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. I'd be interested in hearing other
people's opinions on this.
Anyway, here's 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) 60.0% R3, 40.0% 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 C C C C C

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

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

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

2  R1 R1 R1 C C C C C C C C C C
[1] (KK) 98.1% R2, 0.9% 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 C 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) 14.1% R1, 85.9% R2
[2] (A5o) 5.4% C, 94.6% R1

Paul Hankin
>Here's some analysis done by Alex Selby (he gave me permission to post it).
>It'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).
Is there a limit to the number of raises allowed?
How do you go about deriving or calculating this optimal strategy?
> Here's some analysis done by Alex Selby (he gave me permission to post it).
> It'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. I'd be interested in hearing other
> people's opinions on this.
I caution against applying it in a normal game with big stacks. The
difference is that you have to pay a lot more to see the river in a
normal game, so sucky hands become not worth playing since you get
penalized for them on every round. On the other hand, when heads up
in the blinds, you will often win the pot by theft or by making any
pair, and the ones that are nearly connected can flop straight draws,
so the sucky hands do have a lot of redeeming value. But on the
balance, you have to play fewer hands when you will have additional
betting after the flop. You should avoid the worst reverse implied
odds hands like J5 when big stacked.
Even so, there are some similarities to my preflop recommendations.
The biggest qualitative difference is that in your mutated game, you have
to play your best hands hard preflop, since there is nothing to be
gained after the flop, and so it's your next best hands that must
serve as the protection for your limping hands. Also, unlike in a
real game when you raise under the gun, you have no fear of stealing
the blinds.
> Anyway, here's 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
Here is your table rewritten in a different format:
F 8283 7273 6263 52 4243 32
C 22 T2sT4s 92s94s 82s85s 72s76s 62s65s 52s54s 42s43s 32s
J2K3 Q2Q4 J2J4 T2T5 9296 8487 7476 6465 5354
R1 3366 A2sA9s K2sK9s Q2sQJs J2sJTs T4sT9s 95s98s 86s87s
A2A9 K4KT Q5QJ J6JT T6T9 9798
R2 77QQ ATsAKs AK AJ
R3 AA
CR1 KTsKQs AQ KQ
*1 KK
*2 KJ
*3 AT
*4 J5
> *1 => (KK) 60.0% R3, 40.0% 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
I haven't rewritten the other tables.
In regards to never folding in the big blind, no hand is more
than a 3:1 dog to any other, so that's why.
I can't vouch for the correctness but it looks correct to me.
Good job, Alex.
The strategy is almost directly applicable to "short stack" strategy,
for tournaments or when you are very low on chips in a ring game.
(Rec.gambler common wisdom is that you should wait until your blinds
to rebuy in a ring game if you're short stacked and there is no rake,
to take advantage of the can'tfoldifallin rule.)

Abdul
Hmm... then I almost certainly made an error at TARGET when
I folded 92 offsuit to a raise on the big blind which left
me with one chip short of a small blind + big blind. Probably
should have taken one of the 98 offsuites that I got before
the big blind hit me again...
Sam

