A technical discussion of pocket pairs and implied odds requirements

[DavidC]

Inspired by my 3.5 bets cold to AKo post:
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Situation:

You're in the SB. You have 22.

A TAA raises UTG, folded to MP maniac re-raises next to him, folded to button who caps it. You're in the SB.

BB's loose and all, but he's not calling 3-cold without a good hand.

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In the original discussion, of which there was a lot, I'd had AKo. There's been some disagreement on whether or not to play this hand, with Shillx taking the side of the Righteous Fold, and Aaron (iirc) taking the side of the Righteous Bitch-slap to those who would cowardly fold.

One of the points that interested me, though, is that Shillx said that you could call it with 22, but that AKo was an easy fold.

AdamL, my impoverished-due-to-variance friend, was a little hesitant to call with 22 in this spot, getting crappy pot odds and being a little unsure of the value and amount of future action.

Somewhere in the 2+2 archives, it mentions that you're looking for not a specific number of opponents, but high implied odds in order to enter pots with pocket pairs.

I'm curious, though... we're 7.5:1 to hit our set, right? That means that when we hit our set, we want to make sure that pot size is more than 7.5x whatever we put into it preflop... This is what I'll call the "Simple Goal", and it seems to be the general approach to the hand here.

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The problem that I find with the Simple Goal is that if you flop a set of 7's on a 872r board, and someone has t9s (no bdf), another guy has AKo, and a third guy has JJ... they call you down and miss: the pot size is 8.5x what you put in preflop... did you win money? Well, not really... because YOU put some of that money into the pot, so you have to look at the pot size, minus what you put into it, and make sure that that's more than 7.5x what you paid preflop. I'll call this the "Exclusive Goal".

So let's say the same hand happened... this time you measure the pot again, but you take out what you put into it before measuring... It shows 8.5x. Did you win? Well... no.

The problem with the "Exclusive Goal" is that all the money that's put into the pot by other people isn't 100% Your Sklansky Bucks... Some of that money is going into other people's pockets. In fact, some of the money that you put in (that we excluded in the Exclusive Goal) is actually not your money either (some portion of that usually goes towards someone else's Sklansky Bucks).

By the way, keep in mind the effects of the rake at all times. Sorry, but it can become important.

If we were to look at the same pot, which was "Exclusively" 8.5x your pf investment, and you saw that 6x your pf investment was put in going from the flop to the river, you would remember that much of that action was put in while the guy with t9 was winning about a third of every bet that was put into the pot. If you were to take away 2x your pf investment here in earnings, then you're left iwth 6.5x pf, and therefore a loss of maybe 1/10th your pf investment on the hand.

So, somehow the opponents' equity in a hand has to be calculated into the implied odds that we will Truly require in making these plays.

In multiway pots, where you have guys with overpairs and lower sets pumping you, or you have two opponents on a straight or flush draw (thus you're getting paid twice and they're getting paid once, for their draw), this wouldn't be such a huge factor, but in short-handed pots, it can be.

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All this sounds pretty reasonable, but I never see it getting discussed, here, and all this thought hasn't really led to any solid conclusions, for me.

I'd appreciate getting a discussion going on what exact scenario we really need to earn profit on pocket pairs. I mean, I've made some money on pocket pairs, so I know that it can be done. I've also got some idea of what sorts of tactics to use to make the most money with them... but from an intellectual / mathematical standpoint, I haven't got a bloody clue as to what's going on.

[numeri]

My thoughts were always that I had to make enough SBs post-flop to make up for what I missed pre-flop. In your case, we'll assume that the others will call, so it's 4-handed to the flop if we call. That means we're paying 3.5 with 13.5 in the pot. As you mentioned, we need ~8:1 for our set. Since we're calling 3.5, we need 8*3.5 = 25.5.

The difference is what we need to make up then, which is 25.5-13.5 = 12SB. So by this idea, we'll need to make up 12SB post-flop. With this much aggression, you can be assured of further betting, so it seems reasonable.

