Pocket Pairs and Implied Odds

[atnels]

I’ve been thinking about this for the last week or so. When I first started playing limit poker about 9 months ago (when I was a charter member of the weak-tight club), I would fold any pocket pair lower than TT to a preflop raise. Then, after reading SSHE for the first time and starting to lurk around 2+2, I began to understand implied odds and began to play pocket pairs for 2 bets when I knew there would be at least 3 others in the pot. However, I’m starting to wonder if this is a winning play in the long run; especially at the loose/passive games I see so much at Party .5/1.

With a pocket pair, you’re looking at 7.5 to 1 to flop a set. Of course, a set alone won’t win every pot so I think I remember seeing somewhere else that 9 to 1 is a common estimate for implied odds calculations. If you see the flop for 1 SB with 3 other limpers, the pot is 4 SB. In order to make playing a pocket pair trying to hit a set profitable, you’ll need to make up 5 more SBs somewhere along the line - not too tough, even in very passive games. Of course, your hand may be best in this situation and you could win unimproved or through your shrewd postflop play.

Raised pots pose another problem. Take the scenario where you hold a mid pair in EP. UTG limps, you limp, then there are folds to the Button who raises. SB folds, BB calls, and UTG calls. This is leaves an ‘easy’ call for you, as conventional wisdom dictates you are closing the action for one more SB. The pot now contains 4.25 BB. So, you’ll need to make up 5.5 BB (4.75 BB + .5 BB to account for the rake, rounded up to an even SB). This needs to be OTHERS bets, not ones you put it - so, we’re not just talking about a 10 BB pot, it will more realistically be at least 13-14 BB including your own bets/raises.

This is where I started to wonder. To make the play worthwhile, you need to drag a large pot, on the order of 15 BB. At Party .5/1 pots larger than $13 are definitely common, but far from the rule. Lots of loose passive players will see the flops, but back off when someone shows aggression or just dump their hands when they miss. To make up 5.5 BB in the example, you’ll need probably 2/3 other players who saw the flop to hang out until the river.

Another common situation I’ve seen is when you limp with a low PP and it is raised and reraised behind you, but there are lots of players coming along. Many 2+2 posts have said calling here is okay because if you hit your set your implied odds will take care of the investment. I thought I would try to quantify this to look at these situations, in addition to seeing the flop with a pocket pair for a preflop cap (4 SB).

In each situation, I assumed that the PP is played in a purely speculative fashion - you’re folding if you don’t hit your set on the flop. Also, I used Party’s low limit rake structure to account for the rake losses (Rake is .5 BB when pot exceeds 5 BB, then an additional .25 BB when it exceeds 12 BB). The applicable rake is shown in parentheses.

0.25 BB is rounded up to 0.5 BB.

Lastly, I tried to account for how “hard” it would be to achieve the necessary pot size needed. To do this, I calculated what I call a “Player Contribution Factor,” or PCF. If all players pay 1 bet each to see the turn, river, and showdown, this is 2.5 BB. This hardly ever happens, but provides a rough quantization of the fact that more players provide more money in the pot.

I couldn’t come up with a way to quantify the aggression of a particular game. Clearly, sets play the best in conditions where you can get as much money into the pot as possible and many players will remain in the hand.

The results:

Paying 2 SB, need to drag 9 BB.
4 players: flop pot = 4 BB (0), need to make up > 5.5 BB (.5). PCF=73%
5 players: flop pot = 4.5 BB (.5), need to make up > 4.5 BB(0). PCF=45%
6 players: flop pot = 5.5 BB (.5), need to make up > 3.5 BB (0). PCF= 28%
7 players: flop pot = 6.5 BB (.5), need to make up > 2.5 BB (0). PCF= 17%

