How to Win Texas Hold em Poker
|
|
- Maximilian Emil Harvey
- 7 years ago
- Views:
Transcription
1 How to Win Texas Hold em Poker Richard Mealing Machine Learning and Optimisation Group School of Computer Science University of Manchester / 44
2 How to Play Texas Hold em Poker Deal private cards per player st (sequential) betting round 3 Deal 3 shared cards ( flop ) 4 nd betting round 5 Deal shared card ( turn ) 6 3rd betting round 7 Deal shared card ( river ) 8 4th (final) betting round If all but player folds, that player wins the pot (total bet) Otherwise at the end of the game hands are compared ( showdown ) and the player with the best hand wins the pot / 44
3 How to Play Texas Hold em Poker 3 / 44
4 How to Play Texas Hold em Poker Ante = forced bet (everyone pays) Blinds = forced bets ( people pay big/small) If players > then (big blind player, small blind player, dealer) If players = ( heads-up ) then (big blind, small blind/dealer) No-Limit Texas Hold em lets you bet all your money in a round Minimum bet = big blind Maximum bet = all your money Limit Texas Hold em Poker has fixed betting limits A $4/$8 game means in betting rounds & bets = $4 and in betting rounds 3 & 4 bets = $8 Big blind usually equals small bet e.g. $4 and small blind is usually 5% of big blind e.g. $ Total number of raises per betting round is usually capped at 4 or 5 4 / 44
5 -Card Poker Trees Game tree - both players private cards are known 5 / 44
6 -Card Poker Trees Public tree - both players private cards are hidden 6 / 44
7 -Card Poker Trees P information set tree - P s private card is hidden 7 / 44
8 -Card Poker Trees P information set tree - P s private card is hidden 8 / 44
9 -Card Poker Trees Game tree - both players private cards are known Public tree - both players private cards are hidden 3 P information set tree - P s private card is hidden 4 P information set tree - P s private card is hidden 9 / 44
10 Heads-Up Limit Texas Hold em Poker Tree Size Cards Dealt F C R F C R F C R F C R F C R F C R F C R F C R F F C C P dealt private cards = ( 5 ) = 36 P dealt private cards = ( 5 ) = 5 st betting round = 9, 9 continuing Flop dealt = ( 48 3 ) = 796 nd betting round = 9, 9 continuing Turn dealt = 45 3rd betting round = 9, 9 continuing River dealt = 44 4th betting round = 9 / 44
11 Heads-Up Limit Texas Hold em Poker Tree Size Player Deal = Player Deal = 36 st Betting Round = 36 * 5 * 9 nd Betting Round = 36 * 5 * 9 * 796 * 9 3rd Betting Round = 36 * 5 * 9 * 796 * 9 * 45 * 9 4th Betting Round = 36 * 5 * 9 * 796 * 9 * 45 * 9 * 44 * 9 Total =.79 8 (quintillion) / 44
12 Abstraction Lossless Suit isomorphism, at the start (pre-flop) two hands are strategically the same if each of their cards ranks match and they are both suited or off-suit e.g. (A K, A K ) or (T J, T J ), 69 equivalence classes reduces possible starting hands from 6435 to 856 Lossy Bucketing (binning) groups hands into equivalence classes e.g. based on their probability of winning at showdown against a random hand Imperfect recall eliminates past information Betting round reduction Betting round elimination / 44
13 Abstraction Heads-up Limit Texas Hold em poker has around 8 states Abstraction can reduce the game to e.g. 7 states Nesterov s excessive gap technique can find approximate Nash equilibria in a game with states Counterfactual regret minimization can find approximate Nash equilibria in a game with states 3 / 44
14 Nash Equilibrium Game theoretic solution Set of strategies per player such that no one can do better by changing their strategy if the others keep their strategies fixed Nash proved that in every game with finite players and pure strategies there is at least (possibly mixed) Nash equilibrium 4 / 44
15 Annual Computer Poker Competition Heads-up Limit Texas Hold em Total Bankroll: Slumbot (Eric Jackson, USA) Little Rock (Rod Byrnes, Australia) and Zbot (Ilkka Rajala, Finland) Bankroll Instant Run-off: Slumbot (Eric Jackson, USA) Hyperborean (University of Alberta, Canada) 3 Zbot (Ilkka Rajala, Finland) Heads-up No-Limit Texas Hold em Total Bankroll: Little Rock (Rod Byrnes, Australia) Hyperborean (University of Alberta, Canada) 3 Tartanian5 (Carnegie Mellon University, USA) Bankroll Instant Run-off: Hyperborean (University of Alberta, Canada) Tartanian5 (Carnegie Mellon University, USA) 3 Neo Poker Bot (Alexander Lee, Spain) 3-player Limit Texas Hold em Total Bankroll: Hyperborean (University of Alberta, Canada) Little Rock (Rod Byrnes, Australia) 3 Neo Poker Bot (Alexander Lee, Spain) and Sartre (University of Auckland, New Zealand) Bankroll Instant Run-off: Hyperborean (University of Alberta, Canada) Little Rock (Rod Byrnes, Australia) 3 Neo Poker Bot (Alexander Lee, Spain) and Sartre (University of Auckland, New Zealand) Source: 5 / 44
16 Annual Computer Poker Competition Total Bankroll = total money won against all agents Bankroll Instant Run-off Set S = all agents Set N = agents in a game 3 Play every ( S N ) possible matches between agents in S storing each agent s total bankroll 4 Remove the agent(s) with the lowest total bankroll from S 5 Repeat steps and 3 until S only contains N agents 6 Play a match between the last N agents and rank them according to their total bankroll in this game 6 / 44
