Tic-Tac-Toe: Understanding the Minimax Algorithm
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Paper and Pencils for Everyone
(CM^2) Math Circle Lesson: Game Theory of Gomuku and (m,n,k-games) Overview: Learning Objectives/Goals: to expose students to (m,n,k-games) and learn the general history of the games through out Asian cultures. SWBAT… play variations of m,n,k-games of varying degrees of difficulty and complexity as well as identify various strategies of play for each of the variations as identified by pattern recognition through experience. Materials: Paper and pencils for everyone Vocabulary: Game – we will create a working definition for this…. Objective – the goal or point of the game, how to win Win – to do (achieve) what a certain game requires, beat an opponent Diplomacy – working with other players in a game Luck/Chance – using dice or cards or something else “random” Strategy – techniques for winning a game Agenda: Check in (10-15min.) Warm-up (10-15min.) Lesson and game (30-45min) Wrap-up and chill time (10min) Lesson: Warm up questions: Ask these questions after warm up to the youth in small groups. They may discuss the answers in the groups and report back to you as the instructor. Write down the answers to these questions and compile a working definition. Try to lead the youth so that they do not name a specific game but keep in mind various games that they know and use specific attributes of them to make generalizations. · What is a game? · Are there different types of games? · What make something a game and something else not a game? · What is a board game? · How is it different from other types of games? · Do you always know what your opponent (other player) is doing during the game, can they be sneaky? · Do all of games have the same qualities as the games definition that we just made? Why or why not? Game history: The earliest known board games are thought of to be either ‘Go’ from China (which we are about to learn a variation of), or Senet and Mehen from Egypt (a country in Africa) or Mancala. -
Bibliography of Traditional Board Games
Bibliography of Traditional Board Games Damian Walker Introduction The object of creating this document was to been very selective, and will include only those provide an easy source of reference for my fu- passing mentions of a game which give us use- ture projects, allowing me to find information ful information not available in the substan- about various traditional board games in the tial accounts (for example, if they are proof of books, papers and periodicals I have access an earlier or later existence of a game than is to. The project began once I had finished mentioned elsewhere). The Traditional Board Game Series of leaflets, The use of this document by myself and published on my web site. Therefore those others has been complicated by the facts that leaflets will not necessarily benefit from infor- a name may have attached itself to more than mation in many of the sources below. one game, and that a game might be known Given the amount of effort this document by more than one name. I have dealt with has taken me, and would take someone else to this by including every name known to my replicate, I have tidied up the presentation a sources, using one name as a \primary name" little, included this introduction and an expla- (for instance, nine mens morris), listing its nation of the \families" of board games I have other names there under the AKA heading, used for classification. and having entries for each synonym refer the My sources are all in English and include a reader to the main entry. -
Ai12-General-Game-Playing-Pre-Handout
Introduction GDL GGP Alpha Zero Conclusion References Artificial Intelligence 12. General Game Playing One AI to Play All Games and Win Them All Jana Koehler Alvaro´ Torralba Summer Term 2019 Thanks to Dr. Peter Kissmann for slide sources Koehler and Torralba Artificial Intelligence Chapter 12: GGP 1/53 Introduction GDL GGP Alpha Zero Conclusion References Agenda 1 Introduction 2 The Game Description Language (GDL) 3 Playing General Games 4 Learning Evaluation Functions: Alpha Zero 5 Conclusion Koehler and Torralba Artificial Intelligence Chapter 12: GGP 2/53 Introduction GDL GGP Alpha Zero Conclusion References Deep Blue Versus Garry Kasparov (1997) Koehler and Torralba Artificial Intelligence Chapter 12: GGP 4/53 Introduction GDL GGP Alpha Zero Conclusion References Games That Deep Blue Can Play 1 Chess Koehler and Torralba Artificial Intelligence Chapter 12: GGP 5/53 Introduction GDL GGP Alpha Zero Conclusion References Chinook Versus Marion Tinsley (1992) Koehler and Torralba Artificial Intelligence Chapter 12: GGP 6/53 Introduction GDL GGP Alpha Zero Conclusion References Games That Chinook Can Play 1 Checkers Koehler and Torralba Artificial Intelligence Chapter 12: GGP 7/53 Introduction GDL GGP Alpha Zero Conclusion References Games That a General Game Player Can Play 1 Chess 2 Checkers 3 Chinese Checkers 4 Connect Four 5 Tic-Tac-Toe 6 ... Koehler and Torralba Artificial Intelligence Chapter 12: GGP 8/53 Introduction GDL GGP Alpha Zero Conclusion References Games That a General Game Player Can Play (Ctd.) 5 ... 18 Zhadu 6 Othello 19 Pancakes 7 Nine Men's Morris 20 Quarto 8 15-Puzzle 21 Knight's Tour 9 Nim 22 n-Queens 10 Sudoku 23 Blob Wars 11 Pentago 24 Bomberman (simplified) 12 Blocker 25 Catch a Mouse 13 Breakthrough 26 Chomp 14 Lights Out 27 Gomoku 15 Amazons 28 Hex 16 Knightazons 29 Cubicup 17 Blocksworld 30 .. -
