Trying to sandwich the caller postflop

[Baloosh]

True. True.

I was planning on him not wanting to give a free card either, and since I only had 75 hands on him, probably should not have checked that turn -- if it was headsup with the TAG then I definitely was leading the turn... but I really wanted UTG+1 out of the hand, and thought my best chance to do that was to face him with 2 cold... Hell maybe even MP1 picked up on my psychic thought beam and we had an unspoken collusion to go Check/Check/Bet/Raise against UTG+1.

[MrWookie]

OK, I've got a challenge for you. It's a little tough, but you'll learn a lot doing it, and so will the forum. I want you to figure out the EV of your c/r versus a bet.

1. Put UTG+1 on a range of hands that make sense for his play. Let's assume that he'll never fold a draw w/ 4 outs or more to a hand better than yours with a bet, and he'll never fold a better hand. At the time of your decision, you can't be sure that he doesn't have a straight, remember. I think we can also assume he doesn't raise you w/o a straight if you bet.

2. Let's assume that MP has JJ-AA. He'll definitely call if you bet, but we'll just call the percentage of the time he bets when checked to X.

3. What's the probability that UTG+1 has a better hand, and how does calling down his bet and a call affect your EV? We'll assume he always bets a better hand in this spot. Would he be willing to bet some worse ones? You make the call on that.

4. Now, if you c/r, we'll assume he'll fold any draw w/ 5 outs or fewer to a better hand. How much does this increase your own equity in the pot (both %, and BB's)? How often does he have such a draw?

5. OK, so what does X have to be in order for a c/r to be +EV?


This is a lot of work, but I really want to see it. Working through all this will really benefit your understanding of the math behind the game, and everyone who reads it will benefit. I'm starting to think this play might be dead sexy, but I want to see the numbers that support it.

[Baloosh]

I wouldn't know where to begin. Can someone point me in the right direction to get me started?

1. UTG+1 can have a wide range of hands... I really could put him on low pair with any kicker all the way up to top pair, medium kicker... gutshots just got there on the turn... Ace-rag? So narrowing it down to "a draw w/4 outs or more" puts him on any pair 6x, 7x, 8x, 9x. Ax is folding, where his x in Ax is not a 6 7 8 or 9. JQ is in, as is 34.

2. I'd go so far as to say that if MP has specifically AA, he'll 3-bet the turn if UTG+1 drops, but just call my c/r if UTG+1 calls two cold. If UTG+1 calls the turn, MP1 and I are both probably behind... and it's a definite if UTG+1 suddenly c/3-bets my c/r on the turn.

3. I don't think UTG+1 is betting a worse hand. He's along for the ride with something, but in reality about half of what I can put him on got there on the turn. So if he bets the turn, he's got a better hand. If he checks he's either still drawing or so passive as to just check/call with a winner. I'm lost on how to calculate the probability that UTG+1 "has a better hand." Help?

4. I'd have to figure out how to do #3 before I can do this one.

5. Like I said, I don't know how to do the actual numbers, but I'd love to try it if someone can help get me started. My instinct tells me that X has to be pretty low, since anytime he bets the turn after I check he's made his hand (which beats mine)... so unless this is one of those Tommy Angelo "poker is counter-intuitive" exercises I think it's close, and by that I mean less than .5BB.

But that's just me pulling numbers out of my ass.

[MrWookie]

OK, working out the probability that you're either ahead of or behind UTG+1 isn't too tough. First off, you enumerate his possible hands (yeah, there are a lot) and count up the number of ways he can have each one. For example, there are 16 ways he could have AK if we put that in his range, but since you're looking at 2 K's, then there are only 4x2 = 8 ways for him to have AK. Then, you add up the numbers, and take the number of hands that beat you divided by the total number of hands, and 1 minus that for the probability that you're ahead.

Also, poker stove (google it) may help you w/ some of the equity calculations.

[Baloosh]

Roger that. So enumerating for all his possible holdings:

6x would be:
62 16 ways
63 16 ways
64 16 ways
65 16 ways
6T 16 ways
6J 16 ways
6Q 16 ways
6K 8 ways
6A 16 ways
Total of 136 ways to have 6x.

Repeat for 7x 8x and 9x.

4*136 = 544 possible hands with a single pair and any other card to hit his 2 pair. Subtract out his 6K hands (and 7K, 8K and 9K) since that will give Hero a set which beats UTG+1 2 pair. 544-32 is 512 hands.