17 Extensive-Form Game A finite set of players N = {,,..., N } {c} A finite set of action sequences or histories e.g. H = {(),..., (A A ),...} Z H terminal histories e.g. Z = {..., (A A, 7, r, F ),...} A(h) = {a : (h, a) H} actions available after history h H\Z P(h) N {c} player who takes an action after history h H\Z u i : Z R utility function for player i 7 / 44
18 Extensive-Form Game f c maps every history h where P(h) = c to an independent probability distribution f c (a h) for all a A(h) I i is an information partition (set of nonempty subsets of X where each element of X is in subset) for player i I j I i is player i s jth information set containing indistinguishable histories e.g. I j = {..., (A A, 7 ),..., (A A, 6 3 ),...} Player i s strategy σ i is a function that assigns a distribution over A(I j ) for all I j I i where A(I j ) = A(h) for any h I j A strategy profile σ is a strategy for each player σ = {σ, σ,..., σ N } 8 / 44
19 Nash Equilibrium Nash Equilibrium: u (σ) max σ Σ u (σ, σ ) u (σ) max σ Σ u (σ, σ ) ɛ-nash Equilibrium: u (σ) + ɛ max σ Σ u (σ, σ ) u (σ) + ɛ max σ Σ u (σ, σ ) 9 / 44
20 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = / 44
21 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = / 44
22 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = / 44
23 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 3 / 44
24 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 4 / 44
25 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 5 / 44
26 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 6 / 44
27 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 7 / 44
28 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 8 / 44
29 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - I = {I, I, I 7, I 8 } and I = {I 3, I 4, I 5, I 6 } A((J, J)) = {C, R} and P((J, J)) = 9 / 44
30 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - f c (J (J)) =.5 and f c (K (J)) =.5 σ (I, C) =.6 and σ (I, R) =.4 3 / 44
31 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - f c (J (J)) =.5 and f c (K (J)) =.5 σ (I, C) =.6 and σ (I, R) =.4 3 / 44
32 Extensive-Form Game.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - f c (J (J)) =.5 and f c (K (J)) =.5 σ (I, C) =.6 and σ (I, R) =.4 3 / 44
33 Counterfactual Regret Minimization Counterfactual regret minimization minimizes the maximum counterfactual regret (over all actions) at every information set Minimizing counterfactual regrets minimizes overall regret In a two-player zero-sum game at time T, if both players average overall regret is less than ɛ, then σ T is a ɛ Nash equilibrium. 33 / 44
34 Counterfactual Regret Minimization Counterfactual Value v i (I j σ) = n I j π σ i(root, n)u i (n) u i (n) = z Z[n] π σ (n, z)u i (z) v i (I j σ) is the counterfactual value to player i of information set I j given strategy profile σ π i σ (root, n) is the probability of reaching node n from the root ignoring player i s contributions according to strategy profile σ π σ (n, z) is the probability of reaching node z from node n according to strategy profile σ u i (n) is the payoff to player i at node n if it is a leaf node or its expected payoff if it is a non-leaf node Z[n] is the set of terminal nodes that can be reached from node n 34 / 44
35 Counterfactual Regret Minimization.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - v (I 8 σ) = π i(root, σ n)u (n) n I 8 =.5.5. (. +. ) (. +. ) =. 35 / 44
36 Counterfactual Regret Minimization Counterfactual Regret r(i j, a) = v i (I j σ Ij a) v i (I j σ) r(i j, a) is the counterfactual regret of not playing action a at information set I j Positive regret means the player would have preferred to play action a rather than their strategy Zero regret means the player was indifferent between their strategy and action a Negative regret means the player preferred their strategy rather than playing action a 36 / 44
37 Counterfactual Regret Minimization.5J.5K.5J.5K.5J.5K I I I I.6C.4R.6C.4R.3C.7R.3C.7R.8C.R.F.C.C.9R.F.C.8C.R.F.C.C.9R.F.C -.F.C.F.C.F.C.F.C - v (I 8 σ F ) =.5.5. (. +. ) (. +. ) =.75 r (I 8 F ) = v (I 8 σ F ) v (I 8 σ) =.75. = / 44
38 Counterfactual Regret Minimization Cumulative Counterfactual Regret R T (I j, a) = T r t (I j, a) t= R T (I j, a) is the cumulative counterfactual regret of not playing action a at information set I j for T time steps Positive cumulative regret means the player would have preferred to play action a rather than their strategy over those T steps Zero cumulative regret means the player was indifferent between their strategy and action a over those T steps Negative cumulative regret means the player preferred their strategy rather than playing action a over those T steps 38 / 44
39 Counterfactual Regret Minimization Regret Matching R T,+ (I j,a) if denominator is positive a σ T + (I j, a) = A(I j ) RT,+ (I j,a ) otherwise A(I j ) R T,+ (I j, a) = max(r T (I j, a), ) 39 / 44
40 Counterfactual Regret Minimization Initialise the strategy profile σ e.g. for all i N, for all I j I i and for all a A(I j ) set σ(i j, a) = A(I j ) For each player i N, for all I j I i and for all a A(I j ) calculate r(i j, a) and add it to R(I j, a) 3 For each player i N, for all I j I i and for all a A(I j ) use regret matching to update σ(i j, a) 4 Repeat from 4 / 44
