EV turn bluff-shove. Put together this would require us to stack off with at least 31% of our continuation betting range to prevent our opponent having a +EV turn bluff-shove.

If on paired boards we C-bet 90% of the time3 (for many players the continuation bet is automatic on say 773 as the pre-flop 3bettor) then we shall need to stack off the top 28% of our range. When I checked one of my 3bet ranges in this spot, I found I had 13% trips on the flop and around 60% overpairs. This means that I only need to stack off with 1/4 of my overpairs by the turn on such a flop to not be exploited. Provided I call the flop with hands with backdoor flush draws, I tend to be able to find sufficient hands to call the turn with some equity when the board comes two tone.

It is worth stating that the calculation I followed above has a flaw from the aggressors’ perspective; whilst the flop and turn bet are +EV in isolation, the whole line is not since the turn bet “wins back” some money invested in the bluff on the flop. In reality, the flop raiser invests 90BB by the turn in order to win the pot of 21BB + the 3bettor’s flop bets of 31BB = 52BB. The turn bet needs to succeed 64% of the time for the line to show a profit. In practice this allows the 3bettor to get away with stacking the top 19-20% of our range by the turn.

Many players are conscious on some level that raising the flop and shoving the turn with air is likely to be spew, which explains why they so often one-and-done the flop. This is one of those spots where it “feels really bad” to call the flop and fold the turn, but in practice the biggest mistake against most regulars is where we bet/fold AAxx to the flop raise. If the board stays rainbow on the turn then I am afraid you are on a guess which overpairs to continue with. Good luck with that.

This article was originally featured as a subscriber-only post on my legacy site Quadrophobia.com.

QUESTION(S) OF THE WEEK: What flop textures give you the most trouble as the pre-flop 3-bettor? Do you find yourself bet/folding hands in the top 25% of your range on these textures? Share your favorite ‘trouble textures’ in the comments below.

Show 3 footnotes

  1. This is a combination of the most pessimistic outcome and the most weak-tight line possible!
  2. A simple way to exploit a player doing this too much is too check more of your air so that you are bet/calling with a very high frequency. At 80% calls he starts losing 15BB per shove. Another way is to reduce your C-bet size.
  3. I do not advocate this approach to paired boards. In the article I am more interested in examining the contingent consequences of a given flop approach than making a value-judgement on the initial decision.
By | 2017-04-10T13:22:52+00:00 June 4th, 2015|

3 Comments

  1. Kyle February 24, 2017 at 9:52 pm - Reply

    f our opponent chooses never to shove over our C-bet then we make 1.1bb per C-bet if he folds the flop 35% of the time..

    0.35x(21) + 0.65x(0.08x(0.66×21 + 21) + 0.92x(-0.66×21) = 0.87

    Where did i miss?

    • Phil Rocquemore February 24, 2017 at 11:49 pm - Reply

      Dear Kyle,

      The calculation should be:
      0.35x(21) + 0.65x(4/45x(0.66×21 + 21) + 41/45x(-0.66×21)) = 1.1

      You missed the card removal effect of us knowing 7 cards (the flop and the 4 cards in our hand) which increases the probability of us hitting our gutter on the turn. You also omitted a bracket.

  2. Andy June 29, 2015 at 6:58 pm - Reply

    A1.The paired boards in HU context give me the most trouble as the pre-flop 3bettor.
    A2.Yes I do.It feels that I should fold even more frequently to XR.

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