Those are just my thoughts - more for myself to see if it makes any sense than from any sort of knowledge. I'm interested to see what other thoughts are.

[AdamL]

Awesome post.

[pryor15]

i've been wondering something along those lines w/ small PP in early position, but i'm thinking there's a flaw in my logic.

it seems to me that in a typical loose SS games, an EP limper is going to usually ensure at least 3-4 to the flop and usually 4-6 (no EP limper brings into play the open-raise from later positions from the better players). it also seems that, for the most part, you can get 2-3 to the turn, etc, so it should be all that hard to make up the 8sb we need in order to play something like 44 profitably.

of course, we will often get raised by someone behind us w/ a bigger hand, and occasionally it'll be 2 back to us, but that first raise shouldn't be too much of a concern since we can usually count on the PFR making at least 1 continuation bet and when we flop our set, we can raise the hell out of him and pump the pot pretty well.

of course, i've also been doing a lot of drinking tonight, so that could be clouding my judgement.

[Aaron W.]

It's about 7.5:1 to make a set, and I believe that sets (not trips) hold up about 80-85% of the time. If someone who has a large database is able to run a SQL script to find out how close this estimate is, I would appreciate it. So if we're 1/8.5 = 11.7% to make a set and we win 80% of those, then pocket pairs make sets and win 9.4% of the time, which is about 9.6:1. At 85%, this is 9.9% or 9.1:1. We'll take 9.5:1 as the estimate because it's a nicer number to work with.

Using this, we can try to estimate if it's worth playing our hand. The preflop investment will be 3.5 SB. The entire preflop pot will be 17 SB, so the amount contributed by others will be 13.5 SB. We need to get the pot to be about 33.25 SB (contributed by others) to make this a winning call. How much action can we expect? Here's a slightly conservative estimate: Two bets from everyone on the flop (6 SB), two bets from one or two players on the turn (2-4 BB = 4-8 SB) and one or two bets on the river (2-4 SB).

So the postflop pot (excluding your contribution) will be roughtly 25.5-31.5 SB, which is a couple BB short of the required value on average. However, the value given was taken to be slightly conservative. Factoring in the maniac read on MP, it's very possisble to pick up those extra bets when maniac goes nuts with top pair. Also, if villain has AA or KK and has an overpair, you will often see an overplayed hand.

Therefore, calling with 22 is likely a profitable call. Can you ever get this exactly right? Nope. There are too many factors to consider. But as a general rule, be more inclined to play when villains spew postflop and less inclined to play if they're weak.

What about the affect of making losing sets? That happens 1.8-2.3% of the time. If you end up losing an extra 8 SB when you make losing sets, your EV drops some .14-.18 SB/hand. This has a small affect, but it is much smaller than you might expect. Why? It doesn't happen that often relative to the number of times you limp. Even though it's a solid 15-20% of the time *when* you hit your set, you don't hit the set very often, so you don't pay off losing sets very often.

Also, the order of magnitude of this change is smaller than the error of guessing the number of bets you'll get postflop. So you don't need to worry about it that much.

[Cosimo]

I'm coming up with 1:7.5 to flop a set. Note that this is set, full house, or quads.

2/50 + 48/50*2/49 + 48/50*47/49*2/48
= 2/50 * (1 + 48/49 * (1 + 47/48))
(the latter is easier to punch on a calculator)
2 / 50 = MS * 48 / 49 = M+ * 47 / 48 = M+ MR

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OK so my point: sets are not gimmes. I've got about a 66% winrate with sets for the past 18k, which might be running bad (I did have 8k of pain), but still.

In the example given, 872r board vs T9, AK, and JJ, there's two shots at six cards that beat you (note that AK is drawing dead), minus the seven cards that improve your hand. Hence you invest 3.5SB and win 1/8.5x80% = 9.411% of the time, which means the pot needs to be 37SB (19BB) bigger than your investment (which I'll conservatively call 13SB). Throw in rake and you've gotta see a 26+BB pot on average to come out ahead. That's huge. I win that once per 8000-ish hands, which is thirty-six pairs of dueces of which maybe four flop sets. This doesn't include all the other low pairs which have nearly the same odds. (Point being that I don't win pots that huge at the party .5-1 often enough to justify chasing sets with low pairs when it's 3.5 SB to me. Whew that's a mouthful.)

