# On the Laziness of Monte-Carlo Game Tree Search in Non-tight Situations

Save this PDF as:

Size: px
Start display at page:

Download "On the Laziness of Monte-Carlo Game Tree Search in Non-tight Situations"

## Transcription

1 Technical Report On the Laziness of Monte-Carlo Game Tree Search in Non-tight Situations September 8, 2008 Ingo Althofer Institute of Applied Mathematics Faculty of Mathematics and Computer Science Friedrich-Schiller University Jena Jena - Germany Key words: Monte-Carlo game tree search, computer Go, experimental analysis. 1 Introduction In many different games, including the heavy-weight classic Go, Monte-Carlo game tree search performs surprisingly well ([Bru 1993], [Cou 2006], [Gel 2007]). Already in several situations it was observed that Monte-Carlo (and variants and extensions of it) seems to play really well when the situation in the game is tight. In non-tight situations, however, Monte-Carlo (called MC for short in the sequel) often shows some laziness. In this report we investigate the very basic MC algorithm in a class of very simple abstract games. Our experiments indicate that MC is at its best when the board situation is balanced or even slightly worse for the player to move.

2 In Section 2 we introduce "Double Step Races", a class of very simple board games without chance and without interaction. The very short Section 3 gives the straightforward optimal strategy for Double Step Races. The basic MC algorithm is described in Section 4. Section 5 specifies the settings for our experiment, Section 6 contains the results and Section 7 the evaluation. We conclude with a short discussion in Section 8. 2 Rules of the Game "Double Step Race" There are two players, Black and White. The players move in turn, Black has the first move. Each player has only one stone. This stone is running in a horizontal lane, consisting of finitely many squares. The stone is starting in the leftmost square of its lane. A legal move consists of moving the stone one or two steps to the right in the lane. The stones of the players are running in different lanes. They do not interfere. The player who first reaches the rightmost square in his lane is winner. Naming: when the lane of player Black contains k+1 squares (inclduing starting square and goal), and the lane of player White has m+1 squares, we call the game DSR--k-vs-m, which is short for "Double Step Race with two players and lanes of lengths k and m". Diagram 2.1 The board for DSR--6-vs-6, including the feasible target squares for the first move of Black. 3 Perfect Play in Double Step Races In each situation it is optimal to move the stone two squares to the right. (The only exception is when the distance between stone and goal is an odd number. Then it is

3 equi-optimal to make a single step in the first move. Of course, in the square next to the goal the single step move is the only feasible move.) Looking at DSR--6-vs-x, Black has a forced win for all x larger or equal to 5. White has a win for all x smaller than 5. 4 Specification of Monte Carlo Game Tree Search In our experiment we will use the most basic version of MC game tree search: In a given situation S all feasible moves are determined. For each move n random games are played. The move with the best score (from the viewpoint of the player who has to move in S) is executed. "n" is called the Monte-Carlo parameter of this algorithm MC(n). 5 Setup of the Experiment In DSR--6-vs-x only for the "intermediate" values from 3 to 10 for x the outcome of the game depends on the skills of the players. We looked exactly at these games. For each such "6-vs-x" rule and each MC parameter n from {1, 2, 4, 8, 16, 32, 64} 10,000 games were played where both players used MC(n) to decide upon their moves. 6 Results of the Experiment Table 1: Number of wins for Black (from 10,000 games) 6-vs-3 6-vs-4 6-vs-5 6-vs-6 6-vs-7 6-vs-8 6-vs-9 6-vs-10 n

