Some specialized draw plays

[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!