Alex Selby

From PokerAI

Jump to: navigation, search

Alex Selby is one of the guys who first programmatically calculated preflop Nash equilibrium for limit poker. He has also published the source code on his web site [1]

Original Usenet Article

Paul Hankin [it was Paul's idea to do this calculation] writes:

Here's an optimal strategy for heads-up limit holdem with 1-2 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 late-position steal raise.

Anyway, here are the details:


OPTIMAL HEADS-UP 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 re-re-raise if raised back.
CR1 => Call-raise
CR2 => Call-raise 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!

See also

Links

  • Alex Selby's Optimal Preflop Holdem
Personal tools