Jeanne's Favorite Games
Jeanne’s Favorite Games Game Maker Cognitive functions/operations For Younger Children Blink Out of the Box games comparing at a high rate of speed Toot and Otto Thinkfun planning, hypothetical thinking, sequencing Barnyard Critters Rio Grande Games logic, hypothetical thinking, non competitive LCR (fun family game) George and Co. LLC spatial sense for left and right Open Sesame Ravensburger memory for spatial relationships Walk the Dogs Simply Fun planning, visual transport The Storybook Game Fundex memory, vocabulary, sequencing, forming relationships Camelot Jr. Smart Games hypothetical thinking, logical evidence, planning, Feed the Kitty Gamewright visual representation, spatial orientation Quick Cups Spin Master visual processing speed, color sequencing Go Ape (great family game) Play Monster identify emotions, reading body language Rat-a-tat-tat (card game) Gamewright memory, numbers, planning, relevant cues Penguins On Ice Smart Games spatial sense, hypothetical thinking, planning Slapzi Carma Games language development, understanding negatives (no or not), relevant cues, processing speed Tenzi (fun for groups) Carma Games matching numbers, systematic search, visual processing speed, tons of fun, buy the additional pack of cards “77 Ways to Play Tenzi” Older Children - Adult Skribble TDC Games visualizing, planning, predicting Take Your Pick Simply Fun point of view, predicting Linkity Simply Fun forming relationships, Squint Junior Out of the Box games visualizing, analyzing-integrating, Khet (a very cool laser game) www.khet.com -
A Scalable Neural Network Architecture for Board Games
A Scalable Neural Network Architecture for Board Games Tom Schaul, Jurgen¨ Schmidhuber Abstract— This paper proposes to use Multi-dimensional II. BACKGROUND Recurrent Neural Networks (MDRNNs) as a way to overcome one of the key problems in flexible-size board games: scalability. A. Flexible-size board games We show why this architecture is well suited to the domain There is a large variety of board games, many of which and how it can be successfully trained to play those games, even without any domain-specific knowledge. We find that either have flexible board dimensions, or have rules that can performance on small boards correlates well with performance be trivially adjusted to make them flexible. on large ones, and that this property holds for networks trained The most prominent of them is the game of Go, research by either evolution or coevolution. on which has been considering board sizes between the min- imum of 5x5 and the regular 19x19. The rules are simple[5], I. INTRODUCTION but the strategies deriving from them are highly complex. Players alternately place stones onto any of the intersections Games are a particularly interesting domain for studies of the board, with the goal of conquering maximal territory. of machine learning techniques. They form a class of clean A player can capture a single stone or a connected group and elegant environments, usually described by a small set of his opponent’s stones by completely surrounding them of formal rules and clear success criteria, and yet they often with his own stones. A move is not legal if it leads to a involve highly complex strategies. -
A Scalable Neural Network Architecture for Board Games
A Scalable Neural Network Architecture for Board Games Tom Schaul, Jurgen¨ Schmidhuber Abstract— This paper proposes to use Multi-dimensional II. BACKGROUND Recurrent Neural Networks (MDRNNs) as a way to overcome one of the key problems in flexible-size board games: scalability. A. Flexible-size board games We show why this architecture is well suited to the domain There is a large variety of board games, many of which and how it can be successfully trained to play those games, even without any domain-specific knowledge. We find that either have flexible board dimensions, or have rules that can performance on small boards correlates well with performance be trivially adjusted to make them flexible. on large ones, and that this property holds for networks trained The most prominent of them is the game of Go, research by either evolution or coevolution. on which has been considering board sizes between the min- imum of 5x5 and the regular 19x19. The rules are simple[4], I. INTRODUCTION but the strategies deriving from them are highly complex. Players alternately place stones onto any of the intersections Games are a particularly interesting domain for studies of of the board, with the goal of conquering maximal territory. machine learning techniques. They form a class of clean and A player can capture a single stone or a connected group elegant environments, usually described by a small set of of his opponent’s stones by completely surrounding them formal rules, have very clear success criteria, and yet they with his own stones. A move is not legal if it leads to a often involve highly complex strategies. -
