CS 427 Research Paper: Artificial Intelligence Applications to Texas Hold em Poker
|
|
- Simon Berry
- 7 years ago
- Views:
Transcription
1 CS 427 Research Paper: Artificial Intelligence Applications to Texas Hold em Poker By: Introduction Attempting to build agents that simulate human players for games that rely on hidden information and are nondeterministic in nature is an active area in Artificial Intelligence research. Texas Hold em poker provides a context in which to study the effectiveness of various implementation techniques and theories because [t]he game of poker has been identified as a beneficial domain for current AI research because of the properties it possesses such as the need to deal with hidden information and stochasticity (Rubin and Watson 958). Texas Hold em poker is targeted specifically because it is a popular variant of the game and contains many human experts that are still able to outperform modern AI implementations. Attempting to solve complicated games like Texas Hold em is important for the field of artificial intelligence because successfully achieving high computer performance in a non-trivial game can be a stepping stone toward solving more challenging real-world problems (Rubin and Watson 958). Problem Complexity: The Complexity of Texas Hold em Poker Poker is a complex game because it is a game of imperfect information, where multiple competing agents must deal with probabilistic knowledge, risk assessment, and possible deception, not unlike decisions made in the real world (Billings et al. 201). Billings et al explains that one way to reduce the complexity of the game is to look at a subset of the game. A common method for simplifying the analysis of Texas Hold em is having only scenarios that involve two-players, since complexity normally increases exponentially for each player added. Nicolai and Hilderman explain what makes the game challenging by stating, the game is difficult for a computer agent to play well, as it has a very large state space, some information is hidden from the players, there is an element of chance due to the order in which the cards are dealt, and opponents often provide false information (125). Their approximation for the tree for Texas Hold em limit poker is about nodes, which is far too large to analyze completely (222). The state space is large because the deck contains 52 unique cards, each player has 2 cards facedown, and there are different rounds of betting per hand. A complete hand involves the flop, which is three cards dealt face up, the turn which is an additional card dealt face up, and the river which is the last card dealt face up. Each player must build the best standard 5-card poker hand with the community cards dealt on the flop, turn, and river and any number of their starting 2 cards. There is also a round of betting between each phase of cards being dealt. The complexity of the game is increased by betting as well, there are variable rounds of betting and these amounts may be fixed or not depending on whether it is a limit or no-limit game that is being played. For a no-limit game, betting complicates things dramatically, because for each round of betting there may be a bet of 0 (called a Check) to a bet of all 1
2 available chips that a player may have (called an All-In ) and any number in between. This means that for no limit games, an agent must be able to analyze the bet being made in addition to the board state. A way that this complexity is handled is by taking a subset of the game and determining strategies for these problem states instead of a general solution for the entire game of Texas Hold em. Miltersen and Sorensen consider in their study a limited Texas Hold em tournament where only heads-up play (2 players) is considered with a fixed small and big blind (1168). This allowed them to apply minimax with unique jam/fold actions to determine near-optimal play (1170). This near-optimal strategy only considers bets where the current posted bet is an all-in for one of the players, they explain the difference in calculating this by stating the reason computing optimal jam/fold strategies turns out to be feasible is that they make analysis of the post-flop game irrelevant, (1170). This is because the expected payoff is collapsed into a singular model that doesn t expand when considering additional betting rounds (1171). Stochastic Methods: How Randomness Affects the Game Texas Hold em Poker is a nondeterministic game, meaning that different outcomes are entirely possible given the same board states. It s something that must be considered when designing and implementing an agent. This is because some outcomes will be very unlikely, such as having a royal flush on the board and other events will be substantially more likely, such as having only a high card hand among the community cards. Although there are many events left to chance, it is possible to have some idea of how good a starting hand is, disregarding the other players hands which are unknown. A very good measurement (feature) of a player s hand before the community cards are shown or any of the opponents actions observed, is the strength of cards in one s possession. Equity Value (EV) of a hand provides a quality description of its strength. Each card pair is associated with a numerical value calculated from the sample of 115 million hands using data mining methods (Bosilj et al. 1629). With games where much of the information is unknown and left to chance, some methods emphasize the information that is