This is mitigated somewhat by the times you'll win with something other than a set, but again you rarely have odds to chase to the turn or river. Twodimes shows a 17% equity for 22 vs the three named hands, which includes all those strange flushes, straights, and late sets. You won't win this hand 17% of the time cuz you'll fold before that.

So low pairs are a toss for 3.5 SB. Even for 2.5 SB, there's always the risk of a reraise, and the pot "only" needs to be about 18 BB. Doable with that much aggro, but still marginal. Considering that it's really high variance, I toss it.

For 1.5BB vs three opponents it's an easier call cuz you're ten SB away from breaking even -- yet at this point it's easy to see that might be hard. Ten SB more when you've got three opponents means two of them are gonna hafta see the turn and there needs to be a raise somewhere. If these guys are loose and aggro, you'll make it (but why no more raising preflop?). If they're weak-tight (of the call-two-cold-too-often type), you won't.

Summary: throw 22 away from the SB if it's 2.5+ cold to you.

[ QUOTE ]
My thoughts were always that I had to make enough SBs post-flop to make up for what I missed pre-flop. In your case, we'll assume that the others will call, so it's 4-handed to the flop if we call. That means we're paying 3.5 with 13.5 in the pot. As you mentioned, we need ~8:1 for our set. Since we're calling 3.5, we need 8*3.5 = 25.5.

The difference is what we need to make up then, which is 25.5-13.5 = 12SB. So by this idea, we'll need to make up 12SB post-flop. With this much aggression, you can be assured of further betting, so it seems reasonable.

Those are just my thoughts - more for myself to see if it makes any sense than from any sort of knowledge. I'm interested to see what other thoughts are.

[/ QUOTE ]

Here's something else to think about. I'm a newb, so my maths could be a little wonky but the principle should still hold up.

If you assume (not too far fetched I hope) that in a capped flop, villians are going to play the top 10% of hands - at least 1\4 of these are pairs.
[more like a third, if you take the top 10% (win rate for 4 player hands)ie.(17\169)hands. 6 of the 17 are pairs.
6\17=35%]
That being said, you can say with a fair amount of certainty one of the three villians has a higher pair.

You are now more likely to hit your set because two of your opponents cards are now somewhat seen (ie. two cards that you don't need). You could take this principle further and say that no one else is going to have a 2 - even if they didn't hold a pair. (Making 6 cards you don't need). But sticking with just two -
Your likelyhood of hitting your set comes just below 7:1. Now we only need to make up 9 SB post flop. Very easily done.

It is the increased likelyhood of PP in capped flops, coupled with the fact that with AK, you are much more likely to make your hand and still lose and be sharing cards with others to make your hand, that makes me agree with Shillx (I'm sure he's stoked).
Low PP are much easier to get away from - you make your hand or you don't.

On a side note...
BUT now the Villian's higher pair also has a higher % chance of hitting. The chance that both of you will hit your set is 60:1.
BUT in ~50% of cases, Villian's PP is going to warrant staying in regardless of whether he hits or not, increasing his chance of making his set and beating you. Fortunately, the chance of you catching a set (and not quads) and him staying in til the river and making his set is still - 35:1. Maybe worth 1 SB\hand at best.

P.S. If you assume that none of the other six cards the villians holds is a 2 (Very, Very reasonable - even more reasonable than one of them holding PP). Your odds of hitting your set becomes better than 6.5:1!!!!

Food for thought, even if I'm totally wrong.