4 Table 2: Number of games with a (bad) single step in the first move of Black 6-vs-3 6-vs-4 6-vs-5 6-vs-6 6-vs-7 6-vs-8 6-vs-9 6-vs-10 n Evaluation of the Experiment In each column of Table 1 the scores change monotonically. For x = 3 and 4 the scores decrease towards the game-theoretic value 0. For all x > 5 the scores increase towards the game theoretic value 10,000. In each row the scores are monotonically increasing, at least in most cases. However, when comparing the columns 6-vs-5 and 6-vs-6, Black turns out to perform better in the more difficult situation (6-vs-5) for MC parameters 8 and larger. This may come as a surprise: Both games are won for Black in case of perfect play, however in 6-vs-5 the opponent is already nearer to his goal, so the situation is more tight for Black. The pairs of games (6-vs-7, 6-vs-8) and (6-vs-9, 6-vs-10) show similar performances (relevant parts printed in fat). This phenomenon is related to the phenomenon observed in Table 2. On Table 2: Single steps in the very first moves are sometimes deadly bad moves: for x = 5 and 6 they turn a won position into a lost position. In the other situations they are only "normally bad", because they decrease Black's chances in a game between non-perfect players. In each column the values are decreasing monotonically, when the MC parameter n becomes larger. This is not at all a surprise: the larger n, the better the MC algorithm. In each row of the table the values (standing for the frequency of bad or lazy first move) are first decreasing and then increasing again. In each row the minimum is achieved either for x=5 or x=6.

5 8 Discussion * We did not investigate "Double Step Races" (DSR) because they are interesting games. Just to the contrary, we invented the trivial DSR to have games as simple as possible for investigating the laziness of Monte-Carlo game tree search in non-tight situations. * The laziness of MC for non-tight instances of Double Step Races is rather easy to explain: in these instances all feasible moves get either scores near to 1 (in positions with clear advantage) or near to 0 (in positions where the player is clearly behind). This makes it more likely that a non-optimal move gets the best MC score and is thus selected instead of the best move. * Procedures like "dynamic komi" (for instance in computer Go) may try to give MC "always" the impression that the situation is tight, resulting in more goal-oriented play. * In a forthcoming paper, we will report on more general experiments with Monte- Carlo for Double Step Races: other lane lengths, MC with heuristic games instead of completely random games, non-independent MC runs, more than two players, asymmetric move rules,... Acknowledgements Sincere thanks go to Joerg Sameith. When duddling around with his nice multi-game program "McRandom" I stumbled upon the phenomenon of MC-laziness. Thanks to Bernd Bruegmann, just for being there at Jena University. He is no longer active in computer go research, but has a chair in theoretical physics. Always, when coming along on the floor at his office I get fresh motivation to think about Monte Carlo game tree search. Thanks also to Gian-Carlo Pascutto for encouraging me to write this quick report by his reply in the computer-go mailing list. (Quote: "However, there is less and less doubt in my mind about that the effect you describe exists and it hurts playing strength", September 8, 2008). References [Bru 1993] B. Bruegmann. Monte Carlo Go. Technical report, 1993, downloadable from

6 [Cou 2006] R. Coulom. Efficient selectivity and backup operators in Monte-Carlo tree search. In "Computer Games 2006" (Editors P. Ciancarini and H. J. van den Herik). Paper downloadable from [Gel 2007] S. Gelly. A contribution to reinforcement learning; application to Computer-Go. Ph.D. thesis, 2007, downloadable from [Sam 2005] J. Sameith. Website on his Monte Carlo game programs Hanfried, McRandom, and Turbo RapaNui; since 2005.

### Monte-Carlo Methods. Timo Nolle

Monte-Carlo Methods Timo Nolle Outline Minimax Expected Outcome Monte-Carlo Monte-Carlo Simulation Monte-Carlo Tree Search UCT AMAF RAVE MC-RAVE UCT-RAVE Playing Games Go, Bridge, Scrabble Problems with

### Hey, That s Not Fair! (Or is it?)

Concept Probability and statistics Number sense Activity 9 Hey, That s Not Fair! (Or is it?) Students will use the calculator to simulate dice rolls to play two different games. They will decide if the

### Gamesman: A Graphical Game Analysis System

Gamesman: A Graphical Game Analysis System Dan Garcia Abstract We present Gamesman, a graphical system for implementing, learning, analyzing and playing small finite two-person

### A Lock-free Multithreaded Monte-Carlo Tree Search Algorithm

A Lock-free Multithreaded Monte-Carlo Tree Search Algorithm Markus Enzenberger and Martin Müller Department of Computing Science, University of Alberta Corresponding author: mmueller@cs.ualberta.ca Abstract.