41 Counterfactual Regret Minimization Cumulative counterfactual regret is bounded by R T i (I j ) (max z u i (z) min z u i (z)) A(I j ) T Total counterfactual regret is bounded by I i (max z u i (z) min z u i (z)) max h:p(h)=i A(h) Ri T T 4 / 44
42 Counterfactual Regret Minimization (a) Number of game states, number of iterations, computation time, and exploitability of the resulting strategy for different sized abstractions (b) Convergence rates for three different sized abstractions, x-axis shows iterations divided by the number of information sets in the abstraction Source: 8 - Regret Minimization in Games with Incomplete Information - Zinkevich et al 4 / 44
43 Summary If you want to win (in expectation) at Texas Hold em poker (against exploitable players) then... Abstract the version of Texas Hold em poker you are interested so it has at most game states Run the counterfactual minimization algorithm on the abstraction for T iterations and obtain the average strategy profile σ abs T 3 Map the average strategy profile σ abs T for the abstracted game to one σ T for the real game 4 Play your average strategy profile σ T against your (exploitable) opponents 43 / 44
44 References Annual Computer Poker Competition Website Regret Minimization in Games with Incomplete Information - Zinkevich et al Robust strategies and counter-strategies Building a champion level computer poker player - Johanson Monte Carlo Sampling and Regret Minimization for Equilibrium Computation and Decision-Making in Large Extensive Form Games - Lanctot 48ae86c-45-4c-b9c-3e7ce9bc9ae 44 / 44
Tartanian5: A Heads-Up No-Limit Texas Hold em Poker-Playing Program
Tartanian5: A Heads-Up No-Limit Texas Hold em Poker-Playing Program Sam Ganzfried and Tuomas Sandholm Computer Science Department Carnegie Mellon University {sganzfri, sandholm}@cs.cmu.edu Abstract We
More informationAdaptive play in Texas Hold em Poker
Adaptive play in Texas Hold em Poker Raphaël Maîtrepierre and Jérémie Mary and Rémi Munos 1 Abstract. We present a Texas Hold em poker player for limit headsup games. Our bot is designed to adapt automatically
More informationLaboratory work in AI: First steps in Poker Playing Agents and Opponent Modeling
Laboratory work in AI: First steps in Poker Playing Agents and Opponent Modeling Avram Golbert 01574669 agolbert@gmail.com Abstract: While Artificial Intelligence research has shown great success in deterministic
More informationEfficient Monte Carlo Counterfactual Regret Minimization in Games with Many Player Actions
Efficient Monte Carlo Counterfactual Regret Minimization in Games with Many Player Actions Richard Gibson, Neil Burch, Marc Lanctot, and Duane Szafron Department of Computing Science, University of Alberta
More informationUniversity of Alberta. Library Release Form
University of Alberta Library Release Form Name of Author: Michael Bradley Johanson Title of Thesis: Robust Strategies and Counter-Strategies: Building a Champion Level Computer Poker Player Degree: Master
More informationA heads-up no-limit Texas Hold em poker player: Discretized betting models and automatically generated equilibrium-finding programs
A heads-up no-limit Texas Hold em poker player: Discretized betting models and automatically generated equilibrium-finding programs Andrew Gilpin Computer Science Dept. Carnegie Mellon University Pittsburgh,
More informationAction Translation in Extensive-Form Games with Large Action Spaces: Axioms, Paradoxes, and the Pseudo-Harmonic Mapping
Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Action Translation in Extensive-Form Games with Large Action Spaces: Axioms, Paradoxes, and the Pseudo-Harmonic
More informationSARTRE: System Overview
SARTRE: System Overview A Case-Based Agent for Two-Player Texas Hold em Jonathan Rubin and Ian Watson Department of Computer Science University of Auckland, New Zealand jrub001@aucklanduni.ac.nz, ian@cs.auckland.ac.nz
More informationDecision Generalisation from Game Logs in No Limit Texas Hold em
Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Decision Generalisation from Game Logs in No Limit Texas Hold em Jonathan Rubin and Ian Watson Department of Computer
More informationTexas Hold em. From highest to lowest, the possible five card hands in poker are ranked as follows:
Texas Hold em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold em is the version played most often at casinos
More informationPotential-Aware Imperfect-Recall Abstraction with Earth Mover s Distance in Imperfect-Information Games
Potential-Aware Imperfect-Recall Abstraction with Earth Mover s Distance in Imperfect-Information Games Sam Ganzfried and Tuomas Sandholm Computer Science Department Carnegie Mellon University {sganzfri,
More informationAccelerating Best Response Calculation in Large Extensive Games
Accelerating Best Response Calculation in Large Extensive Games Michael Johanson johanson@ualberta.ca Department of Computing Science University of Alberta Edmonton, Alberta, Canada Michael Bowling bowling@ualberta.ca
More informationRafael Witten Yuze Huang Haithem Turki. Playing Strong Poker. 1. Why Poker?