Adversarial Search (Game)
Adversarial Search CE417: Introduction to Artificial Intelligence Sharif University of Technology Spring 2019 Soleymani “Artificial Intelligence: A Modern Approach”, 3rd Edition, Chapter 5 Most slides have been adopted from Klein and Abdeel, CS188, UC Berkeley. Outline } Game as a search problem } Minimax algorithm } �-� Pruning: ignoring a portion of the search tree } Time limit problem } Cut off & Evaluation function 2 Games as search problems } Games } Adversarial search problems (goals are in conflict) } Competitive multi-agent environments } Games in AI are a specialized kind of games (in the game theory) 3 Adversarial Games 4 Types of Games } Many different kinds of games! } Axes: } Deterministic or stochastic? } One, two, or more players? } Zero sum? } Perfect information (can you see the state)? } Want algorithms for calculating a strategy (policy) which recommends a move from each state 5 Zero-Sum Games } Zero-Sum Games } General Games } Agents have opposite utilities } Agents have independent utilities (values on outcomes) (values on outcomes) } Lets us think of a single value that one } Cooperation, indifference, competition, maximizes and the other minimizes and more are all possible } Adversarial, pure competition } More later on non-zero-sum games 6 Primary assumptions } We start with these games: } Two -player } Tu r n taking } agents act alternately } Zero-sum } agents’ goals are in conflict: sum of utility values at the end of the game is zero or constant } Perfect information } fully observable Examples: Tic-tac-toe, -
Chapter 6 Two-Player Games
Introduction to Using Games in Education: A Guide for Teachers and Parents Chapter 6 Two-Player Games There are many different kinds of two-person games. You may have played a variety of these games such as such as chess, checkers, backgammon, and cribbage. While all of these games are competitive, many people play them mainly for social purposes. A two-person game environment is a situation that facilitates communication and companionship. Two major ideas illustrated in this chapter: 1. Look ahead: learning to consider what your opponent will do as a response to a move that you are planning. 2. Computer as opponent. In essence, this makes a two-player game into a one- player game. In addition, we will continue to explore general-purpose, high-road transferable, problem-solving strategies. Tic-Tac-Toe To begin, we will look at the game of tic-tac-toe (TTT). TTT is a two-player game, with players taking turns. One player is designated as X and the other as O. A turn consists of marking an unused square of a 3x3 grid with one’s mark (an X or an O). The goal is to get three of one’s mark in a file (vertical, horizontal, or diagonal). Traditionally, X is the first player. A sample game is given below. Page 95 Introduction to Using Games in Education: A Guide for Teachers and Parents X X X O X O Before X's O's X's game first first second begins move move move X X X X O X X O X O O O X O X O X O X O X O O's X's O's X wins on second third third X's fourth move move move move Figure 6.1. -
Inventaire Des Jeux Combinatoires Abstraits
INVENTAIRE DES JEUX COMBINATOIRES ABSTRAITS Ici vous trouverez une liste incomplète des jeux combinatoires abstraits qui sera en perpétuelle évolution. Ils sont classés par ordre alphabétique. Vous avez accès aux règles en cliquant sur leurs noms, si celles-ci sont disponibles. Elles sont parfois en anglais faute de les avoir trouvées en français. Si un jeu vous intéresse, j'ai ajouté une colonne « JOUER EN LIGNE » où le lien vous redirigera vers le site qui me semble le plus fréquenté pour y jouer contre d'autres joueurs. J'ai remarqué 7 catégories de ces jeux selon leur but du jeu respectif. Elles sont décrites ci-dessous et sont précisées pour chaque jeu. Ces catégories sont directement inspirées du livre « Le livre des jeux de pions » de Michel Boutin. Si vous avez des remarques à me faire sur des jeux que j'aurai oubliés ou une mauvaise classification de certains, contactez-moi via mon blog (http://www.papatilleul.fr/). La définition des jeux combinatoires abstraits est disponible ICI. Si certains ne répondent pas à cette définition, merci de me prévenir également. LES CATÉGORIES CAPTURE : Le but du jeu est de capturer ou de bloquer un ou plusieurs pions en particulier. Cette catégorie se caractérise souvent par une hiérarchie entre les pièces. Chacune d'elle a une force et une valeur matérielle propre, résultantes de leur capacité de déplacement. ELIMINATION : Le but est de capturer tous les pions de son adversaire ou un certain nombre. Parfois, il faut capturer ses propres pions. BLOCAGE : Il faut bloquer son adversaire. Autrement dit, si un joueur n'a plus de coup possible à son tour, il perd la partie. -
Best-First Minimax Search Richard E