Paying 3 SB, need to drag 13.5 BB.
4 players: flop pot = 5.5 BB (.5), need > 8.5 BB (.25). PCF= 113%
5 players: flop pot = 7 BB (.5), need to make up > 7 BB (.25). PCF= 70%
6 players: flop pot = 8.5 BB (.5), need to make up > 5.5 BB (.25). PCF= 44%
7 players: flop pot = 10 BB(.5), need to make up > 4 BB (.25). PCF= 27%

Paying 4 SB, need to drag 18 BB.
4 players: flop pot = 7.5 BB (.5), need to make up > 11 BB (.25). PCF= 147%
5 players: flop pot = 9.5 BB (.5), need to make up > 9 BB (.25). PCF= 90%
6 players: flop pot = 11.25 BB (.75), need to make up > 7 BB (0). PCF= 57%
7 players: flop pot = 13.25 BB (.75), need to make up > 5 BB (0). PCF= 33%

I’m not quite sure the correct way to interpret these results. I’d like to coordinate actual game statistics to look at, for example, the average pot size with four players with a preflop raise, but my own personal database only has ~15k hands (thanks to my old hard drive crashing) so the results would be trivial without a much, much larger sample to draw from.

My theory is that lots of players are playing speculative hands - not just pocket pairs but they are the easiest to look at - in small pots without seeing returns in the long run due to dragging smaller pots than probability requires. I’m not sure what the solution is, because folding pocket pairs to a single raise even 4-handed seems too weak tight to me.

If you’ve read this far, that’s impressive. Everyone’s opinions and criticism of either particulars in here or the concept in general are welcomed and appreciated.

Thanks!
Andy

[adsman]

You obviously put a lot of time into your post. I goota admit, I didn't get to the end. I just jumped into my PT stats to find some incredible data that would sever you off at the knees. Instead I discovered that all my small pockets up to 88 are losers, at around an average of 0.25BB per hand. 99 and above are all major winners. So maybe you've got something. I only flopped one set today and nobody paid me off, (6BB for the hand). Interesting. As everybody will say, "sample size!!" but I'm going to keep an eye on this.

[SlantNGo]

[ QUOTE ]
Raised pots pose another problem. Take the scenario where you hold a mid pair in EP. UTG limps, you limp, then there are folds to the Button who raises. SB folds, BB calls, and UTG calls. This is leaves an ‘easy’ call for you, as conventional wisdom dictates you are closing the action for one more SB. The pot now contains 4.25 BB.

[/ QUOTE ]

When you limp with a PP in EP and it's raised behind you, that SB you put in already is no longer yours; it's part of the pot. So getting 7.5:1 back to you, it's an easy call not only because you are closing the action for 1 SB, but because you're 8:1 to hit the set. You don't need much in terms of implied odds to make this call.

[ QUOTE ]
Lastly, I tried to account for how “hard” it would be to achieve the necessary pot size needed. To do this, I calculated what I call a “Player Contribution Factor,” or PCF.

[/ QUOTE ]

I think you've got the right idea, but a better way to look at it is how many others need to contribute the 2.5 BB rather than the percentage factor, for example:

Paying 2 SB, need to drag 9 BB.
4 players: flop pot = 4 BB (0), need to make up > 5.5 BB (.5). Need 5.5/2.5 = 2.2 callers
5 players: flop pot = 4.5 BB (.5), need to make up > 4.5 BB(0). Need 4.5/2.5 = 1.8 callers
6 players: flop pot = 5.5 BB (.5), need to make up > 3.5 BB (0). Need 3.5/2.5 = 1.4 callers
7 players: flop pot = 6.5 BB (.5), need to make up > 2.5 BB (0). Need 2.5/2.5 = 1 caller

And this doesn't account for one of the raisers hitting top pair when you hit your set, in which case, you'll most certainly extract more than 2.5 BB.

[Saint_D]

One other thing you don't take into account is the times that your pocket pairs win the hand without improving. If you play them with some skill after the flop, they can make up a little ground.

hole cards 7 [img]/images/graemlins/spade.gif[/img] 7 [img]/images/graemlins/heart.gif[/img] on the button. 4 players limp to you and you limp.