Rafael Witten Yuze Huang Haithem Turki Playing Strong Poker 1. Why Poker? Chess, checkers and Othello have been conquered by machine learning - chess computers are vastly superior to humans and checkers
More informationGame theory and AI: a unified approach to poker games
Game theory and AI: a unified approach to poker games Thesis for graduation as Master of Artificial Intelligence University of Amsterdam Frans Oliehoek 2 September 2005 ii Abstract This thesis focuses
More informationComputing Approximate Nash Equilibria and Robust Best-Responses Using Sampling
Journal of Artificial Intelligence Research 42 (20) 575 605 Submitted 06/; published 2/ Computing Approximate Nash Equilibria and Robust Best-Responses Using Sampling Marc Ponsen Steven de Jong Department
More informationChampion Poker Texas Hold em
Champion Poker Texas Hold em Procedures & Training For the State of Washington 4054 Dean Martin Drive, Las Vegas, Nevada 89103 1 Procedures & Training Guidelines for Champion Poker PLAYING THE GAME Champion
More informationMeasuring the Size of Large No-Limit Poker Games
Measuring the Size of Large No-Limit Poker Games Michael Johanson February 26, 2013 Abstract In the field of computational game theory, games are often compared in terms of their size. This can be measured
More informationA competitive Texas Hold em poker player via automated abstraction and real-time equilibrium computation
A competitive Texas Hold em poker player via automated abstraction and real-time equilibrium computation Andrew Gilpin and Tuomas Sandholm Computer Science Department Carnegie Mellon University {gilpin,sandholm}@cs.cmu.edu
More informationCreating a NL Texas Hold em Bot
Creating a NL Texas Hold em Bot Introduction Poker is an easy game to learn by very tough to master. One of the things that is hard to do is controlling emotions. Due to frustration, many have made the
More informationPlaying around with Risks
Playing around with Risks Jurgen Cleuren April 19th 2012 2011 CTG, Inc. Introduction Projects are done in a probabilistic environment Incomplete information Parameters change over time What is true in
More informationBest-response play in partially observable card games
Best-response play in partially observable card games Frans Oliehoek Matthijs T. J. Spaan Nikos Vlassis Informatics Institute, Faculty of Science, University of Amsterdam, Kruislaan 43, 98 SJ Amsterdam,
More informationNikolai Yakovenko, PokerPoker LLC CU Neural Networks Reading Group Dec 2, 2015
Deep Learning for Poker: Inference From Patterns in an Adversarial Environment Nikolai Yakovenko, PokerPoker LLC CU Neural Networks Reading Group Dec 2, 2015 This work is not complicated Fully explaining
More informationA Simulation System to Support Computer Poker Research
A Simulation System to Support Computer Poker Research Luís Filipe Teófilo 1, Rosaldo Rossetti 1, Luís Paulo Reis 2, Henrique Lopes Cardoso 1 LIACC Artificial Intelligence and Computer Science Lab., University
More informationCombinatorics 3 poker hands and Some general probability
Combinatorics 3 poker hands and Some general probability Play cards 13 ranks Heart 4 Suits Spade Diamond Club Total: 4X13=52 cards You pick one card from a shuffled deck. What is the probability that it
More informationSYMMETRIC FORM OF THE VON NEUMANN POKER MODEL. Guido David 1, Pearl Anne Po 2
International Journal of Pure and Applied Mathematics Volume 99 No. 2 2015, 145-151 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v99i2.2
More informationLecture V: Mixed Strategies
Lecture V: Mixed Strategies Markus M. Möbius February 26, 2008 Osborne, chapter 4 Gibbons, sections 1.3-1.3.A 1 The Advantage of Mixed Strategies Consider the following Rock-Paper-Scissors game: Note that
More informationTHREE CARD BRAG (FLASH)
THREE CARD BRAG (FLASH) Introduction Three Card Brag is a gambling game played with a standard 52 card pack without jokers. The cards in each suit rank in the usual order from high to low: A-K-Q-J-10-9-8-7-6-5-4-3-2.
