The EV of a continuation bet

[adanthar]

I've recently noticed that a lot of people seem to have no idea about when a continuation bet is +EV. There are several types of continuation bets, but the classic example is when you flop overcards on a raggedy board. Most of the EV of that bet comes from the times that every other player folds and you take the pot.

The good thing is that figuring out if a bet that is designed solely to fold your opponents is +EV is extremely simple. Just take the ratio of your flop bet to the chips already in the pot, divide by the number of your opponents, and express it as a percentage. If they fold more than that percentage of the time (plus some small percentage for when they fold a small pair/just call and you hit the perfect turn card), the bet is good.

For example, if you have AK on a QTx flop and bet half the pot into 2 opponents, they will both have to fold roughly 33% of the time [adjusted for the times JT just calls/would have bet if you checked, and the turn card is an ace, so it's actually around 30%] for the bet to make any sense. If you bet the pot, they must both fold slightly under 50% of the time.

To translate this into the real world quickly, put them both on a range of hands and then figure out how likely they are to have hit. On this QTx flop, for example, it is wildly unlikely that two decent (but not great) players both missed. On the other hand, on a flop of 222, it is at least conceivable that a solid continuation bet from KQ will be able to fold out both AQ and a disciplined 55. Therefore, a bet of 2/3 of the pot into two opponents, designed to get them both to fold around 40% of the time, is probably a good one. Also note that while it is actually relatively rare for a continuation bet into 2 people with overcards to be good and a bet into 3 people usually sucks, a bet against only one opponent is almost always profitable.

Of course, stack sizes, how loose the callers are, if a second barrel on the turn might work, etc. also factor into this. But the rule of thumb is that you should take the 'fold percentage', subtract a little bit for lucky breaks and free cards, and then go with that.

[DJ Sensei]

very good post, I think too many decent players read HOH or some of many continuation bet posts on here and just assume that they should bet half the pot whenever their AK misses. I must admit, I was guilty of this myself once. But now, against players who don't seem overly clever, I'm making a lot of money when I see these seemingly forced continuation bets.

[ QUOTE ]
For example, if you have AK on a QTx flop and bet half the pot into 2 opponents, they will both have to fold roughly 33% of the time [adjusted for the times JT just calls/would have bet if you checked, and the turn card is an ace, so it's actually around 30%] for the bet to make any sense. If you bet the pot, they must both fold slightly under 50% of the time.

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I'm not following you.

You bet so as to offer opponents 3:1 and they both have to fold 1/3 of the time to make the bet profitable.

You bet so as to offer opponents 2:1 and they both have to fold 1/2 of the time to make the bet profitable.

Yet not accounting for other factors, this event will only happen roughly 1/6 of the time (2 players each with 3 possible decisions {call, raise, fold}, 6 possible outcomes one of which is both fold).

On this basis alone, are either of the bets rational?

I have to believe that players are more inclined to fold if one bets the pot (or better still overbets the pot) rather than half the pot. But facing two opponents one would have to have some combination of factors working in his or her favor (e.g., much bigger stack, proven ability to knock people out with semi-bluffs, or simply being in the outright lead at that point).

A lot would seem to depend on the preflop action and subjective probabilities of their likelihood of folding versus caling or raising.

In other words, using your example with AK, presumably you raised and were called by two players preflop. The flop more or less completely missed you.

So what is the probability that flop hit one of the others? Or what is the probability that you were behind from the start (eg., one or both opponents is on a medium pair) and are now down to perhaps 10 outs at best with the QT flop.

Under these circumstances, wouldn't you need a higher probability of both opponents folding in order to make your bet profitable?

I guess what I'm getting at is that this is a much more complicated situation facing two opponents, one of whom presumably cold called and the other overcalled preflop. Head-up is a whole different and probably simpler story.

I tend to be thick in the head on theory, so maybe I'm not looking at this in the right way. Hence my response in an effort to exchange dialog and learn.

[SossMan]

so much of it really comes down to a few things that help you determine your FE,
in order of importance:

# of opponents you are betting into
your relative position on these opponents
have you recently (in the last few orbits) either a) taken down a pot with a flop bet and your hand went unshown or b) checked the flop w/ a monster and it was eventually shown or c) failed to put any more money in the pot after raising preflop either on the flop or turn

in addition to reads on the players and relative stack sizes, these are the most important things that you should be looking at in terms of calculating FE.

Also, be careful when cont betting with a whiffed hand that your cont bet wont commit you to calling a shorter stack's all in when you likely only have 6 outs max. generally, early in a tournament, I won't cont bet if a shorter stacked opponent will put me in and i will be forced to call. I may push into them, but i don't want them to think that they have some FE w/ their middling pair after i make the half pot cont bet.

Nice post, though, andathar

[adanthar]

You're off on the wrong track - it's not nearly this complicated. Say the pot is 100 chips on the flop and you bet 50 with a second (or third, or sixth) best hand; it follows that the bet is +EV if you win the 100 chips more than 1 in 3 times. The number of opponents you face is irrelevant, except that the more of them there are the less likely they are to oblige you and fold.

Now that you know your 1 opponent, 2 opponents, or 17 opponents need to fold 33% of the time combined, put them all on a range and very quickly run through if the flop is likely to have hit at least one of them and if they are the kind to call 88/raise A2o on it (assuming, of course, that you must fold to their raise). There are a dozen other things you can consider here, but the primary one is just how likely you are to get a fold.