[DavidC]

[ QUOTE ]
i've been wondering something along those lines w/ small PP in early position, but i'm thinking there's a flaw in my logic.

it seems to me that in a typical loose SS games, an EP limper is going to usually ensure at least 3-4 to the flop and usually 4-6 (no EP limper brings into play the open-raise from later positions from the better players). it also seems that, for the most part, you can get 2-3 to the turn, etc, so it should be all that hard to make up the 8sb we need in order to play something like 44 profitably.


[/ QUOTE ]

I think you should read this post just one more time.

... But I'm glad you've enjoyed your evening. /images/graemlins/smile.gif

[DavidC]

[ QUOTE ]
My thoughts were always that I had to make enough SBs post-flop to make up for what I missed pre-flop. In your case, we'll assume that the others will call, so it's 4-handed to the flop if we call. That means we're paying 3.5 with 13.5 in the pot. As you mentioned, we need ~8:1 for our set. Since we're calling 3.5, we need 8*3.5 = 25.5.

The difference is what we need to make up then, which is 25.5-13.5 = 12SB. So by this idea, we'll need to make up 12SB post-flop. With this much aggression, you can be assured of further betting, so it seems reasonable.


[/ QUOTE ]

You miss the point. I'm speculating here that we would need to make up more than that. Please re-read the post.

[lautzutao]

[ QUOTE ]
OK so my point: sets are not gimmes. I've got about a 66% winrate with sets for the past 18k, which might be running bad (I did have 8k of pain), but still.


[/ QUOTE ]

I did a bit of pokerstovin, and you're not that far off...

Against 5 random hands on the flop you're anywhere from %70-76 to win, depending on board texture complementing your set. Against 4 random hands it's about %78-83.

So you're running a little low but not really that far off. This is a neat post, keep it going:)

[DavidC]

[ QUOTE ]
You are now more likely to hit your set because two of your opponents cards are now somewhat seen (ie. two cards that you don't need).

[/ QUOTE ]

Excellent observation! I'm not going to say that the maniac couldn't have a 2, though. /images/graemlins/smile.gif

[TripleH68]

It is good to work through some numbers even if math is not your strong suit. That said...

Ed Miller says in SSH p53 "small pocket pairs...can afford to pay two bets to see the flop(though they are often more profitable for one bet.)"

Hero is dealt 22 in the SB.

Capped preflop four ways hero pays 3.5sb into a pot that is likely 13.5sb. 3.86:1.
Hero needs to make up a deficit of 12.75sb postflop!

3-bet preflop four ways hero pays 2.5sb into a pot that is likely 10.5sb(with the risk of another raise behind him). 4.2:1.
Hero needs to make up a deficit of 8.25sb postflop.

2-bet preflop four ways with BB folding hero pays 1.5sb into a pot that is 7.5sb(with only the risk of BB raising). 5:1 at least.
Hero needs to make up a deficit of just 3.75sb postflop. A big difference.

Is my simple math correct?

[DavidC]

[ QUOTE ]
Ed Miller says in SSH p53 "small pocket pairs...can afford to pay two bets to see the flop(though they are often more profitable for one bet.)"

[/ QUOTE ]

Miller's a very good player, writer, and communicator of poker theory / practice.

But this isn't something I want to take on authority.

With stuff like pot odds, he shows the math and no one can argue. I don't recall seeing anything highly technical in his analysis of pocket pairs's implied odds requirements in his book.

1) I could be way wrong here, because it's been a while since I've read it.

2) Even if it's true, I'm not saying that his advice is incorrect... I'm just saying that I don't know WHY it's correct, and I'd like to.

[ QUOTE ]
Hero needs to make up a deficit of 12.75sb postflop!

[/ QUOTE ]

Maybe, maybe more... I don't know, do you? /images/graemlins/smile.gif

Aaron seems to be getting closer to the answer, saying that we take our average equity if we hit, and use that to figure out what we really want our implied odds to be.

Another guy points out that our hands win less often vs 5 opponents when we hit than vs 4...

(Now, does that mean that we want 5 players or 4 in the pot when we hit? 5, due to EV, rather than 4, due to equity!!)