More informationMonte Carlo Tree Search and Opponent Modeling through Player Clustering in no-limit Texas Hold em Poker
Monte Carlo Tree Search and Opponent Modeling through Player Clustering in no-limit Texas Hold em Poker A.A.J. van der Kleij August 2010 Master thesis Artificial Intelligence University of Groningen, The
More informationBetter Automated Abstraction Techniques for Imperfect Information Games, with Application to Texas Hold em Poker
Better Automated Abstraction Techniques for Imperfect Information Games, with Application to Texas Hold em Poker Andrew Gilpin Computer Science Department Carnegie Mellon University Pittsburgh, PA, USA
More informationNash Equilibria and. Related Observations in One-Stage Poker
Nash Equilibria and Related Observations in One-Stage Poker Zach Puller MMSS Thesis Advisor: Todd Sarver Northwestern University June 4, 2013 Contents 1 Abstract 2 2 Acknowledgements 3 3 Literature Review
More informationSource. http://en.wikipedia.org/wiki/poker
AI of poker game 1 C H U N F U N G L E E 1 0 5 4 3 0 4 6 1 C S E 3 5 2 A R T I F I C I A L I N T E L L I G E N C E P R O. A N I T A W A S I L E W S K A Source CAWSEY, ALISON. THE ESSENCE OF ARTIFICAL INTELLIGENCE.
More informationArtificial Intelligence Beating Human Opponents in Poker
Artificial Intelligence Beating Human Opponents in Poker Stephen Bozak University of Rochester Independent Research Project May 8, 26 Abstract In the popular Poker game, Texas Hold Em, there are never
More informationIn this variation of Poker, a player s cards are hidden until showdown. Prior to receiving cards, you must place an initial wager known as an ante.
1 POKER Poker is one of the most skilful and fascinating games ever devised, offering players a chance to test their skills against other players rather than the house (casino). Poker is as much a game
More information1 Representation of Games. Kerschbamer: Commitment and Information in Games
1 epresentation of Games Kerschbamer: Commitment and Information in Games Game-Theoretic Description of Interactive Decision Situations This lecture deals with the process of translating an informal description
More informationMinimax Strategies. Minimax Strategies. Zero Sum Games. Why Zero Sum Games? An Example. An Example
Everyone who has studied a game like poker knows the importance of mixing strategies With a bad hand, you often fold But you must bluff sometimes Lectures in Microeconomics-Charles W Upton Zero Sum Games
More informationUltimate Texas Hold'em features head-to-head play against the player/dealer and an optional bonus bet.
*Uultimate Texas Hold'em is owned, patented and/or copyrighted by ShuffleMaster Inc. Please submit your agreement with Owner authorizing play of Game in your gambling establishment together with any request
More informationBAD BEAT. Bad Beat will reset at $10,000 with a qualifier of four deuces beaten. Every Monday at 6:00 AM the Bad Beat will increase by $10,000.
BAD BEAT Bad Beat will reset at $10,000 with a qualifier of four deuces beaten. Every Monday at 6:00 AM the Bad Beat will increase by $10,000. OFFICIAL RULES Horseshoe may change or cancel this promotion
More informationWe employed reinforcement learning, with a goal of maximizing the expected value. Our bot learns to play better by repeated training against itself.
Date: 12/14/07 Project Members: Elizabeth Lingg Alec Go Bharadwaj Srinivasan Title: Machine Learning Applied to Texas Hold 'Em Poker Introduction Part I For the first part of our project, we created a
More informationRULES FOR TEXAS HOLD EM POKER
RULES FOR TEXAS HOLD EM POKER DEFINITIONS In these rules Action means a player acting in turn All-in means a player has invested all of his/her remaining chips in the outcome of a hand. His/her wager cannot
More informationHeads-up Limit Hold em Poker is Solved
Heads-up Limit Hold em Poker is Solved Michael Bowling, 1 Neil Burch, 1 Michael Johanson, 1 Oskari Tammelin 2 1 Department of Computing Science, University of Alberta, Edmonton, Alberta, T6G2E8, Canada
More informationGame Theory and Algorithms Lecture 10: Extensive Games: Critiques and Extensions
Game Theory and Algorithms Lecture 0: Extensive Games: Critiques and Extensions March 3, 0 Summary: We discuss a game called the centipede game, a simple extensive game where the prediction made by backwards
More informationBayesian Nash Equilibrium
. Bayesian Nash Equilibrium . In the final two weeks: Goals Understand what a game of incomplete information (Bayesian game) is Understand how to model static Bayesian games Be able to apply Bayes Nash
More informationA Near-Optimal Strategy for a Heads-Up No-Limit Texas Hold em Poker Tournament
A Near-Optimal Strategy for a Heads-Up No-Limit Texas Hold em Poker Tournament Peter Bro Miltersen University of Aarhus Åbogade 34, Århus, Denmark bromille@daimi.au.dk Troels Bjerre So/ rensen University
More information2016 POKER TOURNAMENT CONTEST RULES
2016 POKER TOURNAMENT CONTEST RULES SECTION 1 Code of Conduct: Temple Etz Chaim will attempt to maintain a pleasant environment for all players, staff and volunteers, but is not responsible for the conduct
More informationAnalysis of poker strategies in heads-up poker
BMI paper Analysis of poker strategies in heads-up poker Author: Korik Alons Supervisor: Dr. S. Bhulai VU University Amsterdam Faculty of Sciences Study Business Mathematics and Informatics De Boelelaan
More informationRULES FOR PLAY TEXAS HOLD EM
RULES FOR PLAY TEXAS HOLD EM The player to the left of the dealer s button places the small blind which will be a stipulated amount. The player seated second from the dealer s left places the big blind
More informationPoker Strategies. Joe Pasquale CSE87: UCSD Freshman Seminar on The Science of Casino Games: Theory of Poker Spring 2006
Poker Strategies Joe Pasquale CSE87: UCSD Freshman Seminar on The Science of Casino Games: Theory of Poker Spring 2006 References Getting Started in Hold em, E. Miller excellent beginner book Winning Low
More informationCasino Gaming Rule 2010
Queensland Office of Liquor and Gaming Regulation Casino Gaming Rule 2010 This Rule is prepared by the Queensland Office of Liquor and Gaming Regulation 2010 V6 1 Queensland Contents Part 1 Preliminary...