Yes, varying the bet size can be slightly more/less likely to get a fold. That's already accounted for in the calculation - if you bet twice the pot your 1, 2 or 17 opponents must fold 66% of the time.

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it follows that the bet is +EV if you win the 100 chips more than 1 in 3 times. The number of opponents you face is irrelevant, except that the more of them there are the less likely they are to oblige you and fold.

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Thanks ... I understood the algebra part of it but was just looking at it from a different perspective. As the previous poster responded, there are other factors to be considered when the situation is other than head up. But yeah, good post nonetheless.

[DJ Sensei]

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For example, if you have AK on a QTx flop and bet half the pot into 2 opponents, they will both have to fold roughly 33% of the time [adjusted for the times JT just calls/would have bet if you checked, and the turn card is an ace, so it's actually around 30%] for the bet to make any sense. If you bet the pot, they must both fold slightly under 50% of the time.

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I'm not following you.

You bet so as to offer opponents 3:1 and they both have to fold 1/3 of the time to make the bet profitable.

You bet so as to offer opponents 2:1 and they both have to fold 1/2 of the time to make the bet profitable.

Yet not accounting for other factors, this event will only happen roughly 1/6 of the time (2 players each with 3 possible decisions {call, raise, fold}, 6 possible outcomes one of which is both fold).

On this basis alone, are either of the bets rational?

I have to believe that players are more inclined to fold if one bets the pot (or better still overbets the pot) rather than half the pot. But facing two opponents one would have to have some combination of factors working in his or her favor (e.g., much bigger stack, proven ability to knock people out with semi-bluffs, or simply being in the outright lead at that point).

A lot would seem to depend on the preflop action and subjective probabilities of their likelihood of folding versus caling or raising.

In other words, using your example with AK, presumably you raised and were called by two players preflop. The flop more or less completely missed you.

So what is the probability that flop hit one of the others? Or what is the probability that you were behind from the start (eg., one or both opponents is on a medium pair) and are now down to perhaps 10 outs at best with the QT flop.

Under these circumstances, wouldn't you need a higher probability of both opponents folding in order to make your bet profitable?

I guess what I'm getting at is that this is a much more complicated situation facing two opponents, one of whom presumably cold called and the other overcalled preflop. Head-up is a whole different and probably simpler story.

I tend to be thick in the head on theory, so maybe I'm not looking at this in the right way. Hence my response in an effort to exchange dialog and learn.

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The fact that they both have to fold 33% of the time when given 3:1 pot odds (with a half pot bet) is based on expected value. Lets assume that if they call or raise, you lose the hand, and if they fold, you win the hand. (A very simple assumption, but surprisingly good in most cases)

So lets say the pot is 1000. You bet 500. Now, lets say they both fold 1/3 of the time and the other 2/3 of the time, one or the other calls or raises.

EV = 1/3 * (+1000) + 2/3 * (-500) = 0

thus, its an neutral EV play if they fold exactly 1/3 of the time. If they fold less than this, you're losing chips in the long run by making the play. If they fold more, you're winning chips.


Now, onto probability of them both folding. It gets more complicated with both people involved, but lets continue to assume that the only way you win the pot is if they both fold. If the two opponents act independently (another very false assumption, but one that is good enough for this simulation). Lets say they both fold 2/3 of the time independently. Then the probability that they both fold is 2/3 * 2/3 = 4/9. This is greater than 1/3, so the play would be profitable. If each of them independently folds 1/2 the time, they both fold 1/2 * 1/2 = 1/4 of the time, thus its a bad play. (Note that this alone is a very good example of why continuation bets work much better against a single opponent)


Now you mentioned that betting the pot is more likely to chase off opponents. This is true, since theyre getting worse pot odds. However, you need to win the pot outright 1/2 the time with a full pot bet to make it profitable to bet that much. (By the same EV calculations as above, betting 1000 into a 1000 pot, your EV = 1/2 * (+1000) + 1/2 * (-1000) = 0 if they both fold half the time, its positive if they fold more than half the time, and negative if they fold less than half the time)
Now, if they're acting independently, you need them each to fold about 3/4 of the time to make a full pot bet profitable: (3/4 * 3/4 = 9/16 chance that both fold, 9/16 is greater than 1/2, so the play is +EV)

As you noted, the probability that any particular play will work is very subjective. As such, these calculations are a good exercise in expected value, but are no substitute for a good read on your opponents.

[johnnybeef]

Great post. One thing I would like to add in here is the value of position. My raises are made from a multitude of factors, but, generally speaking, from lp, i will make a continuation bet every time it is checked to me if i raised pf. There are so many other factors that i'd like to discuss here as continuation bets are imo the most important weapon in a tournament players arsenal, but im just too damn lazy (and i kinda hope others will touch upon them.)

[curtains]

The AK example on the QTx flop isn't so clear as your hand has serious value if you check behind. Thus if a bet is only slightly above breakeven, and a lot of the time you will be checkraised if you bet, you may choose to check instead to take a free turn card.

Most of the time I check behind 2 opponents on this type flop with AK, but of course this depends on a lot of factors.

[adanthar]

I was actually thinking of you being in EP to make it easier, but you are right, that's something else to consider.