HU Limit Preflop Equilibrium

ok|select · **Posts:** 47 **Favourite Bot:** LGC

This is long answer to a question in the Exploitative/Equilibrium thread, which I apologise for hijacking.

Indiana asked to see a HU limit preflop strategy from a pseudo optimal bot, so below is the preflop strategy of my strongest bot to date. The strategy is almost pure, so I've taken the liberty of imagining we always pick the 'most likely action' at every point (which will give a pure strategy) as this is much easier to describe.

The game assumes a 10/20 structure (with 5/10 blinds) and a 4-bet cap on each round.

Some general observations and how they relate to the play of typical strong human players.

We never limp from the small blind (standard)
We never 4-bet from the small blind (unusual)
We never fold after volunteering money into the pot (standard)
If the SB open raises, we 3-bet from the big blind quiet frequently (pretty common, but opinions differ)
If the SB open limps we raise very frequently too (this looks way more aggressive than typical play)

There are some actions that look anomalous. It seems to have a particular fondness for Q9 !?

Marv

Code:

                        
                        Hands are ordered as follows: (AA is bottom right corner and lower diagonal is suited)

                        22  32o 42o 52o 62o 72o 82o 92o T2o J2o Q2o K2o A2o

                        32s 33  43o 53o 63o 73o 83o 93o T3o J3o Q3o K3o A3o

                        42s 43s 44  54o 64o 74o 84o 94o T4o J4o Q4o K4o A4o

                        52s 53s 54s 55  65o 75o 85o 95o T5o J5o Q5o K5o A5o

                        62s 63s 64s 65s 66  76o 86o 96o T6o J6o Q6o K6o A6o

                        72s 73s 74s 75s 76s 77  87o 97o T7o J7o Q7o K7o A7o

                        82s 83s 84s 85s 86s 87s 88  98o T8o J8o Q8o K8o A8o

                        92s 93s 94s 95s 96s 97s 98s 99  T9o J9o Q9o K9o A9o

                        T2s T3s T4s T5s T6s T7s T8s T9s TT  JTo QTo KTo ATo

                        J2s J3s J4s J5s J6s J7s J8s J9s JTs JJ  QJo KJo AJo

                        Q2s Q3s Q4s Q5s Q6s Q7s Q8s Q9s QTs QJs QQ  KQo AQo

                        K2s K3s K4s K5s K6s K7s K8s K9s KTs KJs KQs KK  AKo

                        A2s A3s A4s A5s A6s A7s A8s A9s ATs AJs AQs AKs AA

                        small blind

                        22  32o 42o 52o 62o 72o 82o 92o T2o J2o Q2o K2o A2o

                          R   f   f   f   f   f   f   f   f   f   R   R   R    2

                          f   R   f   f   f   f   f   f   f   R   R   R   R    3

                          R   R   R   R   R   f   f   f   R   R   R   R   R    4

                          R   R   R   R   R   R   R   R   R   R   R   R   R    5

                          R   R   R   R   R   R   R   R   R   R   R   R   R    6

                          f   R   R   R   R   R   R   R   R   R   R   R   R    7

                          R   R   R   R   R   R   R   R   R   R   R   R   R    8

                          R   R   R   R   R   R   R   R   R   R   R   R   R    9

                          R   R   R   R   R   R   R   R   R   R   R   R   R    T

                          R   R   R   R   R   R   R   R   R   R   R   R   R    J

                          R   R   R   R   R   R   R   R   R   R   R   R   R    Q

                          R   R   R   R   R   R   R   R   R   R   R   R   R    K

                          R   R   R   R   R   R   R   R   R   R   R   R   R    A

                        big blind after small blind calls

                        22  32o 42o 52o 62o 72o 82o 92o T2o J2o Q2o K2o A2o

                          R   k   k   k   k   k   k   k   k   k   k   k   R    2

                          k   R   k   k   k   k   k   k   k   k   k   k   R    3

                          k   R   R   R   R   k   k   k   k   k   k   R   R    4

                          R   R   R   R   R   R   k   k   k   k   R   R   R    5

                          k   R   R   R   R   R   R   R   k   k   R   R   R    6

                          k   k   R   R   R   R   R   R   R   R   R   R   R    7

                          k   k   R   R   R   R  RR   R   R   R   R   R   R    8

                          k   k   R   R   R   R  RR  RR   R   R   R   R   R    9

                          k   k   R   R   R   R  RR  RR  RR   R   R   R   R    T

                          k   R   R   R   R   R   R  RR  RR  RR   R   R   R    J

                          R   R   R   R   R   R   R  RR   R  RR  RR   R   R    Q

                          R   R   R   R   R   R   R   R   R   R  RR  RR  RR    K

                          R   R   R  RR   R   R   R   R   R   R  RR  RR  RR    A

                        RR = raise and raise again if re-raised

                        big blind after small blind raises