FLOP:
A [img]/images/graemlins/heart.gif[/img] A [img]/images/graemlins/spade.gif[/img] 6 [img]/images/graemlins/heart.gif[/img]

Everyone checks to you. You play the hand with some skill based on reads and win.

Somewhere around 2% of your pocket pairs will be winnable without hitting the set. Be sure too keep a close eye on the flop and the action when and don't just automuck when you miss the set.

Now if only I played my own pocket pairs with this much subtlety...

-D

[atnels]

I'd like to see if this gets any more discussion today than it did yesterday with a different crowd. If it doesn't this time, I'll take it as a hint that the idea doesn't have much merit and let it die.

[Aaron W.]

[ QUOTE ]
With a pocket pair, you’re looking at 7.5 to 1 to flop a set. Of course, a set alone won’t win every pot so I think I remember seeing somewhere else that 9 to 1 is a common estimate for implied odds calculations. If you see the flop for 1 SB with 3 other limpers, the pot is 4 SB. In order to make playing a pocket pair trying to hit a set profitable, you’ll need to make up 5 more SBs somewhere along the line - not too tough, even in very passive games. Of course, your hand may be best in this situation and you could win unimproved or through your shrewd postflop play.

[/ QUOTE ]

If you can find that 9:1 number somewhere, it would be helpful. It seems a little conservative at first glance. And you're talking about playing for ONLY a set, and not on any intrinsic value of the pair itself, right?

[ QUOTE ]
Raised pots pose another problem. Take the scenario where you hold a mid pair in EP. UTG limps, you limp, then there are folds to the Button who raises. SB folds, BB calls, and UTG calls. This is leaves an ‘easy’ call for you, as conventional wisdom dictates you are closing the action for one more SB. The pot now contains 4.25 BB. So, you’ll need to make up 5.5 BB (4.75 BB + .5 BB to account for the rake, rounded up to an even SB). This needs to be OTHERS bets, not ones you put it - so, we’re not just talking about a 10 BB pot, it will more realistically be at least 13-14 BB including your own bets/raises.

[/ QUOTE ]

Once you've limped in, you're calling the raise because you're now comitted to seeing the flop based on the immediate odds given to you. If you could look into the future and see that it was going to get raised, then you would be more inclined to fold the first time through. But lacking such clairvoyant skills, you're stuck calling and calling the raise.

[ QUOTE ]
Another common situation I’ve seen is when you limp with a low PP and it is raised and reraised behind you, but there are lots of players coming along. Many 2+2 posts have said calling here is okay because if you hit your set your implied odds will take care of the investment. I thought I would try to quantify this to look at these situations, in addition to seeing the flop with a pocket pair for a preflop cap (4 SB).

[/ QUOTE ]

Again, you need to take into account the immediate odds you are getting. Those immediate odds (which tend to be something like 5:1) balanced with the implied odds you will get on future action give you enough reason to call.

[ QUOTE ]
In each situation, I assumed that the PP is played in a purely speculative fashion - you’re folding if you don’t hit your set on the flop. Also, I used Party’s low limit rake structure to account for the rake losses (Rake is .5 BB when pot exceeds 5 BB, then an additional .25 BB when it exceeds 12 BB). The applicable rake is shown in parentheses.

<< A BUNCH OF NUMBERS >>


[/ QUOTE ]

When I look at those numbers, it seems you are being too hard on Hero. Again, this is because you are not taking into account the immediate odds Hero has when he makes his preflop call (plus you've never acconted for 'bonus' money from the blinds when they fold). Here's a specific example:

[ QUOTE ]
Paying 2 SB, need to drag 9 BB.
4 players: flop pot = 4 BB (0), need to make up > 5.5 BB (.5). PCF=73%

[/ QUOTE ]

If Hero is trying to cold-call two bets, then your numbers make sense. But if Hero limped UTG with 77, MP raises, both blinds call, then Hero is looking at a 7:1 call. He's only short 2 SB of the 9:1 number you gave at the top. This means his decision *NOW* is whether he believes he can make up the 2 SB he's short *NOW* in postflop play. This is a no-brainer call.

