Some specialized draw plays

[bigpooch]

In the last month or so, I have visited the old haunt of the
Paradise draw games. When I returned I was surprised to see
so many tables during the evenings (not quite the way I
remembered it when I played it last year!). I thought I
would share with you some technical hands that happened
lately that were kind of interesting (the ones I remembered
offhand) in a quiz format. I'll post the answers shortly if
there is any interest at all!

BTW, I'm not sure if the spot cards are correct, but it
really doesn't matter (it's not bridge!).

Some Limit draw questions:

1) The limp: I was in the cutoff with KKKAA and decided to
limp and was heads-up with the BB (not a success) who now
draws three. (I had another situation that was virtually
identical.)

2) The hazards of 7-/8-/9-tabling: I must have been on
PStars as well as a Party skin when I accidently reraised
an utg raise with AAK54 in the cutoff and got heads up with
the opponent drawing three. This is bread and butter for
Gardena players.

3) Did you do your homework? I raised utg with AAKQ2 and
get heads up with the SB (who is fairly solid) who draws
three.

4) Trouble at the river's edge: You got reraised and are
up against someone standing pat and are heads up. Are you
better off drawing two to a straight or flush than drawing
to your longshot boat (or quads)? Let's assume your lone
opponent isn't snowing!

5) Posting? A player posts and it's folded to you in the
SB after the poster checks his option. How liberal are you
in raising here?

6) 12 or 15 outs: You get to have fun when it's four (or
more) players that will take the draw, so when do you jam?

7) Pat with position: It's capped before the draw and your
heads up with the BB and you both have pat hands. Now your
opponent leads out (say he's not hopeless), so what's your
minimal raising hand? (The worst hand that would make it
technically correct for a raise to be +EV; don't forget
that the BB could reraise!).

[schubes]

1) Open limping makes me feel sick. I'd probably discard 1 ace, limping and drawing 2 looks more conspicuous. If I raised I would draw 1 or 2 depending on other factors (If it gets capped I stand pat even if they draw 3).

2-3) I'd keep the K both times. Wiesenberg discussed a hand in CP where he kept both K and Q (the "homework"?), but in that case his opponent drew 2 not 3.

4) Draw 3 to the full

5) I think my mininum would be TT-88 depending on how tight BB is.

6) 12 outs you're winning your fair share with only 3 opponents, but it's pretty thin. The possibility of opponents dropping and maybe even rake suggests against jamming.
With 15 you'll hit almost a third of the time, so jamming 3 opponents looks better. If you can bluff after the draw raising only 2 players looks good.

7) If they will bet any pat hand oblivious to you I'd raise a medium flush or better. Otherwise I'd need to have some idea of their betting requirements. I'm not sure, but I'll guess that the minimum raising hand against an optimal player who value bets correctly is an A or maybe K high flush. Note there are a few rocks in these games who won't bet without a full.

Disclaimer- I'm not quite comfortable answering a draw quiz where your opponents' strategies and your table image aren't discussed. Although I can understand how these small factors become less important when you play an absurd amount of tables. [img]/images/graemlins/tongue.gif[/img]

[TheShootah]

1. I would toss one, faking some kind of draw.
2. Toss the 54, hoping to check it down I think.
3. Why not just toss all three. Maybe just toss the deuce....I don't know about this one.
4. Draw three.
5. I think pairs 99 and up...
6. I am too lazy to do the math, but 12 seems like enough against 4 guys...
7. Ace High Flush is my guess....I did no math....

PS - I don't claim to be a decent draw player at all! So be gentle.....

[crabbypatty]

1) no joker correct? best play to try and get a call now is discard one of your aces and bet after draw.

My question for you. why would you limp in this situation? this is open or fold correct? someone could very likely make it 3 bets after you and then you could cap it(if you are expecting them to raise for you anyways). at least you should get one call and as long as it is not the button you still have option to make the same play depending on cards that they draw. playing it the way you did seems like a <font color="red"> negative ev </font> <font color="black"> </font> play to me.

2) seems like there are 2 options for this one.
You can discard the 54 and draw 2 (i would bet after draw no matter what, if checked to me) or you can rap pat and show it down.

3) feels a lot like #2. you could hold AAKQ but, there isn't much point (only if he drew 2) if you knew he would only call with AA or better (have to define solid player) and check it down. you could also hold only AAK and then play it like #2 (more action, but will have higher variance) if this solid player will call you with other than AA then you should draw 3 and try to improve and raise if he improves also. if you know this solid player will bet out unimproved after the draw often then one of the first 2 options would be best. you could even draw 3 and and try to induce a bluff from KK or worse.