                        22  32o 42o 52o 62o 72o 82o 92o T2o J2o Q2o K2o A2o

                          C   f   f   f   f   f   f   f   C   C   C   C   C    2

                          C   R   C   C   C   f   f   C   C   C   C   C   C    3

                          C   C   R   C   C   C   C   C   C   C   C   C   C    4

                          C   C   R   R   C   C   C   C   C   C   C   C   R    5

                          C   C   C   R   R   C   C   C   C   C   C   C   C    6

                          C   C   C   R   R   R   C   C   C   C   C   C   R    7

                          C   C   C   R   R   R   R   R   C   C   C   C   R    8

                          C   C   C   R   R   R   R   R   R   C   R   C   R    9

                          C   C   C   C   R   R   R   R   R   R   C   C   R    T

                          C   C   C   R   C   R   R   R   R   R   C   R   R    J

                          C   C   C   R   R   R   R   R   R   R   R   R   R    Q

                          C   C   C   R   R   R   R   R   R   R   R   R   R    K

                          C   R   R   R   R   R   R   R   R   R   R   R   R    A

indiana · **Posts:** 7622 **Favourite Bot:** V12

This is quite different than Alex Selby's charts (Which is the optimal preflop play, in game setup without postflop betting).

E.g., the small blind play on your side is 90+% raise hands (I was surprised that it's so much, btw). Compared to Selby's below, where ~30% are call hands.

Code:

                        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

spears · **Posts:** 794 **Favourite Bot:** my bot

So no sign of bluffing the weakest hands as in this idealized game? Do you see this phenomenom in NL heads up?

Steven · **Posts:** 37 **Favourite Bot:** KdKs

This post is great, I think no one has published such information till now.

It is indeed different to Selby's optimal preflop charts. Marv, what flop/turn/river abstractions did you use to calculate this preflop strategy, or this already goes beyond what you are willing to share? I will be interested to discuss this with you (also in private, not neccessarily here).

ok|select · **Posts:** 47 **Favourite Bot:** LGC

Steven wrote:

This post is great, I think no one has published such information till now.

It is indeed different to Selby's optimal preflop charts. Marv, what flop/turn/river abstractions did you use to calculate this preflop strategy, or this already goes beyond what you are willing to share? I will be interested to discuss this with you (also in private, not neccessarily here).

I use buckets on (HS, HS^2) to incorporate 'private' information like the UofA, but I define the buckets differently from them.

I also take some flop, turn and river board features into account by grouping flops into clusters using kmeans, and then generating all the turn boards arising form all the flops in one cluster and splitting that into more clusters to get the turn 'public' information.

For example, one of my flop clusters contains all the monotone flops plus a few highly co-ordinated two-tone flops like 6h 7h 4d and 8h Jh Td .

I don't want to give too much more away, but I'll say that the solution file is about 3GB.

Marv

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

I'm curious if you think aggressive preflop strategies would work well at lower limits where players are more likely to call to the river?

ok|select wrote:

I use buckets on (HS, HS^2) to incorporate 'private' information like the UofA, but I define the buckets differently from them.

Could you elaborate on this?

ok|select · **Posts:** 47 **Favourite Bot:** LGC

c2008 wrote:

I'm curious if you think aggressive preflop strategies would work well at lower limits where players are more likely to call to the river?

ok|select wrote:

I use buckets on (HS, HS^2) to incorporate 'private' information like the UofA, but I define the buckets differently from them.

Could you elaborate on this?

I don't want to go into more detail than this. As the number of buckets you can afford to use is so small, I doubt it matters much how you divide up (HS, HS^2) space provided you don't create buckets which are ridiculously too big or too small.

Marv

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

I'm not asking you to divulge any secrets, I just wasn't sure what you were referring to in regard to the UofA?

ok|select · **Posts:** 47 **Favourite Bot:** LGC

c2008 wrote:

I'm not asking you to divulge any secrets, I just wasn't sure what you were referring to in regard to the UofA?

Oh, I just meant that Polaris 1's abstraction just looks at the current and past values of two quantites:
HS = equity vs a random opponent
HS^2 = expected value of (river equity)^2, where our river equity is our equity versus a random opponent once all the remaining board cards are revealed.