### Introduction. Math Bingo Rules. Math Bingo Notes

Game Information 1 Introduction Math is an activity that is perfect for reviewing key mathematics vocabulary in a unit of study. It can also be used to review any type of mathematics problem. Math provides

### A Knowledge-based Approach of Connect-Four

A Knowledge-based Approach of Connect-Four The Game is Solved: White Wins Victor Allis Department of Mathematics and Computer Science Vrije Universiteit Amsterdam, The Netherlands Masters Thesis, October

### Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

### University of Amsterdam. Encyclopedia of AI project. Tic-Tac-Toe

University of Amsterdam Encyclopedia of AI project. Tic-Tac-Toe Tic-tac-toe on fields of size n with m dimensions Comparisons and analysis of different sizes, dimensions, heuristics and search depths.

Math Board Games For School or Home Education by Teresa Evans Copyright 2005 Teresa Evans. All rights reserved. Permission is given for the making of copies for use in the home or classroom of the purchaser

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### Counting the Score: Position Evaluation in Computer Go

Counting the Score: Position Evaluation in Computer Go Martin Müller Department of Computing Science, University of Alberta Edmonton, Canada T6G 2E8 mmueller@cs.ualberta.ca Abstract Position evaluation

### Section 7C: The Law of Large Numbers

Section 7C: The Law of Large Numbers Example. You flip a coin 00 times. Suppose the coin is fair. How many times would you expect to get heads? tails? One would expect a fair coin to come up heads half

### SCRATCHING THE SURFACE OF PROBABILITY. Robert J. Russell, University of Paisley, UK

SCRATCHING THE SURFACE OF PROBABILITY Robert J. Russell, University of Paisley, UK Scratch cards are widely used to promote the interests of charities and commercial concerns. They provide a useful mechanism

### COMP-424: Artificial intelligence. Lecture 7: Game Playing

COMP 424 - Artificial Intelligence Lecture 7: Game Playing Instructor: Joelle Pineau (jpineau@cs.mcgill.ca) Class web page: www.cs.mcgill.ca/~jpineau/comp424 Unless otherwise noted, all material posted

### The New Mexico Lottery

The New Mexico Lottery 26 February 2014 Lotteries 26 February 2014 1/27 Today we will discuss the various New Mexico Lottery games and look at odds of winning and the expected value of playing the various

### Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

### From the probabilities that the company uses to move drivers from state to state the next year, we get the following transition matrix:

MAT 121 Solutions to Take-Home Exam 2 Problem 1 Car Insurance a) The 3 states in this Markov Chain correspond to the 3 groups used by the insurance company to classify their drivers: G 0, G 1, and G 2

### MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL

MAKING MATH MORE FUN BRINGS YOU FUN MATH GAME PRINTABLES FOR HOME OR SCHOOL THESE FUN MATH GAME PRINTABLES are brought to you with compliments from Making Math More Fun at and Math Board Games at Copyright

### During 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 Infinite Games JOHN H. CONWAY

More Games of No Chance MSRI Publications Volume 42, 2002 More Infinite Games JOHN H. CONWAY Abstract. Infinity in games introduces some weird and wonderful concepts. I examine a few here. My PhD thesis

### Session 8 Probability

Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

### A security analysis of the SecLookOn authentication system

Institut f. Statistik u. Wahrscheinlichkeitstheorie 1040 Wien, Wiedner Hauptstr. 8-10/107 AUSTRIA http://www.statistik.tuwien.ac.at A security analysis of the SecLookOn authentication system K. Grill Forschungsbericht

### LET S MAKE A DEAL! ACTIVITY

LET S MAKE A DEAL! ACTIVITY NAME: DATE: SCENARIO: Suppose you are on the game show Let s Make A Deal where Monty Hall (the host) gives you a choice of three doors. Behind one door is a valuable prize.

