[From rec.gambling.poker  Usenet, February 1999]
Paul Hankin [it was Paul's idea to do this calculation] writes...
Here's an optimal strategy for headsup limit holdem with 12 blinds but with
no betting on the flop, turn or river (so there's only betting preflop).
Despite the (relative) simplicity of the game, the optimal strategy brings
up several interesting points:
1. You *never* fold in the big blind, and rarely in the small blind.
2. You can value bet some really quite weak hands because the other
player has to call with such good pot odds.
3. There are similarities between the small blind play and Abdul's
preflop strategy. In particular, you're calling with KK 40% of
the time to trap the big blind for 2 bets.
Although this isn't a strategy for real holdem, it's probably pretty good, and
could well provide a basis for blind play, and the advice for playing the big
blind might also be applicable to defending against a lateposition steal
raise.
Anyway, here are the details:

OPTIMAL HEADSUP PREFLOP HOLDEM WITH $1$2 BLINDS WITH $2 BETS
Above the \ diagonal in the table are the suited hands, below unsuited
hands. The letters in the table represent a strategy for playing the
hand. In all the strategies (except for F) you *never* fold. For the
entries marked * you should randomize your play, choosing a strategy
(with the given probability) from the choices below the table.
F => Fold
C => Call
R1 => Raise
R2 => Raise, and reraise if raised back.
R3 => Raise, reraise and rereraise if raised back.
CR1 => Callraise
CR2 => Callraise and reraise if raised back.
SMALL BLIND PLAY
suited
 A K Q J T 9 8 7 6 5 4 3 2

A  R3 R2 R2 R2 R2 R1 R1 R1 R1 R1 R1 R1 R1

K  R2 *1 CR1 CR1 CR1 R1 R1 R1 R1 R1 R1 R1 R1

Q  CR1 CR1 R2 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1

J  R2 *2 R1 R2 R1 R1 R1 R1 R1 R1 R1 R1 R1
u 
n T  *3 R1 R1 R1 R2 R1 R1 R1 R1 R1 R1 C C
s 
u 9  R1 R1 R1 R1 R1 R2 R1 R1 R1 R1 C C C
i 
t 8  R1 R1 R1 R1 R1 R1 R2 R1 R1 C C C C
e 
d 7  R1 R1 R1 R1 R1 R1 C R2 C C C C C

6  R1 R1 R1 R1 R1 C C C R1 C C C C

5  R1 R1 R1 *4 C C C C C R1 C C C

4  R1 R1 C C C C C C C C R1 C C

3  R1 C C C C C F F F C F R1 C

2  R1 C C C C C F F F F F F C
[1] (KK) 60.1% R3, 39.9% CR2
[2] (KJo) 62.7% R1, 37.3% CR1
[3] (ATo) 73.0% R2, 27.0% CR1
[4] (J5o) 64.3% C , 35.7% R1
BIG BLIND PLAY
When the small blind has called
suited
 A K Q J T 9 8 7 6 5 4 3 2

A  R3 R2 R2 R2 R2 R1 R1 R1 R1 R1 R1 R1 R1

K  R2 *1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1

Q  R2 R1 R2 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1

J  R2 R1 R1 R2 R1 R1 R1 R1 R1 R1 R1 R1 R1
u 
n T  R1 R1 R1 R1 R2 R1 R1 R1 R1 R1 R1 R1 C
s 
u 9  R1 R1 R1 R1 R1 R2 R1 R1 R1 R1 C C C
i 
t 8  R1 R1 R1 R1 R1 R1 R2 R1 R1 C C C C
e 
d 7  R1 R1 R1 R1 R1 R1 *2 *3 C C C C C

6  R1 R1 R1 R1 R1 R1 C C R1 C C C C

5  R1 R1 R1 R1 R1 C C C C R1 C C C

4  R1 R1 R1 R1 C C C C C C R1 C C

3  R1 R1 R1 R1 C C C C C C C R1 C

2  R1 R1 R1 C C C C C C C C C R1
[1] (KK) 98.1% R2, 1.9% R3
[2] (87o) 26.0% C, 74.0% R1
[3] (77) 99.8% R1, 0.2% R2
When the small blind has raised
suited
 A K Q J T 9 8 7 6 5 4 3 2

A  R3 R2 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1

K  *1 R2 R1 R1 R1 R1 R1 C C C C C C

Q  R1 R1 R2 R1 R1 C C C C C C C C

J  R1 R1 R1 R2 C C C C C C C C C
u 
n T  R1 R1 C C R1 C C C C C C C C
s 
u 9  R1 R1 C C C R1 C C C C C C C
i 
t 8  R1 C C C C C R1 C C C C C C
e 
d 7  R1 C C C C C C R1 C C C C C

6  R1 C C C C C C C R1 C C C C

5  *2 C C C C C C C C R1 C C C

4  C C C C C C C C C C C C C

3  C C C C C C C C C C C C C

2  C C C C C C C C C C C C C
[1] (AKo) 14.2% R1, 85.8% R2
[2] (A5o) 5.4% C, 94.6% R1
Alex Selby writes...
One thing I found interesting, was that although in principle the optimal
strategy could have been very mixed, it turned out to be almost completely
pure. I.e. there were only a few hands where you are meant to make a random
choice. This made the computation much easier.
Another thing I found interesting about the limit holdem result, was that it
turned out that the EV to BB is 0.02232 bets. So the BB is +EV even after
putting more money in to start with. The positional advantage (& live blind)
must more than make up for it. This surprised me a bit. Maybe it would make a
good proposition bet!