Artificial Intelligence ELSEVIER Artificial Intelligence 84 ( 1996) 299-337 Best-first minimax search Richard E. Korf *, David Maxwell Chickering Computer Science Department, University of California, Los Angeles, CA 90024, USA Received September 1994; revised May 1995 Abstract We describe a very simple selective search algorithm for two-player games, called best-first minimax. It always expands next the node at the end of the expected line of play, which determines the minimax value of the root. It uses the same information as alpha-beta minimax, and takes roughly the same time per node generation. We present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth, but at significant time cost. Our actual implementation saves the subtree generated for one move that is still relevant after the player and opponent move, pruning subtrees below moves not chosen by either player. We also show how to efficiently generate a class of incremental random game trees. On uniform random game trees, best-first minimax outperforms alpha-beta, when both algorithms are given the same amount of computation. On random trees with random branching factors, best-first outperforms alpha-beta for shallow depths, but eventually loses at greater depths. We obtain similar results in the game of Othello. Finally, we present a hybrid best-first extension algorithm that combines alpha-beta with best-first minimax, and performs significantly better than alpha-beta in both domains, even at greater depths. In Othello, it beats alpha-beta in two out of three games. 1. Introduction and overview The best chess machines, such as Deep-Blue [lo], are competitive with the best humans, but generate billions of positions per move. -
Exploding Offers with Experimental Consumer Goods∗
Exploding Offers with Experimental Consumer Goods∗ Alexander L. Brown Ajalavat Viriyavipart Department of Economics Department of Economics Texas A&M University Texas A&M University Xiaoyuan Wang Department of Economics Texas A&M University August 29, 2014 Abstract Recent theoretical research indicates that search deterrence strategies are generally opti- mal for sellers in consumer goods markets. Yet search deterrence is not always employed in such markets. To understand this incongruity, we develop an experimental market where profit-maximizing strategy dictates sellers should exercise one form of search deterrence, ex- ploding offers. We find that buyers over-reject exploding offers relative to optimal. Sellers underutilize exploding offers relative to optimal play, even conditional on buyer over-rejection. This tendency dissipates when sellers make offers to computerized buyers, suggesting their persistent behavior with human buyers may be due to a preference rather than a miscalculation. Keywords: exploding offer, search deterrence, experimental economics, quantal response equi- librium JEL: C91 D21 L10 M31 ∗We thank the Texas A&M Humanities and Social Science Enhancement of Research Capacity Program for providing generous financial support for our research. We have benefited from comments by Gary Charness, Catherine Eckel, Silvana Krasteva and seminar participants of the 2013 North American Economic Science Association and Southern Economic Association Meetings. Daniel Stephenson and J. Forrest Williams provided invaluable help in conducting the experimental sessions. 1 Introduction One of the first applications of microeconomic theory is to the consumer goods market: basic supply and demand models can often explain transactions in centralized markets quite well. When markets are decentralized, often additional modeling is required to explain the interactions of consumers and producers. -
Annai Women's College
Annai Women’s College (Arts and Science) Affiliated to Bharathidasan University – Tiruchirapalli. TNPL Road ,Punnam Chattram, Karur-639 136 Department of Computer Science, Computer Applications & IT Faculty Name : Mrs.V.Bhuvaneswari,Msc., M.phil., Major : I M.Sc(Computer Science) Paper Code : P16CSE2B Title of the Paper : Artificial Intelligence INDEX S.No Topics P.No 1 UNIT I: Introduction: AI Problems - Al techniques - Criteria for 3 to 24 success. Problems, Problem Spaces, Search: State space search - Production Systems 2 UNIT II: Heuristic Search techniques: Generate and Test - Hill 25 to 35 Climbing- Best-First - Means-end analysis. Knowledge representation issues: Representations and mappings - Approaches to Knowledge representations -Issues in Knowledge representations - Frame Problem. 3 UNIT III: Using Predicate logic: Representing simple facts in logic 36 to 43 - Representing Instance and is a relationships - Computable functions and predicates - Resolution. 4 UNIT III: Using Predicate logic: Representing simple facts in logic 44 to 53 - Representing Instance and is a relationships - Computable functions and predicates - Resolution. 5 UNIT V: Game playing – The mini max search procedure – Expert 54 to 82 System - Perception and Action Artificial Intelligence Annai Women’s College,Karur. (Arts and Science) Department of Computer Science. Class: I M.Sc (Computer Science) Staff Incharge: V.Bhuvaneswari M.Sc.,M.Phil., AP in CS Department. Artificial Intelligence UNIT I: Introduction: AI Problems - Al techniques - Criteria for success. Problems, Problem Spaces, Search: State space search - Production Systems UNIT II: Heuristic Search techniques: Generate and Test - Hill Climbing- Best-First - Means-end analysis. Knowledge representation issues: Representations and mappings - Approaches to Knowledge representations -Issues in Knowledge representations - Frame Problem.