### How to Beat Online Roulette!

Martin J. Silverthorne How to Beat Online Roulette! Silverthorne Publications, Inc. How to Beat Online Roulette! COPYRIGHT 2015 Silverthorne Publications Inc. All rights reserved. Except for brief passages

### KNOWLEDGE-BASED MONTE-CARLO TREE SEARCH IN HAVANNAH. J.A. Stankiewicz. Master Thesis DKE 11-05

KNOWLEDGE-BASED MONTE-CARLO TREE SEARCH IN HAVANNAH J.A. Stankiewicz Master Thesis DKE 11-05 Thesis submitted in partial fulfilment of the requirements for the degree of Master of Science of Artificial

### Term Project: Roulette

Term Project: Roulette DCY Student January 13, 2006 1. Introduction The roulette is a popular gambling game found in all major casinos. In contrast to many other gambling games such as black jack, poker,

### Computer-Aided Game Inventing

Computer-Aided Game Inventing Ingo Althofer Faculty of Mathematics and Computer Science Friedrich-Schiller University Jena 07740 Jena -- Germany althofer @ minet.uni-jena.de 01 October, 2003 Abstract "Morphling"

### Artificial Intelligence techniques

Chapter 1. Artificial Intelligence techniques used in Go Chapter 1 Artificial Intelligence techniques used in Go Abstract This paper surveys the most used artificial intelligence techniques used in Computer

### Expected Value and the Game of Craps

Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the

### A Defensive Strategy Combined with. Threat-Space Search for Connect6

A Defensive Strategy Combined with Threat-Space Search for Connect6 k 2005 2006 ICGA Kagami ICGA Abstract The study of k-in-a-row games has produced a number of interesting results, and one of its sub-category

### THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/

THE WINNING ROULETTE SYSTEM by http://www.webgoldminer.com/ Is it possible to earn money from online gambling? Are there any 100% sure winning roulette systems? Are there actually people who make a living

### Game 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

### How To Increase Your Odds Of Winning Scratch-Off Lottery Tickets!

How To Increase Your Odds Of Winning Scratch-Off Lottery Tickets! Disclaimer: All of the information inside this report reflects my own personal opinion and my own personal experiences. I am in NO way

### Expected Value. 24 February 2014. Expected Value 24 February 2014 1/19

Expected Value 24 February 2014 Expected Value 24 February 2014 1/19 This week we discuss the notion of expected value and how it applies to probability situations, including the various New Mexico Lottery

### Predicting the World Cup. Dr Christopher Watts Centre for Research in Social Simulation University of Surrey

Predicting the World Cup Dr Christopher Watts Centre for Research in Social Simulation University of Surrey Possible Techniques Tactics / Formation (4-4-2, 3-5-1 etc.) Space, movement and constraints Data

### CASINO GAMING AMENDMENT RULE (No. 1) 2002

Queensland Subordinate Legislation 2002 No. 100 Casino Control Act 1982 CASINO GAMING AMENDMENT RULE (No. 1) 2002 TABLE OF PROVISIONS Section Page 1 Short title....................................................