Samuel S. Paik  http://www.webnexus.com/users/paik/
3D and multimedia, architecture and implementation
Solyent Green is kitniyot!
Is this game related to Roshambo?
In effect no. Increasing the maximum number of raises beyond three doesn't make
any difference here. Obviously you can keep raising forever with AA, but you
won't get any extra EV since every other hand should have folded or called by
the third raise.
>
> How do you go about deriving or calculating this optimal strategy?
Two player poker games are known in the jargon as "zerosum imperfect
information games". These sorts of games always have optimal strategies if you
allow people to do random things ("mixed strategies"), and these can be found
by the simplex algorithm.
For example take the "scissorspaperstone" game. Two players simultaneously
choose "scissors", "paper" or "stone". Stone beats (blunts) scissors, scissors
beats (cuts) paper, and paper beats (wraps) stone. The optimal strategy is to
choose each with probability 1/3. If you make your probabilities unequal, say
you choose "scissors" with prob. 0.4, "paper" with prob. 0.4, and "stone" with
prob. 0.2, then you can be exploited (beaten on average) by someone always
choosing "scissors". You can check that any unequal strategy can be beaten.
Incidentally, this example illustrates a slightly unsatisfying aspect of
"optimal strategies". If you use the optimal strategy of choosing "scissors",
"paper", "stone" with probability 1/3 each, your EV is 0 against _any_ other
strategy. This means that in this example you won't exploit your opponent's
mistakes. In general, optimal strategies usually exploit some, but not all
mistakes. It's the price you pay for being completely unexploitable
yourself. For this reason the optimal strategy might better be called "optimal
defensive strategy".
Two player poker can be thought of as an elaborate version of
scissorspaperstone. It has the added complication that your choice of thing
to do can depend on the situation you're in (what your cards are, how your
opponent has acted). But this can be handled in moreorless the same way.
The limiting factor in practice is that if there are too many situations
then the computations will overload your computer. Solving full twoplayer
holdem (flop, turn, river betting) like this would be very difficult.
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 0.04464. 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. (It didn't surprise the
original poster, Paul, whose poker intuition is better than mine, and whose
idea it was to do this calculation.)
I can give more details (and the program used) if anyone is interested.
Alex Selby
(Remove the obvious thing from email address.)
In the Big blind, when small blind has called, the small pairs 2266
should R1. Also KK should R3 with prob. 1.9% (there is actually a small
amount of freedom to choose different strategies here, because the SB
shouldn't be calling then rereraising.)
In the Big blind, when small blind has raised, the pair 55 should R1.
Corrected tables are given below.
In article <7gpect$39k$1...@pegasus.csx.cam.ac.uk>, pd...@cus.cam.ac.uk (Paul Hankin) writes:
> Here's some analysis done by Alex Selby (he gave me permission to post it).
> It'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 essentially *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. I'd be interested in hearing other
> people's opinions on this.
>
> Anyway, here's 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.
R3F => Raise, reraise and rereraise if raised back, but fold if raised back again.
> CR1 => Callraise
> CR2 => Callraise and reraise if raised back.
>
[Corrected tables follow.]
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
> 
> Paul Hankin
Alex Selby
(Remove obvious thing from email address.)
Sounds like reasonable advice!
>
> Even so, there are some similarities to my preflop recommendations.
> The biggest qualitative difference is that in your mutated game, you have
> to play your best hands hard preflop, since there is nothing to be
> gained after the flop, and so it's your next best hands that must
> serve as the protection for your limping hands. Also, unlike in a
> real game when you raise under the gun, you have no fear of stealing
> the blinds.
>
> > Anyway, here's 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
>
> Here is your table rewritten in a different format:
>
> F 8283 7273 6263 52 4243 32
> C 22 T2sT4s 92s94s 82s85s 72s76s 62s65s 52s54s 42s43s 32s
> J2K3 Q2Q4 J2J4 T2T5 9296 8487 7476 6465 5354
> R1 3366 A2sA9s K2sK9s Q2sQJs J2sJTs T4sT9s 95s98s 86s87s
> A2A9 K4KT Q5QJ J6JT T6T9 9798
> R2 77QQ ATsAKs AK AJ
> R3 AA
> CR1 KTsKQs AQ KQ
> *1 KK
> *2 KJ
> *3 AT
> *4 J5
>
> > *1 => (KK) 60.0% R3, 40.0% 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
>
> I haven't rewritten the other tables.
>
> In regards to never folding in the big blind, no hand is more
> than a 3:1 dog to any other, so that's why.
That's not true. There are many examples of pairs of hands where one is than a
3:1 favourite over the other.
e.g. KK vs K2o is 17.6:1,
AA vs 65s is 3.44:1 and AA vs (any other hand) is more than that,
or K2o vs Q2o is 3.05:1
>
> I can't vouch for the correctness but it looks correct to me.
> Good job, Alex.
>
> The strategy is almost directly applicable to "short stack" strategy,
> for tournaments or when you are very low on chips in a ring game.
> (Rec.gambler common wisdom is that you should wait until your blinds
> to rebuy in a ring game if you're short stacked and there is no rake,
> to take advantage of the can'tfoldifallin rule.)
>
> 
> Abdul
> In article <yerbtfz...@shell9.ba.best.com>, Abdul Jalib <Abd...@PosEV.com> writes:
> > In regards to never folding in the big blind, no hand is more
> > than a 3:1 dog to any other, so that's why.
>
> That's not true. There are many examples of pairs of hands where one is than a
> 3:1 favourite over the other.
> e.g. KK vs K2o is 17.6:1,
> AA vs 65s is 3.44:1 and AA vs (any other hand) is more than that,
> or K2o vs Q2o is 3.05:1
Oops, sorry, I meant no hand is more than a 3:1 dog to a *random* hand.
I hope that's true. However here since you are limping with some of
your hands in the small blind, you don't have a random hand when you
raise in the small blind. Therefore, it's not immediately obvious that
the big blind must not be worse than a 3:1 dog, but it's at least plausible
since the small blind raises with so many hands.

Abdul
>Oops, sorry, I meant no hand is more than a 3:1 dog to a *random* hand.
>I hope that's true.
True. The worst is 32o which is about 2.1:1

"Erik Reuter" <ere...@uiuc.edu>  http://www.uiuc.edu/ph/www/ereuter/