AK is a possible hand. Like you said, 8 ways for that one. Back to 512 total hands.
AQ is possible. 16 ways. 528 hands.
AJ is possible. 16 ways. 544 hands.
QJ is possible. 16 ways. 560 hands.
34 is possible. 16 ways. 576 hands.
A-rag is out, fewer than 4 outs, with the exceptions noted above.

Did I miss any? Let's include the times UTG+1 flops a set and decides to call it the whole way.

pockets 6s, 7s, 8s and 9s are all 3 ways each, correct? 12 more hands. 588 total hands.

I'm ahead of all of those except the last 12, so I'm ahead of 576 hands out of 588. 1 - (12/588) = 0.979591

I have myself ahad of UTG+1 about 98% of the time, so I must be doing something wrong. Based on this range, I want him in there calling a bet if these numbers are correct.

But if he bets the turn, I'm more than 98% sure I'm beaten. So if I check the turn and he suddenly bets, I'm folding, regardless of what MP1 does.

What have I missed so far?

[MrWookie]

OK, good start. However, this isn't correct. First of all, there aren't 16 ways to have 62. There are only 12, becase you're looking at 1 of the 6's. This means there are also only 6 ways for him to have K6.

Next critcism: we need to rule out hands that are unreasonable given preflop action. Even the most passive players will raise their good A's, so I think AK and AQ are out. This guy may be loose passive, but even loose passive players don't always limp 62 UTG. I'd rule out 62-64 an T6, and similar for the 7x, 8x, and 9x hands. What was this guy's VPIP? You may want to rule out even more hands than that.

Lastly, I think 34 is out. The "outs" that this has are only to a chop.

Also, you're willing to give him a set, but not two pair? And what about Tx and 5x? Speaking of which, some of your "lone pair" hands are actually straights.

[Baloosh]

Sonofabitch. I jacked this one up pretty good. Fixing:

6x would be:
62 12 ways
63 12 ways
64 12 ways
65 12 ways
6T 12 ways
6J 12 ways
6Q 12 ways
6K 6 ways
6A 12 ways
Total of 102 ways to have 6x. Repeat for 7x 8x and 9x. 408 total hands. Of which I'm beating all but the 65 and 6T hands (likewise the 75/7T 85/8T and 95/9T). So 24*4 hands beat me out of that group (96 hands) and 312 hands I beat.

AK and AQ are out. AJ in. 16 ways I beat.

96 beat me and 328 I beat.

62 through 64 is out, and 6T (repeating for the 7x 8x and 9x). So 144 hands are removed from the hands I beat and
48 are removed from the hands that beat me.

48 beat me and 184 I beat.

I didn't have any stats on him at the time -- haven't played with him enough, just my read on him given the time at that table so far.

If I check the turn, he's probably betting with a Tx and checking with the ass-end of the straight with 5x. A raggedy two pair (68, 78, 79) is definitely possible:

67 9 ways
68 9 ways
69 9 ways
78 9 ways
79 9 ways
89 9 ways

54 more hands that beat me.

New totals: 102 hands that beat me and 328 I beat. 12 more hands that beat me for his set hands yields 114 hands that beat me and 328 I beat.

74.2081% is the probability I'm ahead given this range. So I can expect to win ~.75BB for each bet he calls.

If he calls one bet, I can expect ~.75BB, if he calls two bets I can expect ~1.5BB, and so on.

Given that he'll only raise my turn bet when he has a straight... uhhh... when I bet on the turn and get raised I lose 1BB. If I check-raise and he folds for 2 cold, I get 0BB. If I bet and he calls, I can expect .75BB. Those times he calls two cold (does he ever?) I can expect 1.5BB. We'll call that occurence negligible.

.75BB > 0BB > -1BB.

Should have bet the turn.

[MrWookie]

This is another good attempt, but you're still not there. First of all, your EV calculation is off. You didn't take into account MP, and that makes a difference. What's the probability that you're ahead of MP? How often do you think he calls our bet? Bets (our variable X) and then calls our check/raise after he bets?