### P vs NP problem in the field anthropology

Research Article P vs NP problem in the field anthropology Michael.A. Popov, Oxford, UK Email Michael282.eps@gmail.com Keywords P =?NP - complexity anthropology - M -decision - quantum -like game - game-theoretical

### Math Circles: Invariants Part 1

Math Circles: Invariants Part 1 March 23, 2011 What is an Invariant? In the simplest terms, an invariant is something that doesn t change. As easy as that sounds, invariance is a very powerful property

### How to Play Chess. The next diagram shows how the pieces are placed at the beginning of a game.

How to Play Chess Setup The players sit facing each other with the board between them. The board consists of 64 alternating colored squares. Regardless of the actual colors, the dark squares are named

### One pile, two pile, three piles

CHAPTER 4 One pile, two pile, three piles 1. One pile Rules: One pile is a two-player game. Place a small handful of stones in the middle. At every turn, the player decided whether to take one, two, or

### Week 5: Expected value and Betting systems

Week 5: Expected value and Betting systems Random variable A random variable represents a measurement in a random experiment. We usually denote random variable with capital letter X, Y,. If S is the sample

### 3.2 Roulette and Markov Chains

238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

### Created by Peter McOwan and Paul Curzon of Queen Mary, University of London with support

The sweet computer: machines that learn Created by Peter McOwan and Paul Curzon of Queen Mary, University of London with support from EPSRC and Google The sweet computer: machines that learn Age group:

### How to Read the Stat Force

How to Read the Stat Force By Gordon Waite, Pico Publishing For twenty years our Stat Attack has helped you to handicap the races, drawing statistics from our huge databases, and showing you information

### Introduction Solvability Rules Computer Solution Implementation. Connect Four. March 9, 2010. Connect Four

March 9, 2010 is a tic-tac-toe like game in which two players drop discs into a 7x6 board. The first player to get four in a row (either vertically, horizontally, or diagonally) wins. The game was first

### Monte-Carlo Hex. Tristan Cazenave 1 and Abdallah Saffidine 2

Monte-Carlo Hex Tristan Cazenave 1 and Abdallah Saffidine 2 1 LAMSADE, Université Paris-Dauphine Place Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France email: cazenave@lamsade.dauphine.fr 2

### Betting on Excel to enliven the teaching of probability

Betting on Excel to enliven the teaching of probability Stephen R. Clarke School of Mathematical Sciences Swinburne University of Technology Abstract The study of probability has its roots in gambling

### Bored MUMSians. Is it possible that he is left with the number 1?

Sam Chow One Player Games Bored MUMSians Han writes the numbers 1, 2,..., 100 on the whiteboard. He picks two of these numbers, erases them, and replaces them with their difference (so at some point in

### SLOT DESIGNER. Elements Of Slot Design 2nd Edition. Tools for professional mathematicians. GameDesignAutomation.com

SLOT DESIGNER Tools for professional mathematicians Elements Of Slot Design nd Edition GameDesignAutomation.com Copyright 03 Game Design Automation Pty Ltd All rights reserved. No part of this publication

### TABLE OF CONTENTS. ROULETTE FREE System #1 ------------------------- 2 ROULETTE FREE System #2 ------------------------- 4 ------------------------- 5

IMPORTANT: This document contains 100% FREE gambling systems designed specifically for ROULETTE, and any casino game that involves even money bets such as BLACKJACK, CRAPS & POKER. Please note although

### Unit 19: Probability Models

Unit 19: Probability Models Summary of Video Probability is the language of uncertainty. Using statistics, we can better predict the outcomes of random phenomena over the long term from the very complex,

### Games ~ 1. Triangle Sum. Poly Pick. In & Out. Skittles. Cover Up. Home Run. Cross Nim. To and Fro. Thirty one. Put Down. One or Two.

Games ~ 1 List of Contents Triangle Sum Poly Pick In & Out Skittles Cover Up Home Run Cross Nim To and Fro Thirty one Put Down One or Two Triangle Sum Players take turns moving a single marker or counter

### 21-110: Problem Solving in Recreational Mathematics

-0: Problem Solving in Recreational Mathematics Homework assignment solutions Problem. An urn contains five red balls and three yellow balls. Two balls are drawn from the urn at random, without replacement.