Obviously, our equity in the hand depends on the texture of hte board, the action pf, and the number and nature of opponents. Furthermore, our EV in the hand is influenced by those things, but more heavily by the nature of opponents (how they play post-flop).

My problem is that if we play with enough opponents and hit a co-ordinated two-tone board, we could have as little as 50% equity... If we put 2 bets into that vs 4 opponents, then I figure we'd have a pf goal of 16SB (not our money), of which 8SB is already eaten up.

Then on the flop forward, since we own 50% of the bets, and have to recover 8sb, we want to get another 16SB of others' money in there before anyone makes a hand... a little more, because half of our money is going to other players also.

I mean, I could be way off here, but this is the way I see the situation developing.

Edit: highly co-ordinated flops are pretty rare, so take that into consideration.

Edit: Also, my example here is one of cold-calling on the button with 4 ops with a pocket pair.

[ QUOTE ]
[ QUOTE ]
You are now more likely to hit your set because two of your opponents cards are now somewhat seen (ie. two cards that you don't need).

[/ QUOTE ]

Excellent observation! I'm not going to say that the maniac couldn't have a 2, though. /images/graemlins/smile.gif

[/ QUOTE ]

Even if you say - 1\10 times (very generous, i think)that a maniac (in a raised flop of three opponents) has a two. That still means that 9\10 times you can say that no opponents have a 2. ie. 9\10 you know that no one is holding a 2 (6 cards that are not the 2 you need).
This means your odds of making your set are still a lot closer to 7:1 than 8:1.
Even in the 1\10 times he holds a 2, that means maniac is drawing to three outs.

BTW. 1\10 times assumes that you are playing against 3 maniacs. In which case, any one of them holds a two, one in 4 times AND THEN ALL of them play their 2's - 2\5 times.

[Aaron W.]

[ QUOTE ]
My problem is that if we play with enough opponents and hit a co-ordinated two-tone board, we could have as little as 50% equity...

[/ QUOTE ]

Or you can have as much as as 80% equity. It's starting to look as if you're chasing worst cases instead of average cases. While I have no numbers to back it up, I suspect your average equity when you flop a set is around 66% for 4 opponents, which is significantly better than 50%.

[ QUOTE ]
Then on the flop forward, since we own 50% of the bets, and have to recover 8sb, we want to get another 16SB of others' money in there before anyone makes a hand... a little more, because half of our money is going to other players also.

[/ QUOTE ]

Taking the 66% number instead of the 50% number, to get an extra 8SB of your equity into the pot, you need only 12 SB = 6 BB = basically two players seeing the showdown. In a big pot, that's a very reasonable thing to expect. (And you get some padding by picking up flop/turn bets, just in case you only get one to the showdown).

I don't think it's as daunting as you imagine it.

[DavidC]

[ QUOTE ]
Or you can have as much as as 80% equity. It's starting to look as if you're chasing worst cases instead of average cases.

[/ QUOTE ]

Not really:
[ QUOTE ]
Edit: highly co-ordinated flops are pretty rare, so take that into consideration.

[/ QUOTE ]

But yeah, I do tend to dwell on it. /images/graemlins/smile.gif

I should also say that any situation where you get people with overlapping outs, and one of them is willing to pump the pot, you're rocking!

However, I'm curious: in the scenario I described... would i be right or wrong about how big you want the pot to become?

I realize that this is an extreme case, but I want to know if it would be accurate.

Good point about 66% equity giving you 1.5x required from the flop forward. Thanks.

I guess you're right that our average equity is something that we have to guess here. I want to revise the formula a bit, though:

((PF investment * 7.5) - (Others' money already in pot))

* the inverse of our average equity = the amount of other people's money we want in the pot going from flop to showdown in order to break even.

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I think that's the answer we're looking for... and it's tough to say what the our average equity will be exactly, as it depends heavily on the number and nature of the opponents, and pf action before it gets to us (so, average postflop equity for a given preflop scenario).