[ QUOTE ]
My theory is that lots of players are playing speculative hands - not just pocket pairs but they are the easiest to look at - in small pots without seeing returns in the long run due to dragging smaller pots than probability requires. I’m not sure what the solution is, because folding pocket pairs to a single raise even 4-handed seems too weak tight to me.

[/ QUOTE ]

Lots of players are playing speculative hands under poor conditions. Limping UTG with 22 in loose passive games (5-6 players seeing a flop) is fine. But when players find themselves at tighter tables or against better opponents, those limps become less profitable. There are a few players here who can still play those hands profitably, but it requires much more postflop skill than no-set-no-bet.

[GrunchCan]

antels -

Good post. Just the fact that you've though about this as much as you have is +EV in itself.

Here's the rule I go by for every PP 88 and lower: postflop I need to make up 10 times whatever I put in preflop. This is becasue flopped sets will win about 75% of all pots, and you'll flop a set about 1:7. Add in a rake factor and I come up with a (possibly conservitave) 10 times preflop investment. I look at the game conditions and decide based on that if its feasable or not.

Your tables of computations seem pretty close to right. But how are you going to use them during play? For one thing, its a lot to recall. For another, you don't account for the nature and position of your opponents - only the number of them.

So deciding weather or not you can play a PP is certianly a lot of science - but its also some art, too.

There are a few things to consider when deciding if you can extract the needed 10 X PF investment from your opponents.

- Fewer opponents means less chance you can breakeven (duh).
- Opponents who are tight postflop make it harder to breakeven

- Opponents who will showdown with virtually any hand make it much easier to breakeven
- Opponents who are aggressive postflop might make it easier to profit if they come with players who will tend to call

Postitional considerations:
- Closer to the button is better.
- Closer to UTG is better than being in the middle, becasue its hard to trap for two from the middle
- If you have a LAG UTG and you are on the button with 2 or more people trapped between, its very easy to make up 20 SB postflop. Trap the loose-passives in the middle on the turn for 2. If the LAG 3-bets, cap.

[atnels]

Thanks for some valid points. I should have specifically said this, but I was thinking more about the purely-speculative pocket pairs like 22-55 that typically need to improve to win a multiway pot. Also, I am not talking about shorthanded play here at all, where a made pair becomes stronger. I feel that when you take these factors into account, an unimproved pocket pair will go to showdown and win in a multiway, raised pot only a small percentage of time.

When I was initially considering this problem, I decided to simplify by assuming the blinds were completing to see the flop. While it is clearly not a realistic simplification, I don't think it is that far off when you are considering a table with weak/tight postflop players. Poor players complete in the blinds with lots of junk, especially against only one raise.

I need to consider the immediate odds and the blinds in my next round of attempted calculations.

Thanks again,
Andy

[walkdoc]

I play a lot of Turbo Texas holdem at limit. Invariably the AI from the software tells me to fold anything under 88 or 77, unless I'm in late position. I initially had success playing my pp's on the program, but lately have gotten no where that way. I'm not sure how, but will look into running some hands w/ PP's from 22-77 to see how they hold up. The software is pretty cool as it will allow you to run thousands of hands, w/o you playing and see the statistal outcomes of those hands. Obviously this is AI, but stats are stats, and we can't always bully our way around w/ pocket 2's. ;-)
Cheers!

[atnels]

Thanks Grunch.

Application: Hero holds 22 in MP1, folds to you in slightly tightish table. No maniacs. Lots of calling stations preflop, but it's tough to get any action postflop unless someone has a big piece of the flop or really nice draw. You have only been at the table for a short time and haven't seen many decent cards and mostly folded - you don't have the image of a LAG or anything.

First question: Fold, call, raise?

Second question, assuming you limp: Folded to CO who raises, BB calls. Pot is 5.25 SB back to you. Action?