Optimal Heads-Up (Preflop) Limit Hold' Em

The Game has changed. As said in The Evolution of Poker: from Super System to math PhD reading Doyle Brunson's SUPER SYSTEM is not enough anymore to give you an edge.  The 2003 Chris Moneymaker's Effect changed all that. Now the common man has the opportunity to play thousands of hands and read all kind of poker books about game theory and exploitative plays. Todays fishes are way more skilled than their counterparts of 20 years ago.

Let me give the highlights of the Optimal Game Theory as ellaborated by highly lucrative online poker pros and some math wizards. These below were particularly devised for Heads-Up Fixed Limit Hold' Em Poker (HUHU / HULHE). HULHE is the simplest form of poker from a programming and mathematical standpoint. Just like chess computer Deep Blue defeated undisputed world chess champion Garry Kasparov. The best bots like Polaris and casino slot machine named "Texas Hold'em Heads Up Poker" AKA "The Brain" routinely beat world-class pro poker players. To this point, HULHE is similar to chess. While it is possible for Artificial Intelligence to play near optimal strategy to beat the best players. Both chess and Heads-Up Limit Hold' Em aren't solved games. 



Exploitative Players were information seekers to keep an edge over their opponents. They are the old guard.
Game Theory Optimal (GTO, optimal) players the highest limits of online plays. They don't much care what their opponent does. They seek to play a strategy designed in the long run to beat any other strategy in the long run.

An optimal player seeks to find the optimal strategy. For any game, there exists at least one optimal strategy. Any GTO poker pro dedicates his time and effort to get as close as possible to the Optimal Strategy. 

However you need to prioritize the practical over the theoretical at all times.

http://www.highstakesdb.com/community/topic/29580-sauce-on-gto/




Game Theoretical Optimum strategy

Button (Small Blind)
rarely fold in the SB (10/169)
23o-28o and 34o-38o

SB never fold to a raise
SB never raise his worst hands


Big Blind (Out Of Position)
NEVER fold in the BB

BB has the edge (act second and is live)
BB never fold to a raise
BB never re-raise with worst hands



Button & BB
R3 - Raise, reraise and re-re-raise if raised back only with AA


Playing different hands in the same way. 
However, Optimal Strategy dictates to vary strategy randomly in order to be deceptive.

The Two Rules

If Villain bet or raise, you should CALL (usually)

If you bet one street and Villain calls, you should BET again on the next card. (usually)


http://www.amazon.com/Pokers-1%25-Secret-Keeps-Players/dp/1496159187/ref=as_sl_pc_tf_til?tag=strenfight-20&linkCode=w00&linkId=QLUJOVHU7GARZLX7&creativeASIN=1496159187



Some of the people who applied professional level mathematics to real world poker. 

Nick Christenson
Alex Selby 
Ben "Sauce123" Sulsky
Matthew Janda
Dr. Darse Billings
Bill Chen
Jerrod Ankenman
Chris Fergusson

http://www.lvrevealed.com/articles/research/selby.html
http://www.archduke.org/simplex/index.html
http://www.archduke.org/simplex/art
http://www.highstakesdb.com/community/topic/29580-sauce-on-gto/

http://forumserver.twoplustwo.com/109/heads-up-limit/optimal-preflop-play-737220/
https://www.google.ca/webhp?sourceid=chrome-instant&rlz=1C1NCHB_enCA593CA593&ion=1&espv=2&ie=UTF-8#safe=off&q=optimal%20heads%20up%20preflop%20holdem

http://www.pokerlistings.com/poker-bots-overrated-says-dr-darse-billings-29734
http://www.pokerlistings.com/strategy/psychology/poker-bots-the-beginning-of-the-end-um-no
http://www.pokerlistings.com/strategy/psychology/bot-this-too-robots-dont-have-girlfriends

http://billchenpoker.com/

Perfect Heads-Up Limit Hold' Em Strategy part 1

STUDYING THE STRATEGIES OF THE FOUR ALL-TIME BEST HULHE PLAYERS, MAN OR MACHINE, TO CREATE YOUR OWN

The best way to find the Perfect Heads-Up Fixed Limit Hold' Em Strategy is to study the strategies used by the great ones.  For me the greats in HULHE are two humans plus two machines.  My two human players are Phil Ivey and Andy Beal.  My two poker bots are Polaris and Texas Hold ‘Em Heads Up Poker.  I will try to enter the brain of those four world-class experts to find and analyze their winning strategies.