known and making decisions based off of that. Bosilj et al go on to explain that as the game progresses, both human and mathematical factors play an equally important role in making decisions about the next move (1629). It has already been mentioned just how complex Texas Hold em poker is, but it is important to note that since the game is nondeterministic, it is also very computationally intensive to determine the probability of other players having certain hands. It must be considered what the probability of each player having a certain hand and the odds that they will fold a hand of that type. This overlaps with opponent modeling, but there is also a mathematical foundation for which hands are generally considered good. Teofilo and Reis mention that usually, the Hand Rank Evaluator takes a poker hand and maps it to a unique integer which is the score of the hand (2). They later go on to state that this normally involves calculating the relative value of a 2 card hand (Chen Formula), Hand Strength, Hand Potential, and Effective Hand Strength (2-3). Randomness and stochasticity must be considered when building high-performing artificial poker agents, given that the nature of the game stems from this analysis of chance. 2
3 Opponent Theory: Human Opponent Modeling Human opponent modeling is a very large research subject for many different Artificial Intelligence applications. Texas Hold em provides interesting insights into how one might model human players by defining traits of players and using various techniques that emulate the ways that humans themselves learn and adapt to uncertainty and changing levels of success and failure. These attempts are not always successful, and, as mentioned below, there are many challenges that have still not been solved. Difficulties and Traits of Human Opponent Models The difficulty of opponent modeling in Texas Hold em Poker comes from the fact that poker is a game of imperfect information and an optimal strategy for an opponent may often involve bluffing that is, betting like they have a better hand than they actually do. Onisawa and Takahashi underline the techniques in calculating relative hand strengths and how these can only be calculated using only probability information (2348). They also state that expected hand strengths change due to the stage of the game hands that are weaker in the beginning may end up being stronger, more playable hands toward the end of the game when there are fewer players in the game (2348). They also go into what makes human opponent modeling both difficult and exciting by explaining, [a]lthough bluff is a useful and an essential strategy in a poker game, it is difficult to incorporate this strategy in a computer poker game since bluff is a kind of strategy that deceives opponent players purposefully and should not be detected by them. In this sense the incorporation of bluff strategy in a playing system is one of exciting problems. Modeling a realistic opponent often involves modeling human opponents mistakes as well as their successes. There has been much effort in simulating different player s temperaments, Teofilo and Reis describe this as an aggressiveness factor. A player s aggression factor is equal to the ratio of the number of raises that player makes to the number of calls that same player makes (2). Depending on the implementation, there may be various other characteristics that players have. Rubin and Watson define something called the exploitability factor. This is defined as a measure that determines how well a poker agent can perform against unknown competition (961). Poker Agent Simulation Techniques Bosilj et al mention some of the methods used for constructing pokerbots, including expertly defined rules, Monte Carlo methods, Case-Based reasoning, and Machine Learning based on modeling of Bayesian networks ( ). It is apparent from this that poker is a very complex game and many different methods of artificial intelligence have been applied to the game to try and simulate a master player. A rule-based expert system is defined as a collection of if-then rules for various scenarios that are likely to occur (Rubin and Watson 961). This can be particularly useful if these scenarios are encountered, but as mentioned before, the inherent complexity of the game will make the number of states that can be encountered in a single game quite large. Luckily there have been many other attempts at player simulation using different methods. Since the state space of this problem is so large it may be necessary to implement many heuristics and evolve these over time, some researchers argue that [e]volutionary methods have been shown to find relatively good results in large state spaces, and neural networks have been shown to be able to find 3