### Free Fall & Harvest Bingo

Free Fall & Harvest Bingo for pre-school through grade one Table of Contents: Pages 3-16 Fourteen different boards, make as many copies of each as wanted. Game boards should be printed in color. Pages

### Gaming the Law of Large Numbers

Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.

### Offline 1-Minesweeper is NP-complete

Offline 1-Minesweeper is NP-complete James D. Fix Brandon McPhail May 24 Abstract We use Minesweeper to illustrate NP-completeness proofs, arguments that establish the hardness of solving certain problems.

### The Poker Squares Challenge. Todd W. Neller

The Poker Squares Challenge Todd W. Neller What is the Poker Squares Challenge? A semester-long contest where Gettysburg College students (individuals and/or teams) compete to develop the best time-limited

### Roman Board Games. I. Lusoria

Roman Board Games I. Lusoria This is an incredibly simple game dating back to ancient Rome. The name is a minor Anglicization of the Latin tabula lusoria, "tablet for (playing) games," a name by which

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

### 1.3. Knights. Training Forward Thinking

1.3 Knights Hippogonal 2,1 movement. Captures by landing on the same square as an enemy piece. Can jump over other pieces Count one, two, turn. There has always been a horse in chess and the chess variants

### Thursday, October 18, 2001 Page: 1 STAT 305. Solutions

Thursday, October 18, 2001 Page: 1 1. Page 226 numbers 2 3. STAT 305 Solutions S has eight states Notice that the first two letters in state n +1 must match the last two letters in state n because they

### VISUAL GUIDE to. RX Scripting. for Roulette Xtreme - System Designer 2.0

VISUAL GUIDE to RX Scripting for Roulette Xtreme - System Designer 2.0 UX Software - 2009 TABLE OF CONTENTS INTRODUCTION... ii What is this book about?... iii How to use this book... iii Time to start...

by Teresa Evans Copyright 2005 Teresa Evans. All rights reserved. Permission is given for the making of copies for use in the home or classroom of the purchaser only. SAMPLE PAGES Please enjoy using these

### Fraction and Decimal

Fraction and Decimal Helping your students convert fractions and decimals, one bingo ball at a time! Created By: Stephanie Moorman Board 1 Board 2 1/2 1/4 3/4 1/3 1/5 1/4 3/5 0.40 2/5 1.00 0.5 0.25 0.75

Artificial Intelligence Adversarial Search Andrea Torsello Games Vs. Planning Problems In many problems you re are pitted against an opponent certain operators are beyond your control: you cannot control

### REWARD System For Even Money Bet in Roulette By Izak Matatya

REWARD System For Even Money Bet in Roulette By Izak Matatya By even money betting we mean betting on Red or Black, High or Low, Even or Odd, because they pay 1 to 1. With the exception of the green zeros,

### The Show Partial Parlay Wager

Frandsen Publishing Presents Favorite ALL-Ways TM Newsletter Articles The Show Partial Parlay Wager Part 1 of 2: Getting Started With the Wager Show bets? Booor-ing. Yaaawn. We think not and suspect you

### The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors.

LIVE ROULETTE The game of roulette is played by throwing a small ball onto a rotating wheel with thirty seven numbered sectors. The ball stops on one of these sectors. The aim of roulette is to predict

### 15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012

15-251: Great Theoretical Ideas in Computer Science Anupam Gupta Notes on Combinatorial Games (draft!!) January 29, 2012 1 A Take-Away Game Consider the following game: there are 21 chips on the table.

### Tactical Analysis Board Games

DM842 Computer Game Programming: AI Lecture 9 Tactical Analysis Board Games Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1. Tactical Pathfinding

### ! HW5 Due Monday 11:59 pm. ! no randomness, e.g., dice. ! Perfect information. ! No hidden cards or hidden Chess pieces. ! Finite

!! Game Trees and Heuristics Announcements! HW5 Due Monday 11:59 pm 15-211: Fundamental Data Structures and Algorithms Margaret Reid-Miller 7 April 2005 2 X O X O X O X O O Games Game Properties!!Two!