4) draw 2 to the flush(hopefully) wouldn't want to draw to a straight here but, still better than trying to fill.

5) would have preferred to be in bb but, i would probably raise with 99 or better(depending on propensity of bb to call) about 1/2 the time one of them will have a pair and about 1/2 of those times they will have a pair bigger than 99. so 1.5 bets to pick up 2.5 hope full half of the time, go for it.

6) with str8 flush draw and getting this many players to call you should raise every single time you are getting correct pot odds with a 12 way draw (will make complete hand 1 time in 4) and 3 callers. It is only gravy with a 15 way draw (will make your hand 1 time in 3). these are fairly rare hands that will also give you a wilder [img]/images/graemlins/crazy.gif[/img]image at the table if you jam with them.

7) since your oppenent is not totally hopeless i would prefer to have a Q or K high flush to raise, correct answer is probably J high but, since i am worried about the reraise i prefer to kick it up a notch [img]/images/graemlins/smile.gif[/img]

[schubes]

[ QUOTE ]
4) Draw 3 to the full

[/ QUOTE ]

I missed a decimal point in my math. I have drawing 2 to a flush at 4.2% (not .4), and 3 to a full or quads at 1.3%, which makes more sense.

I still wouldn't draw to a low flush or a straight.

[bigpooch]

First of all, thanks to crabbypatty (where from?), schubes,
and TheShootah for their responses.

Second, the games discussed don't have a bug/joker and have
two blinds as in LHE to the left of the button.

Answers:

1) Discarding an ace is best. Your opponent is a 286-1
shot to outdraw your trips and you can get a suspicious
opponent to think you missed your draw. Also, in answer to
crabbypatty, limping in this spot is clearly dependant on
the field. If the blinds defend tenaciously or the button
cold calls with hands like shorts, cathops or draws, you
should punish them with a raise. On the other hand, if your
opponents are quite reasonable and won't often play back
with hands like medium 2-pairs, it is much better to just
limp even though it looks suspicious. Besides, there are
some hands that do seem to play better by limping than
either folding or raising (but I won't say but perhaps you
can think of them!).

In the situation where the players are reasonable to tough,
you can only convince yourself after dealing out this
situation a few hundred (or for the doubting Thomas, a few
thousand) times. One of the most common situations is that
the SB completes for another 0.25 BBs and another common
occurrence is that the button limps or plays a flush draw.
A big chunk of the equity comes from the SB completing and
that swamps other less common situations by a big margin.
I estimate that limping in this spot rather than raising is
between 0.1 and 0.2 BBs better against reasonable but not
pathetic opposition.

I had almost another identical situation with AAA22 and here
the limp is a tiny bit better since I have only 3 key cards
taken up in the hand as opposed to five in the previous one.


2) Quite standard to draw two to AAK since your opponent is
supposedly holding AA or KK according to "Wiesenberg 101".
If you didn't know this, back to the basics! [img]/images/graemlins/smile.gif[/img]

BTW, this is normally a fold in the 3-6 game (do you see
why?).

3) Although schubes mentioned that Wiesenberg mentioned
this in CP, it also comes up in Malmuth's "Winning Concepts
in Draw and Lowball" on p.360 "Examining Kickers". Bear in
mind that the hand discussed there had a deck that includes
a bug.

First of all, your opponent better have AA or KK because if
there is any chance at all he has less, he's blowing some
equity with those "shorts" in this spot and if that were the
case you should draw three because of the postdraw action.

Although drawing one to AAKQ maximizes your chances of
holding the best hand after the draw, I did a little bit of
homework to see if that would translate to a better EV since
the postdraw action is also important. My conclusion was
that drawing one to AAKQ was a tiny bit better from an EV
calculation than drawing three against the typical tough
opponent.

4) If you draw three to a pair, your odds of making a boat
or better is around 210/16215 = 0.01295 (I don't know if
in reality it's higher or lower due to another hand being
pat!). Also, you'll be drawing to a pretty big pair, so
when you get lucky, you have a comfortable raise. Of
course, even if you make Kings full, there is still about a
2% chance your opponent will have better! Now, if you do
raise, some of your opponents will muck their straight
since you "obviously filled or made quads", in which case
you don't get that extra bet all of the time, plus there is
that 2% chance they'll come over the top and you'll be
staring at the nuts.