Phil Ivey's HUHU FL HE strategy:

Step 1. Raise pre-flop in position
Raise pre-flop in the big blind with (1:1)
Ax, Kx, Q3s, Q6, J8, T7s, T9, 98s, 33+
Step 2. Continuation bet on the flop and turn 100% of the time
Step 3. Leaving after losing one buy-in
Keep playing when winning

Why is Phil Ivey so good?

a) hyper aggressive
b) incredible focus and concentration
c) uncanny ability to detect and exploit opponents's weaknesses and betting patterns
d) zero tilt factor
e) intimidating and fearsome reputation

f) unlimited bankroll

Andy Beal High Stakes Heads-Up Limit Hold' Em strategy

Step 1. Play nearly every hands
Raise (most of the time) every pot he enters
Step 2. Check-call all the way down to the river with any A-high hand
8-bets the turn with two pair
Step 3. Obsession with preventing tells

Why is Andy Beal so good?

a) Obsession

b) Specialization
c) Mathematical approach

d) High stakes

e) Ultra-aggressive
f) Impossible to read

Polaris's Strategies: 

Polaris have a total of 5 different strategies.

The Nash Equilibrium plus 4 other to use against different types of opponents. Polaris identifies which common poker strategy a human is using and switches its own strategy to counter.

1- The first approach is to approximate a Nash equilibrium strategy which is robust against any opponent.

2- The second approach is to find an exploitive counter-strategy to an opponent. We will show that these counter-strategies are brittle: they can lose to arbitrary other opponents.

3- The third approach is a compromise of the first two, to find robust counter-strategies.

4- The four approach is to combine several of these agents into a team, and learn during a game which to use.

Why is Polaris so good?

a) Adaptation. Polaris doesn't have a "best way" to play; it has a "best way" to adapt.
b) Specialization
c) No emotion
d) Aggression

Texas Hold ‘Em Heads Up Poker "The Brain" strategy

Pre-flop
Hyper-LAG / maniac
+ Raise almost every time from the Button
+ Raise called button from over 75% of the time from the BB.
... Sometime 3-bet weak hands like T5off (negligible)
... Occasionally fold on the Button or when raised in the BB (negligible)
- Fold more hands from the SB on most streets than from the BB

Flop
Very LAG
+ Routinely raise with middle pair and a weak kicker
+ Call most of the time a check-raise+ Rarely 3-bet the flop with weak overcard, no straight, flush or backdoor possibilities
+ Often fold to a bet if a high card comes on the flop
+ But he will bet the high card on the flop
+ Most likely to fold on the flop

Turn
Tightens up
+ Check-call with middle pair and weak kicker
+ Check the turn most of the time 

River
+ Call with a piece of the board
+ Call down with Ace high unless the board is scary
+ Likely to call with King high
+ Sometimes call down with weaker hands (J high) on a bimodal (monster or miss) board.  Like a pair on the board.

Overall
+ Reasonable adjustment to what the opponent might hold based on the board.
+ Seem to play top pair / top kicker more slowly on board with flushes and straights possible
+ Seem to play high card / no pair more aggressively on paired flops.  Check-raise bluff
+ More likely to check-raise if opponent put the last bet on the previous street.
+ Doesn't fold too often.

Why is "The Brain" so good?

a) Defense. The basic idea behind its play technique is to do everything it can "to prevent itself from being exploited".  "The theory behind it is almost paranoid," 

b) Unpredictable. The pokerbot use knowledge gained from billions of staged rounds of poker fed through neural networks, and the result is an unpredictable poker player that can win almost every time.  Three different banks of knowledge are used depending on the gameplay scenario, but the basic idea behind its play technique is, as I said above,  "to prevent itself from being exploited." "The theory behind it is almost paranoid." So "The Brain" unpredictibility and three different game plays are part of his perfectly defensive game. 
   
Here are his 3 tag-team fighters alternating against an opponent.
    * The first is a neural net with optimal number of bluffs and can do anything in anyone hand.
    *The second play a slightly different style.
    *The third come into play when the opponent has a reduced stack.