4 solutions to non-linear search problems such as Poker (Nicolai and Hilderman 125). Nicolai and Hilderman used an evolutionary approach by allowing agents to be mutated and have successful traits be selected from multiple agents, who in turn are combined as parents and then these child agents are tested for their viability (126). They used the metric that reinforces learning as the ranking of particular agents during tournament play (127). They describe how tournament ranking is not the sole way that surviving agents are determined by their co-evolution approach where: Co-evolution uses multiple, genetically distinct populations in direct competition in an effort to increase the genetic diversity of the overall playing population. Competing populations play games against each other, but are evolved separately. In this way, agents that perform at a level that would be deemed poor at the global level may still survive within their respective population. Furthermore, as the generations progress, agents enter a so-called arms race, where each population orients itself towards evolving a strategy that can defeat opponents from the other population. As these opposing strategies meet and pass each other, the overall skill level of the entire population increases (127). In their suggestion where the work should progress from here, they suggest the possibility of using a strategy using an infinite pool of their best agents (called the hall of fame) with random natural selection (130). Conclusion After analyzing the complexity, stochasticity, and the various implementation methods of Texas Hold em poker agents it is safe to say that it is still provides a domain of Artificial Intelligence that is applicable to many other problems through the techniques and the analysis applied. The derivation of a single cohesive solution seems to be far from being actualized due to the inherent complexity of the game, but the given implementations offer various benefits and disadvantages to consider for varying application scenarios. This means that although many of these agents may fail to defeat the top players in the world, there is progress being made on the methods that may allow future, more powerful hardware to have a better chance at emulating and outperforming human players. There is a great deal of uncertainty whether or not artificial pokerbots will be able to match human players. Billings et al summarizes the differences in human players to their computer counterparts by saying: Human poker players are very good at understanding their opponent, often forming an accurate model based on a single data point (and occasionally before the first hand is dealt!). Programs may never be able to match the best players in this area, but they must at least try to reduce the gap, since they can clearly be superior in other aspects of the game (235). In many ways this represents Artificial Intelligence research as a whole, where there are inherent aspects of problems that humans excel at and computers have difficulty at and many scenarios where the opposite is true. 4
5 Bibliography Billings, D., Davidson, A., Schaeffer, J., & Szafron, D. (2002). The challenge of poker. Artificial Intelligence, 32, 12, Bosilj, P., Palasek, P., Popvic, B., & Stefic, D. (2011). Simulation of a texas hold'em poker player. MIPRO 2011, 11, Gilpin, A., Sandholm, T., & Sorensen, T. B. (2008). A heads-up no-limit texas hold em poker player: Discretized betting models and automatically generated equilibrium-finding programs. Carnegie Mellon University computer science periodical, Koller, D., & Pfeffer, A. (1997). Representations and solutions for game-theoretic problems. Artificial intelligence, 94, Miltersen, P. B., & Sorensen, T. B. (2007). Near-optimal strategy for a heads-up no-limit texas hold'em poker tournament. The sixth intl. joint conf. on autonomous agents and multi-agent systems, Nicolai, G., & Hilderman, R. J. (2009). No-limit texas hold em poker agents created with evolutionary neural networks IEEE symposium on computational intelligence and games, Onisawa, T., & Takahashi, C. (2004). Poker playing system considering opponent player's strategy. SICE annual conference in Sapporo, Rubin, J., & Watson, I. (2011). Computer poker: A review. Artificial intelligence, 175, 74, Teofilo, L., & Reis, L. (2010). Universidade do Porto, Portugal. PHD Dissertation
Laboratory 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 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 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 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 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 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 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 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 informationTartanian5: 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 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 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 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 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 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 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 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 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 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 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 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 informationHow to Win Texas Hold em Poker
How to Win Texas Hold em Poker Richard Mealing Machine Learning and Optimisation Group School of Computer Science University of Manchester / 44 How to Play Texas Hold em Poker Deal private cards per player
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 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 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 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 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 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 informationUsing Artifical Neural Networks to Model Opponents in Texas Hold'em
CMPUT 499 - Research Project Review Using Artifical Neural Networks to Model Opponents in Texas Hold'em Aaron Davidson email: davidson@cs.ualberta.ca November 28th, 1999 Abstract: This paper describes