If you have a three flush, even if it's a small one to draw
to, your chance of making it is a bit higher than C(10,2)/
C(47,2)=45/1081 or about 0.04163 (the reason it's higher is
that often the pat hand is a flush in a different suit) and
you often have a big flush when you make it so you can make
a comfortable raise. Clearly, you should draw to a three
flush if you have one and you don't even have to even raise
unless your flush is king high or better.

What about drawing two to a big straight? (Obviously, you
will draw three to a hand like KK753 which will become
clearer later). You will have either a one-way, two-way or
three-way straight draw (the last kind will be with a hand
like QQJT2) and the probability of completing your hand will
be either 16, 32 or 48 out of 1081. When you complete your
hand, you can't raise (except with AKQJT against opponents
that will always pay off, even with a wheel), so you can
forget about the one-way draw. In the other cases, they are
reasonable since you won't be raising and will be getting
there more than twice as often as compared to drawing three.
The downside is that you'll often end up with something like
QJT98 and you'll just lose another bet. I haven't done the
dirty work for equity calculations but I would guess that
you would draw three to your pair unless you had a QJT hand
to draw to, in which case you would want to draw to that.
Your chance of making this draw is around 0.0444 so you
should be happy with that since you win about 40% of the
times you make it.


5), 6) and 7) later! (Time for dinner!).

[schubes]

[ QUOTE ]
If the blinds defend tenaciously or the button
cold calls with hands like shorts, cathops or draws, you
should punish them with a raise. On the other hand, if your
opponents are quite reasonable and won't often play back
with hands like medium 2-pairs, it is much better to just
limp even though it looks suspicious. Besides, there are
some hands that do seem to play better by limping than
either folding or raising (but I won't say but perhaps you
can think of them!).


[/ QUOTE ]

I really think you're approaching the game wrong on this point. If it's not "reasonable" for someone to 3-bet you with a medium 2 pair when you raise from CO you are playing way too tightly - getting 3-bet by a big pair isn't unusual for me. If you are not playing too tightly, and are not dealing with a strange sort of clueless loose players who turn weak-tight after a raise, I am quite certain raising is more +EV than open limping with this hand (and almost every playable hand).

[ QUOTE ]
Although drawing one to AAKQ maximizes your chances of
holding the best hand after the draw, I did a little bit of
homework to see if that would translate to a better EV since
the postdraw action is also important. My conclusion was
that drawing one to AAKQ was a tiny bit better from an EV
calculation than drawing three against the typical tough
opponent.


[/ QUOTE ]

I'm not sure if I agree with this one either. Let's say SB has AAxxx. Drawing one gives you 6/45 = 13.3% to improve, but drawing two gives you 3/45+42/45*6/44 = 19.4%. The percentages would be even more in the favor of drawing two if SB sometimes has KKxxx, because of the possibility of making trips. However, it's true that by drawing two you give SB an extra chance to draw a better AAK while you don't improve, and drawing one gives you an edge in those situations where you both make aces up. But if these small factors really do swing the percentages to drawing one's side, it can't be by much.

So the real test is play after the draw. Again your "typical tough opponent" may be different from mine, but when my opponent checks to me after the draw, they almost never have aces up, and they can't have trips unless they started with KK. If I now bet my unimproved AAK, my tough opponent will call me with AA the majority of the time (think about this, this is really the perfect situation for value betting a pair). Had I drawn 1, I can no longer value bet unimproved, because I am less likely to get called by AA, and it's more likely SB is checking aces up.

[bigpooch]

Continued from my last post:

5) I like crabbypatty comments on this one and schubes notes
that the BB will be taking a big chunk of the equity if he
plays correctly (for example, if he properly reraises you
with AA or better, you can forget about a hand like 66A!).
I thought maybe a hand like 66A32 would be good enough since
the Zadeh recommendation in a three handed situation is that
the hand A77+ is worth a raise on the button. There are
other significant concerns: you will be first to act, and
you will sometimes be repopped by the BB with AA or better.
In addition, if the poster is knowledgable (or loose)
enough, he is going to take the draw correctly with a hand
like AKxxx trying to flop something to big slick! In any
case, stubborn as I was, I checked out how 66A32 did in this
spot by raising and it was a tiny bit worse than calling,
but the sample size was only a few hundred. In any case,
77A and 88 (with any higher kicker) are the reasonable
minimum hands to consider raising with and it will depend
somewhat on your two opponents.

6) schubes is dead on here! Also, raising with three other
players is excellent if you think the last player to defend
the pot after your postdraw bet will be mucking too often,
in which case you could take a stab at the pot even if you
miss. (I did win a pot like that long ago and even showed
my hand voluntarily after everyone mucked!).