Aggression level might change at random moment

c) Aggression.  "The Brain" is the aggressor. Almost never check-calls or simply matches an opponent's bet without a raise. But give credit to your hand when you raise and reraise.  Far too aggressive and steals far too many pots to get beat on a regular basis.


Now that we have a complete overview of the strategies and strengths of our four Heads-Up Fixed Limit bosses.  We will compare their respective style to each other in the second part of this guide.

PYGOD

Perfect Heads-Up Limit Hold' Em Strategy part 2

HEADS-UP LIMIT HOLD' EM POKER

DO NOT COPY AND PASTE MY WORK INTO YOUR WEBSITE WITHOUT GIVING ME THE CREDIT

Polaris

Polaris is the Deep Blue of poker.

The patience of a monk or the fierce aggression of a tiger, changing gears in a single heartbeat. Polaris can make a pro's head spin. Bluff, trap, check-raise bluff, big lay-down -- name your poison.


Polaris is a Texas hold 'em poker playing program developed by the computer poker research group at the University of Alberta, a project that has been under way for 16 years as of 2007. Polaris is a composite program consisting of a number of bots, including Hyperborean08, the winner of the limit equilibrium series in the 2008 Association for the Advancement of Artificial Intelligence (AAAI) Computer Poker Competition. Polaris also contains a number of other fixed strategies, and chooses between these strategies during a match. Polaris requires little computational power at match time, so it is run on an Apple MacBook Pro laptop during competitions. Polaris plays only heads-up (two player) Limit Texas hold'em.



Limit Hold'em (LH) Heads-up Duplicate poker


On July 23–24, 2007, Polaris played against poker professionals Phil Laak and Ali Eslami at the Hyatt Regency Hotel in Vancouver, B.C.  The competition consisted of four duplicate matches, with 500 hands per match. In each duplicate match, the same cards were dealt to both pairs of players, human and bot, but with the seating reversed.  After roughly 16 hours of play over two days, Polaris tied the first round, won the second and lost the last two.

Final result -- Polaris record: 1-2-1


On July 3–6, 2008, Polaris competed against six human professional poker players in the Second Man-Machine Poker Championship, held in Las Vegas at the 2008 Gaming Life Expo. Polaris defeated the human players with three wins, two losses and one tie. Each of the six sessions was a duplicate match of 500 hands against two different players, resulting in six thousand hands played.

Across all six sessions, Polaris won 195 big blinds. The version of Polaris used in the 2008 match was much stronger than the 2007 version, both in the quality of the component strategies and in its ability to learn which component strategy to use.

Final result -- Polaris record: 3-2-1

Overall Polaris's record vs. humans: 4-4-2


Polaris and Deep Blue

Comparing Polaris results with IBM chess computer Deep Blue versus reigning world chess champion Garry Kasparov in two games under tournament regulations.

Result: Kasparov–Deep Blue (4–2)

Result: Deep Blue–Kasparov (3½–2½)

Overall Deep Blue record versus Kasparov (5½–6½)

Polaris (4-4-2 or 5-5) vs. Deep Blue (5½–6½)

So with that being said. We can see that Polaris had better results versus humans than Deep Blue.

Polaris's Strategies:

Polaris have a total of 5 different strategies. 
The Nash Equilibrium plus 4 other to use against different types of opponents.  Polaris identifies which common poker strategy a human is using and switches its own strategy to counter.

1- The first approach is to approximate a Nash equilibrium strategy which is robust against any opponent.
2- The second approach is to find an exploitive counter-strategy to an opponent. We will show that these counter-strategies are brittle: they can lose to arbitrary other opponents.
3- The third approach is a compromise of the first two, to find robust counter-strategies.
4- The four approach is to combine several of these agents into a team, and learn during a game which to use.

Why is Polaris so good?

a) Adaptation. Polaris doesn't have a "best way" to play; it has a "best way" to adapt.
b) Specialization
c) No emotion
d) Aggression

http://en.wikipedia.org/wiki/Polaris_(poker_bot)

http://webdocs.cs.ualberta.ca/~games/poker/man-machine/

http://webdocs.cs.ualberta.ca/~games/poker/publications/johanson.msc.pdf


Originally published in: http://www.heads-up-fixed-limit-texas-hold-em-poker.com/p/polaris.html