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 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 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 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 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 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 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 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 informationCourse Outline Department of Computing Science Faculty of Science. COMP 3710-3 Applied Artificial Intelligence (3,1,0) Fall 2015
Course Outline Department of Computing Science Faculty of Science COMP 710 - Applied Artificial Intelligence (,1,0) Fall 2015 Instructor: Office: Phone/Voice Mail: E-Mail: Course Description : Students
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 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 informationPoker with a Three Card Deck 1
Poker with a Three Card Deck We start with a three card deck containing one Ace, one King and one Queen. Alice and Bob are each dealt one card at random. There is a pot of $ P (and we assume P 0). Alice
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 informationMathematical Analysis Of Packs Poker. September 22, 2014. Prepared For John Feola New Vision Gaming 5 Samuel Phelps Way North Reading, MA 01864
Mathematical Analysis Of Packs Poker September 22, 2014 Prepared For John Feola New Vision Gaming 5 Samuel Phelps Way North Reading, MA 01864 Office: 978 664-1515 Fax: 978-664 - 5117 www.newvisiongaming.com
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 informationMIT 15.S50 LECTURE 2. Wednesday, January 16 th, 2013
MIT 15.S50 LECTURE 2 Wednesday, January 16 th, 2013 OUTLINE: POSTFLOP PLAY 1 st Half: Continuation Betting, Fundamental Theorem of Poker, Flop Analysis 2 nd Half: Implied Odds, Set Mining, Bluffing (with
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 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 informationA Sarsa based Autonomous Stock Trading Agent
A Sarsa based Autonomous Stock Trading Agent Achal Augustine The University of Texas at Austin Department of Computer Science Austin, TX 78712 USA achal@cs.utexas.edu Abstract This paper describes an autonomous
More informationWhen Machine Learning Meets AI and Game Theory
1 When Machine Learning Meets AI and Game Theory Anurag Agrawal, Deepak Jaiswal Abstract We study the problem of development of intelligent machine learning applications to exploit the problems of adaptation
More informationCapturing The Information Conveyed By Opponents Betting Behavior in Poker
Capturing The Information Conveyed By Opponents Betting Behavior in Poker Eric Saund 469 Clifton Avenue San Carlos, CA 94070 saund@alum.mit.edu Abstract This paper develops an approach to the capture and
More informationVideo Poker in South Carolina: A Mathematical Study
Video Poker in South Carolina: A Mathematical Study by Joel V. Brawley and Todd D. Mateer Since its debut in South Carolina in 1986, video poker has become a game of great popularity as well as a game
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 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 informationGenetic Algorithms and Sudoku
Genetic Algorithms and Sudoku Dr. John M. Weiss Department of Mathematics and Computer Science South Dakota School of Mines and Technology (SDSM&T) Rapid City, SD 57701-3995 john.weiss@sdsmt.edu MICS 2009
More informationCurrent California Math Standards Balanced Equations
Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000.
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 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 informationOptimal Drawing Strategy for Deuces Wild Video Poker Stephen Jensen
Optimal Drawing Strategy for Deuces Wild Video Poker Stephen Jensen Abstract Video poker, an extremely popular computerized version of the traditional game of poker, offers many opportunities for mathematical
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 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 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 informationUnderstanding Proactive vs. Reactive Methods for Fighting Spam. June 2003
Understanding Proactive vs. Reactive Methods for Fighting Spam June 2003 Introduction Intent-Based Filtering represents a true technological breakthrough in the proper identification of unwanted junk email,
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 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 informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationAN ANALYSIS OF A WAR-LIKE CARD GAME. Introduction
AN ANALYSIS OF A WAR-LIKE CARD GAME BORIS ALEXEEV AND JACOB TSIMERMAN Abstract. In his book Mathematical Mind-Benders, Peter Winkler poses the following open problem, originally due to the first author:
More informationStandards of play: Type of gaming table utilized for this game:
*Let It Ride Bonus is owned, patented and/or copyrighted by SHFL Entertainment, Inc. Please submit your agreement with Owner authorizing play of Game in your gambling establishment together with any request
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 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 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 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 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 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 informationProblem of the Month: Fair Games