Even if you had taken this into accout, though, you'd still be wrong. You see, we're not at the river. There are still cards to come. Even if we're beating UTG+1 and MP now, we might not be at the river. If we assume that UTG has a 5 out draw to beat you, how often does that draw come in? How big a share of the pot does that draw represent? If he calls, he maintains that share of his equity. If he folds it, that equity is redistributed to you and MP depending on your relative probability to win the hand. When the pot is this big, it may be more +EV for him to fold his share than it is for him to put more money in when he calls your bet. THIS is what I want you to work out next.

[Baloosh]

The probability of me being ahead of MP1 depends on how he reacts to my check-raise. At least to me, if he 3-bets me again, he's either got Aces or Kings (only one way to have Kings). He's got an overpair, and against his range I'm only losing to 6 ways of AA. All the JJ, QQ and one way of KK I'm ahead of (or chopping the one way to KK).

With the overpair, I'd expect him to go into calldown mode (if he doesn't 3-bet the turn).

Okay.. now for the hard part. Assuming UTG+1 has a 5-out draw:

Versus me and MP1 headsup:

Text results appended to pokerstove.txt

836 games 0.005 secs 167,200 games/sec

Board: 8d 6s 9h 7s
Dead:

equity (%) win (%) tie (%)
Hand 1: 60.0478 % 50.24% 09.81% { KcKd }
Hand 2: 39.9522 % 30.14% 09.81% { JJ+ }

With UTG+1 in there with a 5 outer:

Text results appended to pokerstove.txt

72,576 games 0.031 secs 2,341,161 games/sec

Board: 8d 6s 9h 7s
Dead:

equity (%) win (%) tie (%)
Hand 1: 49.5122 % 42.86% 06.66% { KcKd }
Hand 2: 32.5479 % 25.60% 06.95% { JJ+ }
Hand 3: 17.9398 % 13.14% 04.79% { A9o-A6o, Q9o-Q6o, J9o-J6o }

My equity goes from about 50% to about 60%, if UTG+1 drops out. that's a 20% increase, so with UTG+1 dropping out on the turn I can expect to take in 20% of 13.25BB (the size of the pot before the turn action) = 3.3125BB. If UTG+1 drops, my equity increases 20%, while MP1's equity increases from 32.5% to 40%, a 23% equity increase. I've improved MP1's share of the pot by 3% more than mine by folding UTG+1. (I assume this is because of his equity increase to a chop).

Again, this is all assuming that UTG+1 doesn't already have a straight with 5x or Tx. But if I check the turn and he then bets, I can fold with almost 100% certainty I'm drawing dead (to a chop).

So what happens when UTG+1 calls my two cold? There are 16.25BB in and it costs him two. His 5 outer will hit on the river 11% of the time, or 8:1 odds to hit. It would appear that his cold call (if he has 5 outs) is EV neutral. He'll hit 1 in 9 times to win ~$0 over the long run. The extra .25BB in there will probably end up going to the rake after hand completes.

11% of the time he hits and wins 16.25BB (+$32.50)
89% of the time he misses and loses 2BB (-$4)

(32.50 x .11) + (-4 x .89) = 3.575 + -3.56 = 0.015BB. With implied odds (at least one of us is going to make a crying call on the river) this leans towards a pretty certain call for UTG+1 if he has 5 outs and it's 2BB back to him. If he had 5 outs or more, he made a FToP mistake by folding to my check-raise.

[MrWookie]

Now we're getting somewhere. However, I'm going to keep nitting at you. Your equity doesn't increase 20%. Well, it does, but that's not the % increase we're looking at. We're gaining an additional 10% share of the pot if UTG+1 folds; MP1 gets an addtional 7.5%. If we c/r and get UTG+1 to fold, we gain 1.625 BB in equity above and beyond what we would gain from additional bets going into the pot with our equity edge. I actually think UTG+1 will fold a single pair to our c/r fairly often. He's not necessarily thinking about what you have. He sees a terrifying board, a screwplay from you, two bets to him, and a pair of 6's in his own hand. That's not the most fun situation in the world.

So, now we've got to put it all together. We need to calculate the change in equity gained by folding UTG+1 againt the entire subset of his range that is a worse hand than ours. Next, we need to assemble the probability of UTG having a worse hand and checking, having a better hand and betting (you say we fold every time to his bet) and the probability of MP1 betting into one big ol' EV calculation for a c/r. Then we need to do a related calculation for the EV of betting. Let's see what you come up with, and I'll be back to pick it apart.