This is all 'private information' since it involves our hole cards. 'Public information' would be things like 'has the board a four-flush', 'has the board a pair' etc. Polaris 1 takes no explicit notice of any public information contained in the board.

I do the same, but the way I divide up (HS,HS^2) space is a bit different from theirs.

Incidentally, I've now run my bots vs Fell Omen2. All matches were duplicate tournaments with 100k total hands. The standard error in the following estimates is about 0.7% to 0.8% sb/hand.

fello2 beats the headsup bot from mabots (a clone of turbo texas holdem) by about 12% sb/hand.

fello2 loses to all my own bots, including ones from a few years ago that I posted about on the Poker Academy forums, by 0.4% sb/hand (NB this is well within 1 std dev of zero).

Against my current strongest bot (the one I published the preflop strategy from) fello2 loses by 1.8% sb/hand, which is less than Hyperborean08 beats fello2 (that was 2.4% sb/hand), but better than any of the other AAAI08 participants.

Marv

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

Nice work!

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

ok|select wrote:

I do the same, but the way I divide up (HS,HS^2) space is a bit different from theirs.

Incidentally, I've now run my bots vs Fell Omen2. All matches were duplicate tournaments with 100k total hands. The standard error in the following estimates is about 0.7% to 0.8% sb/hand.

fello2 beats the headsup bot from mabots (a clone of turbo texas holdem) by about 12% sb/hand.

fello2 loses to all my own bots, including ones from a few years ago that I posted about on the Poker Academy forums, by 0.4% sb/hand (NB this is well within 1 std dev of zero).

Against my current strongest bot (the one I published the preflop strategy from) fello2 loses by 1.8% sb/hand, which is less than Hyperborean08 beats fello2 (that was 2.4% sb/hand), but better than any of the other AAAI08 participants.

Marv

I'm curious how my strat(s) would perform against your strongest...

abanger · **Posts:** 49 **Favourite Bot:** Hyperborean

Upper diagonal is suited:

Attachment:

File comment: big blind after small blind calls (B = R and RR = R)

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Attachment:

File comment: big blind after small blind raises

Marv_O.bmp [ 392.02 KB | Viewed 1541 times ]

Attachment:

File comment: small blind

Marv_I.bmp [ 393.35 KB | Viewed 1539 times ]

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

Pretty nice!

Mystic River · **Posts:** 31 **Favourite Bot:** my one ;-)

I would still be very interested in simulation results / information how the rake will change this optimal strategy. For example i would like to see the results for 1.8BB rake (typical for 3/6 with 2.7BB rake and around one third rakeback).

One interesting question is how this will reduce the ranges, for example the 3bet %. Or how much more it would fold in the big blind after an open raise, because many strong human players fold more than 20% in the big blind at these limits.

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

Mystic River wrote:

many strong human players fold more than 20% in the big blind at these limits.

Got any stats to back that up?

abanger · **Posts:** 49 **Favourite Bot:** Hyperborean

Marv, would you mind posting a yearly update of your HU Limit Preflop Equilibrium? I'm curious to see how much improvement is left to be made :xx23

ok|select · **Posts:** 47 **Favourite Bot:** LGC

abanger wrote:

Marv, would you mind posting a yearly update of your HU Limit Preflop Equilibrium? I'm curious to see how much improvement is left to be made :xx23

It's pretty much the same.

Mystic River · **Posts:** 31 **Favourite Bot:** my one ;-)

c2008 wrote:

Mystic River wrote:

many strong human players fold more than 20% in the big blind at these limits.

Got any stats to back that up?

No sorry, i checked my database and the 20% is not true.

But what i found out is that most human player open raise with much more, for example 87 or even 90% open raise in 10/20 games.

I only observed 2/4-5/10 so far with 2-3BB rake and what i can say is that open raising as tight as 77-80% seems to deliver the best results, but folding more often in the big blind does not improve the results. Although the sample sizes are only a few 10k hands, this seems to back your 83% o.r. strategy for 10/20.

c2008 · **Posts:** 732 **Favourite Bot:** Johnny #5

So Marv,

Your preflop is never capping the pot from the small blind?

ok|select · **Posts:** 47 **Favourite Bot:** LGC

c2008 wrote:

So Marv,

Your preflop is never capping the pot from the small blind?

That's correct. Since that post, I've created four stronger equilibrium bots whose abstractions allowed 4 bets/raises per round (like the one I posted about), but the preflop strategies are very similar, and they all just call the small blind if the big blind 3-bets.

Marv

HU Limit Preflop Equilibrium

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