More informationBonus Maths 2: Variable Bet Sizing in the Simplest Possible Game of Poker (JB)
Bonus Maths 2: Variable Bet Sizing in the Simplest Possible Game of Poker (JB) I recently decided to read Part Three of The Mathematics of Poker (TMOP) more carefully than I did the first time around.
More informationThis section of the guide describes poker rules for the following cash game types:
Poker Rules What has changed in version 1.2 of the Poker Rules: Rake on games with stakes of 0.02/0.04 was lowered from 2% to 1%. This section of the guide describes poker rules for: Cash Games Tournaments
More informationObject of the Game The object of the game is for each player to form a five-card hand that ranks higher than the player-dealer s hand.
*Ultimate Texas Hold em is owned, patented and/or copyrighted by Bally Technologies, Inc. Please note that the Bureau is making the details of this game available to the public as required by subdivision
More information6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games
6.254 : Game Theory with Engineering Applications Lecture 2: Strategic Form Games Asu Ozdaglar MIT February 4, 2009 1 Introduction Outline Decisions, utility maximization Strategic form games Best responses
More informationIdentifying Player s Strategies in No Limit Texas Hold em Poker through the Analysis of Individual Moves
Identifying Player s Strategies in No Limit Texas Hold em Poker through the Analysis of Individual Moves Luís Filipe Teófilo and Luís Paulo Reis Departamento de Engenharia Informática, Faculdade de Engenharia
More informationIMPORTANT: These are the rules for Youth Home s Texas Hold em Poker Tournament. Entry into the charity casino night event is separate and plays by
IMPORTANT: These are the rules for Youth Home s Texas Hold em Poker Tournament. Entry into the charity casino night event is separate and plays by different rules and for different prizes. Poker Event
More informationGame Theory for Humans. Matt Hawrilenko MIT: Poker Theory and Analytics
Game Theory for Humans Matt Hawrilenko MIT: Poker Theory and Analytics 1 Play Good Poker Read-Based Approach Game Theoretic Approach Why Game Theory? Computer Scientists Humans Audience Theory Practice
More information6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation
6.207/14.15: Networks Lecture 15: Repeated Games and Cooperation Daron Acemoglu and Asu Ozdaglar MIT November 2, 2009 1 Introduction Outline The problem of cooperation Finitely-repeated prisoner s dilemma
More informationSequential lmove Games. Using Backward Induction (Rollback) to Find Equilibrium
Sequential lmove Games Using Backward Induction (Rollback) to Find Equilibrium Sequential Move Class Game: Century Mark Played by fixed pairs of players taking turns. At each turn, each player chooses
More informationDartmouth College. Sandbagging in One-Card Poker
Dartmouth College Hanover, New Hampshire Sandbagging in One-Card Poker A Thesis in Mathematics By: Daniel Hugh Miao Advisor: Peter Doyle 04 Abstract In 950, Kuhn developed a simplified version of poker
More informationClassification/Decision Trees (II)
Classification/Decision Trees (II) Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Right Sized Trees Let the expected misclassification rate of a tree T be R (T ).