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
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 informationBCS HIGHER EDUCATION QUALIFICATIONS Level 6 Professional Graduate Diploma in IT. March 2013 EXAMINERS REPORT. Knowledge Based Systems
BCS HIGHER EDUCATION QUALIFICATIONS Level 6 Professional Graduate Diploma in IT March 2013 EXAMINERS REPORT Knowledge Based Systems Overall Comments Compared to last year, the pass rate is significantly
More informationMath 728 Lesson Plan
Math 728 Lesson Plan Tatsiana Maskalevich January 27, 2011 Topic: Probability involving sampling without replacement and dependent trials. Grade Level: 8-12 Objective: Compute the probability of winning
More informationHow to Use Problem-Solving Simulations to Improve Knowledge, Skills, and Teamwork
How to Use Problem-Solving Simulations to Improve Knowledge, Skills, and Teamwork By Janet L. Szumal, Ph.D. Human Synergistics/Center for Applied Research, Inc. Reprinted from Mel Silberman and Pat Philips
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 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 informationA No Limit Texas Hold em Poker Playing Agent
Imperial College of Science, Technology and Medicine (University of London) Department of Computing A No Limit Texas Hold em Poker Playing Agent Lawrence R. Booker lrb03@doc.ic.ac.uk Project supervisor
More informationDetection. Perspective. Network Anomaly. Bhattacharyya. Jugal. A Machine Learning »C) Dhruba Kumar. Kumar KaKta. CRC Press J Taylor & Francis Croup
Network Anomaly Detection A Machine Learning Perspective Dhruba Kumar Bhattacharyya Jugal Kumar KaKta»C) CRC Press J Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor
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 informationFighting an Almost Perfect Crime
Fighting an Almost Perfect Crime Online Poker Fraud Detection Philipp Schosteritsch, MSc Consultant for Software Development and Technical Advisor on Cyber Security & Privacy Foundation Graz, 2014 Contents
More informationScaling Bayesian Network Parameter Learning with Expectation Maximization using MapReduce
Scaling Bayesian Network Parameter Learning with Expectation Maximization using MapReduce Erik B. Reed Carnegie Mellon University Silicon Valley Campus NASA Research Park Moffett Field, CA 94035 erikreed@cmu.edu
More informationAdvanced Techniques for Solving Optimization Problems through Evolutionary Algorithms
PhD Final Presentation Advanced Techniques for Solving Optimization Problems through Evolutionary Algorithms Advisor: Giovanni Squillero PhD Candidate: Marco Gaudesi CAD Group DAUIN February 27 th, 2015
More informationAN A R TI F I C I AL I N TE L L IG E N C E AG E N T F OR TE X A S HOL D E M P O KE R P ATRI C K M C CU R LEY 0 62 4 91 7 90
AN A R TI F I C I AL I N TE L L IG E N C E AG E N T F OR TE X A S HOL D E M P O KE R P ATRI C K M C CU R LEY 0 62 4 91 7 90 2 An Artificial Intelligence Agent for Texas Hold em Poker I declare tha t this
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 informationARTIFICIAL INTELLIGENCE: DEFINITION, TRENDS, TECHNIQUES, AND CASES
ARTIFICIAL INTELLIGENCE: DEFINITION, TRENDS, TECHNIQUES, AND CASES Joost N. Kok, Egbert J. W. Boers, Walter A. Kosters, and Peter van der Putten Leiden Institute of Advanced Computer Science, Leiden University,
More informationNEUROEVOLUTION OF AUTO-TEACHING ARCHITECTURES
NEUROEVOLUTION OF AUTO-TEACHING ARCHITECTURES EDWARD ROBINSON & JOHN A. BULLINARIA School of Computer Science, University of Birmingham Edgbaston, Birmingham, B15 2TT, UK e.robinson@cs.bham.ac.uk This
More informationLab 11. Simulations. The Concept
Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that
More informationIn the situations that we will encounter, we may generally calculate the probability of an event
What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead
More informationConvergence of online gambling and social media
Convergence of online gambling and social media 8th European Conference on Gambling Studies and Policy Issues 14.-17.9.2010, Vienna Jani Kinnunen University of Tampere, Finland Game Research Lab Social
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 informationAbout the Author. The Role of Artificial Intelligence in Software Engineering. Brief History of AI. Introduction 2/27/2013
About the Author The Role of Artificial Intelligence in Software Engineering By: Mark Harman Presented by: Jacob Lear Mark Harman is a Professor of Software Engineering at University College London Director
More informationBowdoin Computer Science
Bowdoin Science What is computer science, what are its applications in other disciplines, and its impact in society? 101: Introduction to CS Pre-requisites: none Assumes no prior knowledge of programming
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 informationCOMPUTING DURATION, SLACK TIME, AND CRITICALITY UNCERTAINTIES IN PATH-INDEPENDENT PROJECT NETWORKS
Proceedings from the 2004 ASEM National Conference pp. 453-460, Alexandria, VA (October 20-23, 2004 COMPUTING DURATION, SLACK TIME, AND CRITICALITY UNCERTAINTIES IN PATH-INDEPENDENT PROJECT NETWORKS Ryan
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 informationThe cloth covering a Three Card Poker 6 Card Bonus table (the layout) shall have wagering areas for eight players.
*Three Card Poker 6 Card Bonus is owned, patented and/or copyrighted by SHFL Entertainment, Inc. Please submit your agreement with Owner authorizing play of Game in your gambling establishment together
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 information