7)
Well, to be honest, even I didn't know the answer to this
when I posted this question! [img]/images/graemlins/smile.gif[/img] It's best to analyze this
a la Zadeh (see Section 4, especially figure A.1 in the
Appendix) and just so we won't be complete nits or rocket
scientists, we'll just approximate the game theoretical play
after the draw and represent the strength of each hand by
a real number between 0 and 1 (which represents the chance
that the opponent has a better hand). (We can assume that
each hand is an independent uniform random variable on the
unit interval.) Also, when I calculated the result, I was
quite surprised how far off this was compared to what I
thought was the approximate answer. I am very surprised
especially since, if my memory serves me, the literature on
this situation appeared to be quite far off as well
(perhaps, they were thinking of all of those hopeless
opponents!).

Let's assume the limit is 5-10 so that the small blind gets
eaten up by the rake and so there is exactly 4 BBs in the
pot before the postdraw action (in the 1-2, there's another
$0.25 out there which translates to 1/8 of a BB) as I don't
like messy fractions. The optimal calling frequency (based
on pot size) is 4/5 and so the second player will call with
a hand of 0.8+ ( anything between 0 and 0.8). You might
think that the first player betting a hand of 0.4 is quite
reasonable since this will be on the borderline at winning
50% of the time when called; unfortunately, there are two
difficulties with that: one may be better off checking and
calling because the second player may "bluff" with a hand
like a baby straight, and there can be a raise. In any
case, let's say the first player does bet with a hand of
0.4+ (which is btw, a bit loose) and some "bluffs", which,
according to optimal bluffing is 1/6th as frequent, or with
hands in (0.93333,1).

Now, if the second player raises, the first player's folding
frequency is 2/7 (since the second player is risking 2 BBs
at a pot which is currently 5 BBs), so that could include
the bluffing hands for player 1 plus hands between 0.33333
and 0.4 ( 2/7 x 0.46667 - 0.06667 = 0.06667 and this is the
slice of legitimate betting hands to fold to a raise) and
hence, the hands that the first player at least calls a
raise is in the range (0,0.33333) (for simplicity, we'll say
there is no bluff reraising). From an axiom that I know,
and Zadeh only alludes to indirectly in his book, the top
40% of the hands give an approximation of the legitimate
hands to raise with. In other words, with the hands 2/15+
or 0.13333+.

What does that translate to in draw? There are

40 straight flushes (including #1)
3744 boats
5108 ordinary flushes
10200 ordinary straights

Altogether, there are 19092 pat hands and so the top 2/15 of
that would be the top 2546 hands: so you would really need
theoretically a hand of 77722 or better! (Of course, it's
better to hold something like 77788) Also, I've been a bit
on the loose side here, so maybe the true minimum is 88822
or so!

This result reminds me of how tight the play ought to be in
PL Jacks-or-better (see Nesmith Ankeny's book!) as compared
to how actual participants play! [img]/images/graemlins/smile.gif[/img]

Before I made this calculation, I was fairly sure that an
ace high flush would be sufficient, even theoretically, but
I guess I must be just another LAG! I think there is a flaw
that comes to mind here: that when you hold a flush, it's
much more likely that your opponent also holds one: when
there are no cards taken out, there are 5148 flushes (incl.
straight flushes) out of 2598960 possible hands or a
probability of 0.0019808 getting one; on the other hand, if
you already hold a flush, there are 3xC(13,5)+C(8,5)= 3917
combinations out of C(47,5)=1533939 so giving a probability
of 0.0025536, almost a 29% increase of these. In practice,
I think an good ace high flush would work out quite well,
so I like TheShootah's answer!

[schubes]

Very interesting analysis for #7. I want to think about this one some more, but I think the reason the results seem so tight is because the nearly optimal players are folding their pat hands more than us mere mortals do. A bet or a raise isn't getting as much value if you will make a portion of hands you beat fold.

I solved for the optimal strategy in an easier situation, involving some very subobtimal (but not too unrealistic) assumptions:

1) Neither player will fold their pat hands, for fear of being bluffed.
2) The first player will never check-raise

Player 2 bets when checked to with the top 7/9
Player 1 bets out with the top 5/9
2 raises with top 2/9 (I think this is a small to medium flush but not sure)
1 reraises with top 1/9

These might be a little looser than seems natural, but I tend to think of optimal play as involving very thin raises for value, so if you know you will get called (and know you will call a bet if you check), the standards make sense.

Good situations and analysis, keep the draw discussion coming!

[TheShootah]

BOOYAH! No need for calculations! [img]/images/graemlins/grin.gif[/img]