More informationTEXAS HOLD EM POKER FOR SIGHT
Lions Club TEXAS HOLD EM POKER FOR SIGHT Official Rules (April 2016) Buy-in/Rebuy/Add-on: The dollar amount of the initial buy-in shall be posted in plain view of the playing table(s). The buy-in ($135)
More informationcachecreek.com 14455 Highway 16 Brooks, CA 95606 888-77-CACHE
Baccarat was made famous in the United States when a tuxedoed Agent 007 played at the same tables with his arch rivals in many James Bond films. You don t have to wear a tux or worry about spies when playing
More informationTerms and Conditions for Charitable Texas Hold em Poker Tournaments
Terms and Conditions for Tournaments Under subsection 34(2) of the Gaming Control Act, a license is subject to terms and conditions as may be imposed by the Registrar. These are the terms and conditions
More informationKnow it all. Table Gaming Guide
Know it all. Table Gaming Guide Winners wanted. Have fun winning at all of your favorite games: Blackjack, Craps, Mini Baccarat, Roulette and the newest slots. Add in seven mouthwatering dining options
More informationKnowledge and Strategy-based Computer Player for Texas Hold'em Poker
Knowledge and Strategy-based Computer Player for Texas Hold'em Poker Ankur Chopra Master of Science School of Informatics University of Edinburgh 2006 Abstract The field of Imperfect Information Games
More informationProbabilities of Poker Hands with Variations
Probabilities of Poker Hands with Variations Jeff Duda Acknowledgements: Brian Alspach and Yiu Poon for providing a means to check my numbers Poker is one of the many games involving the use of a 52-card
More informationComputational Game Theory and Clustering
Computational Game Theory and Clustering Martin Hoefer mhoefer@mpi-inf.mpg.de 1 Computational Game Theory? 2 Complexity and Computation of Equilibrium 3 Bounding Inefficiencies 4 Conclusion Computational
More informationWill Tipton began playing poker online in 2007. He steadily moved up in stakes in online HUNL tournaments to become a regular winner in the high
Will Tipton began playing poker online in 2007. He steadily moved up in stakes in online HUNL tournaments to become a regular winner in the high stake games. He currently lives in Ithaca, NY, and is a
More informationDuring the last several years, poker has grown in popularity. Best Hand Wins: How Poker Is Governed by Chance. Method
Best Hand Wins: How Poker Is Governed by Chance Vincent Berthet During the last several years, poker has grown in popularity so much that some might even say it s become a social phenomenon. Whereas poker
More informationSCHEDULE OF PLAY LEVEL ANTE BLINDS
WSOP WARMUP TO THE COLOSSUS II $75 BUY-IN NO-LIMIT HOLD EM (UNLIMITED RE-ENTRY ALLOWED DURING REGISTRATION PERIOD) MAY 31 & JUNE 1, 2016 AT 1 PM LEVEL ANTE BLINDS 1-25-25 2-25-50 3-50-100 4-100-200 15
More informationNewPokerSoft. Texas Holdem Poker Game Simulator
NewPokerSoft poker for life Texas Holdem Poker Game Simulator www.newpokersoft.com Poker training simulator for Texas Holdem Here, we present the simulator of the Texas Holdem PokerGame. It can be used
More informationTexas Hold em No Limit Freeze Poker Tournament
Texas Hold em No Limit Freeze Poker Tournament Tournament Overview and Rules & Regulations South Beach Casino and Resort will be conducting Texas Hold em No Limit Poker Tournaments. The entry fee for each
More informationCS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM
CS 341 Software Design Homework 5 Identifying Classes, UML Diagrams Due: Oct. 22, 11:30 PM Objectives To gain experience doing object-oriented design To gain experience developing UML diagrams A Word about
More informationIntelligent Agent for Playing Casino Card Games
Intelligent Agent for Playing Casino Card Games Sanchit Goyal Department of Computer Science University of North Dakota Grand Forks, ND 58202 sanchitgoyal01@gmail.com Ruchitha Deshmukh Department of Computer
More informationGame Theory and Poker
Game Theory and Poker Jason Swanson April, 2005 Abstract An extremely simplified version of poker is completely solved from a game theoretic standpoint. The actual properties of the optimal solution are
More informationBayesian Tutorial (Sheet Updated 20 March)
Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that
More informationPLACE BETS (E) win each time a number is thrown and lose if the dice ODDS AND LAYS HARDWAYS (F) BUY & LAY BETS (G&H)
craps PASS LINE BET (A) must be rolled again before a 7 to win. If the Point is and the shooter continues to throw the dice until a Point is established and a 7 is rolled before the Point. DON T PASS LINE
More informationon a table having positions for six players on one side of the table 1. A true to scale rendering and a color photograph of the
Full text of the proposal follows (additions indicated in boldface thus; deletions indicated in brackets [thus]): 13:69E 1.13M Boston 5 stud poker table; boston 7 stud poker table; physical characteristics
More informationA Comedy of Errors - Light!
