Poker for Fun and Profit(and intellectual challenge)

Robert Holte

Computing Science Dept.

University of Alberta

Poker

World Series of Poker

Poker Research Group - core

• Darse Billings (Ph.D.) • Aaron Davidson M.Sc., Poki • Neil Burch P/A, PsOpti• Terence Schauenberg (M.Sc.), Adapti

• Advisors: J Schaeffer, D Szafron

Poker Research Group – new arrivals

• Bret Hoehn (M.Sc.)• Finnegan Southey (postdoc)

• Michael Bowling• Dale Schuurmans• Rich Sutton• Robert Holte

Our Goal

PsOpti2 vs. “theCount”

                          

Play Us Online

http://games.cs.ualberta.ca/poker/

Poki’s Poker Academy

http://poki-poker.com

Poker Variants

• Many different variants of poker

• Texas Hold’em the most skill-testing

• No-Limit Texas Hold’em used to determine the world champion

• Our research: Limit Texas Hold’em

• Current focus: 2-player (heads up)

Bet Sequence

Initial

Flop

Bet Sequence

Turn

Bet Sequence

River

1,624,350

9 of 19

9 of 19

45

9 of 19

44

17,296

19 Bet Sequence

O(1018)

2-player, limit, Texas Hold’em

2 private cards to each player

3 community cards

1 community card

1 community card

Research Issues

1. Chance events2. Imperfect Information3. Sheer size of the game tree4. Opponent modelling is crucial5. How best to use domain knowledge ?6. Experimental method

Variants have even more challenges:– More than 2 players (up to 10) – “No limit” (bid any amount)

Issues: Chance Events

• Utility of outcomes– currently just reason about expected payoff– short-term vs. long-term

• High variance– was the outcome due to luck or skill ?– experiment design

Issues: Imperfect Information

• Probabilistic strategies are essential

• Cannot construct your strategy in a bottom-up manner, as is done with perfect information games

Issues: Size of the game

• 2-player, Limit, Texas Hold’em game tree has about 1018 states

• Linear Programming can solve games with 108 states

Issues: Opponent Modelling

• Nash equilibrium not good enough– Static– Defensive

• Even the best humans have weaknesses that should be exploited

• How to learn very quickly, with very noisy information ?– Expoitation vs. exploration

• How not to be exploited yourself ?

Issues: Using Expert Knowledge

• We are fortunate to have unlimited access to a poker-playing expert (Darse)

• How best to use his knowledge ?– Expert system (explicitly encoded

knowledge) was not effective– Used his knowledge to devise abstractions

that reduced the game size with minimal impact on strategic aspects of the game

– Use him to evaluate the system

Experimental Method

• High variance

• ‘bot play not the same as human play

• Very limited access to expert humans other than our own expert

Coping with very large games

Full game treeT

StrategyFor T

StrategyFor T*

Abstract game treeT*

abstraction

Solve (LP)

(reversemapping)

(lossy)

too big to solve

Abstraction

• Texas Hold'em 2-player game tree is too big for current LP –solvers (1,179,000,604,565,715,751)

• Many ways of doing the abstractions– We require coarse-grained abstractions– Avoiding a severe loss of accuracy

• Abstract to a set of smaller problems 108 states, 106 equations and unknowns

Alternate Game Structures

• Truncation of betting rounds• Bypassing betting rounds• Models with 3 rounds, 2 rounds, or 1 round

• Many-to-one mapping of game-tree nodes to single nodes in the abstract game tree– How you do the mapping determines the overall

accuracy (few good and many bad mappings)– This is the limiting factor of the method

Bet Sequence

Initial

Flop

Bet Sequence

Turn

Bet Sequence

River

1,624,350

9 of 19

9 of 19

45

9 of 19

44

17,296

19 Bet Sequence

TexasHold'emO(1018)

3-roundModel

(expected valueleaf nodes)

Bet Sequence

Initial

Flop

Bet Sequence

Turn

Bet Sequence

River

1,624,350

9 of 19

9 of 19

45

9 of 19

44

17,296

19 Bet Sequence

TexasHold'emO(1018)

3-roundPostflopModel

(single flop)

1-roundPreflopModel

Abstractions

• Board Q – 7 – 2 • Compare 1.A–3 2.A–4 3.A–K

– Suit isomorphism (24X) (exact)– Rank near-equivalence (small error)

• Bucketing Hands are mapped to a small set of buckets

depending on• Current hand strength• Potential for improvement in hand strength

Bucketing

• Reduce branching factor at chance nodes• Partition hands into six classes per player• Overlaying strategically similar sub-trees

1,1 1,2 1,3 6,6

1,1 1,2 1,3 .…

OriginalBucketing

Next RoundBucketing

Transition Probabilities

….

6,6

Bet Sequence

Initial

Flop

Bet Sequence

Turn

Bet Sequence

River

1,624,350

9 of 19

9 of 19

45

9 of 19

44

17,296

w2 (36)

7 of 15

7 of 15

7 of 15

19 Bet Sequence

15

x2 (36)

z2 (36)

y2 (36)

TexasHold'emO(1018) Abstract

PostflopModelO(107)

AbstractPreflopModelO(107)

Reverse Mapping

• Bucket splitting– LP solution gives a strategy (recipe)– Each partition class split strong / weak– Split the randomized mixed strategy– {0, 0.2, 0.8} => {0, 0, 1.0} & {0, 0.4, 0.6}

• Better hand selection (with some risk)

Putting It All Together – PsOpti1

Bets2 4 6 8

Preflop

Flop

Turn

River

Selby preflop model

Post Post Post Post

Putting It All Together – PsOpti2

Preflop

Flop

Turn

River

Bets +model

3-roundpreflop model

Post Post Post Post Post Post Post

2 4 4 6 6 8 8

Conclusions

• Game Theory can be applied to large problems and practical systems

• Nash Equilibrium (minimax) too defensive, does not exploit the opponent’s weaknesses

• Current work involves opponent modelling– Preliminary results are very promising

• We hope to beat the best poker players in the world in the near future