A Comedy of Errors - Light! Crushing SNGs by Exploiting your Opponents Mistakes We all know where the money comes from in Poker right? It comes from your opponents mistakes! Fortunately there is one area
More informationSlots... 1. seven card stud...22
GAMING GUIDE table of contents Slots... 1 Blackjack...3 Lucky Ladies...5 Craps...7 Roulette... 13 Three Card Poker... 15 Four Card Poker... 17 Texas Hold em Bonus Poker... 18 omaha Poker... 21 seven card
More informationPhantom bonuses. November 22, 2004
Phantom bonuses November 22, 2004 1 Introduction The game is defined by a list of payouts u 1, u 2,..., u l, and a list of probabilities p 1, p 2,..., p l, p i = 1. We allow u i to be rational numbers,
More informationGames of Incomplete Information
Games of Incomplete Information Jonathan Levin February 00 Introduction We now start to explore models of incomplete information. Informally, a game of incomplete information is a game where the players
More informationUNDERGROUND TONK LEAGUE
UNDERGROUND TONK LEAGUE WWW.TONKOUT.COM RULES Players are dealt (5) five cards to start. Player to left of dealer has first play. Player must draw a card from the deck or Go For Low. If a player draws
More informationVon Neumann and Newman poker with a flip of hand values
Von Neumann and Newman poker with a flip of hand values Nicla Bernasconi Julian Lorenz Reto Spöhel Institute of Theoretical Computer Science ETH Zürich, 8092 Zürich, Switzerland {nicla jlorenz rspoehel}@inf.ethz.ch
More informationPoker-CNN: A Pattern Learning Strategy for Making Draws and Bets in Poker Games Using Convolutional Networks
Poker-CNN: A Pattern Learning Strategy for Making Draws and Bets in Poker Games Using Convolutional Networks Nikolai Yakovenko PokerPoker LLC, Las Vegas nickyakovenko@gmail.com Liangliang Cao Columbia
More information2 nd Year Software Engineering Project Final Group Report. automatedpoker player
2 nd Year Software Engineering Project Final Group Report automatedpoker player Supervisor: Graham Kendall Group: gp-gxk2 Group Members: Michael James Pope Neil Oscar Collins Tippett Oliver Philip Turley
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1
More informationOn the Existence of Nash Equilibrium in General Imperfectly Competitive Insurance Markets with Asymmetric Information
analysing existence in general insurance environments that go beyond the canonical insurance paradigm. More recently, theoretical and empirical work has attempted to identify selection in insurance markets
More informationPerfect Bayesian Equilibrium
Perfect Bayesian Equilibrium When players move sequentially and have private information, some of the Bayesian Nash equilibria may involve strategies that are not sequentially rational. The problem is
More informationHow to Play. Player vs. Dealer
How to Play You receive five cards to make your best four-card poker hand. A four-card Straight is a Straight, a four-card Flush is a Flush, etc. Player vs. Dealer Make equal bets on the Ante and Super
More informationBaccarat Gold. 1 Copyright 2007 Dolchee, LLC. All rights reserved.
Baccarat Gold 1 Copyright 2007 Dolchee, LLC. All rights reserved. SUMMARY OF GAME The object of the game is to assemble two hands of two or three cards with a point value as close to nine as possible.
More informationk o G ob in d m n a a G H
Gaming Handbook Welcome to a world of excitement This MGM Grand Detroit Gaming Guide is designed to enhance your gaming fun and total casino experience. The guide provides you with concise, easy-to-understand
More informationGovernment of Nunavut. Community and Government Services. Consumer Affairs Division
Government of Nunavut Community and Government Services Consumer Affairs Division Terms and Conditions for Texas Hold em Poker Tournaments in Nunavut Table of Contents 1. General Provisions.3 2. Penalties...3
More informationTraining Manual. Shuffle Master Gaming Three Card Poker Training Manual R20020325
Training Manual 2001 Shuffle Master, Inc. First Printing All Rights Reserved Printed in the United States of America No part of this publication may be reproduced or distributed in any form or by any means,
More informationUsing Probabilistic Knowledge and Simulation to Play Poker
Using Probabilistic Knowledge and Simulation to Play Poker Darse Billings, Lourdes Peña, Jonathan Schaeffer, Duane Szafron Department of Computing Science, University of Alberta Edmonton, Alberta Canada
More informationMATH 340: MATRIX GAMES AND POKER
MATH 340: MATRIX GAMES AND POKER JOEL FRIEDMAN Contents 1. Matrix Games 2 1.1. A Poker Game 3 1.2. A Simple Matrix Game: Rock/Paper/Scissors 3 1.3. A Simpler Game: Even/Odd Pennies 3 1.4. Some Examples
More informationHow to Play Blackjack Alex Powell, Jayden Dodson, Triston Williams, Michael DuVall University of North Texas TECM 1700 10/27/2014
How to Play Blackjack Alex Powell, Jayden Dodson, Triston Williams, Michael DuVall University of North Texas TECM 1700 10/27/2014 Blackjack Blackjack is a fun game that can be played either for money at
More informationDynamics and Equilibria
Dynamics and Equilibria Sergiu Hart Presidential Address, GAMES 2008 (July 2008) Revised and Expanded (November 2009) Revised (2010, 2011, 2012, 2013) SERGIU HART c 2008 p. 1 DYNAMICS AND EQUILIBRIA Sergiu
More informationRoulette Wheel Selection Game Player
Macalester College DigitalCommons@Macalester College Honors Projects Mathematics, Statistics, and Computer Science 5-1-2013 Roulette Wheel Selection Game Player Scott Tong Macalester College, stong101@gmail.com
More informationNo. 2008 106 MEASURING SKILL IN MORE-PERSON GAMES WITH APPLICATIONS TO POKER. By Ruud Hendrickx, Peter Borm, Ben van der Genugten, Pim Hilbers
No. 00 06 MEASURING SKILL IN MORE-PERSON GAMES WITH APPLICATIONS TO POKER By Ruud Hendrickx, Peter Borm, Ben van der Genugten, Pim Hilbers December 00 ISSN 0-7 Measuring skill in more-person games with
More informationA Graph-Theoretic Network Security Game
A Graph-Theoretic Network Security Game Marios Mavronicolas 1, Vicky Papadopoulou 1, Anna Philippou 1, and Paul Spirakis 2 1 Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus.
More information