### Blunder cost in Go and Hex

Fuego MoHex computing univ alberta 2011 november Fuego MoHex 1 2 Fuego 3 MoHex 4 outline Fuego MoHex def n: to stumble blindly an ignorant move (bad when good available) a random move end game: usually

### STRIKE FORCE ROULETTE

STRIKE FORCE ROULETTE Cycles, cycles, cycles... You cannot get away from them in the game of Roulette. Red, black, red, black... Red, red, red, red. Black, black, black... Red, red, black, black... 1st

### THE PROBABILITY OF CARDS MEETING AFTER A SHUFFLE

International Journal of Pure and Applied Mathematics Volume 85 No. 5 2013, 849-857 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v85i5.3

### Worksheet for Teaching Module Probability (Lesson 1)

Worksheet for Teaching Module Probability (Lesson 1) Topic: Basic Concepts and Definitions Equipment needed for each student 1 computer with internet connection Introduction In the regular lectures in

### NAME: NUMERACY GAMES WITH DICE AND CARDS. A South African Numeracy Chair initiative

NAME: NUMERACY GAMES WITH DICE AND CARDS A South African Numeracy Chair initiative CONTENTS Games in the classroom 3 Benefits 3 Mathematical games are 'activities' which: 3 More benefits 3 Competition

### Statistics and Random Variables. Math 425 Introduction to Probability Lecture 14. Finite valued Random Variables. Expectation defined

Expectation Statistics and Random Variables Math 425 Introduction to Probability Lecture 4 Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan February 9, 2009 When a large

### Bingo Guide January 2015

What is Bingo? Conventional 75 number bingo is played with bingo cards that have a grid of five horizontal rows and five vertical columns. The columns are lettered B I N G O from left to right across the

### CONTENT LAND BASED GAMES

CONTENT LAND BASED GAMES Live Keno 4 Street Racer 6 Darts Bet 8 Western Keno 10 Kings Jewels 12 Circus Fortune 14 Sports Betting 16 Keno Bet 18 Racing Bet 20 Poker Bet (Single Table) 22 Poker Bet + (3

### We 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

### CS188 Spring 2011 Section 3: Game Trees

CS188 Spring 2011 Section 3: Game Trees 1 Warm-Up: Column-Row You have a 3x3 matrix of values like the one below. In a somewhat boring game, player A first selects a row, and then player B selects a column.

### Mus is the most popular Spanish game, due to its attractiveness and characteristics. This game was born in Basque Country

Mus is the most popular Spanish game, due to its attractiveness and characteristics. This game was born in Basque Country Object of the game The winner of the game is the team who wins three sets. The

### How to Read and Write Algebraic Chess Notation

How to Read and Write Algebraic Chess Notation Algebraic notation (or AN) is a method for recording and describing the moves in a game of chess. It is now standard among all chess organizations and most

### Random-Turn Hex and Other Selection Games

Random-Turn Hex and Other Selection Games Yu-Han Lyu March 10, 2011 Abstract In this report, we summarize the results in [6]. We define a mathematical formulation for all selection games. When considering

### This Method will show you exactly how you can profit from this specific online casino and beat them at their own game.

This Method will show you exactly how you can profit from this specific online casino and beat them at their own game. It s NOT complicated, and you DON T need a degree in mathematics or statistics to

### Math 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

### Hooray for the Hundreds Chart!!

Hooray for the Hundreds Chart!! The hundreds chart consists of a grid of numbers from 1 to 100, with each row containing a group of 10 numbers. As a result, children using this chart can count across rows

### Learn How to Use The Roulette Layout To Calculate Winning Payoffs For All Straight-up Winning Bets

Learn How to Use The Roulette Layout To Calculate Winning Payoffs For All Straight-up Winning Bets Understand that every square on every street on every roulette layout has a value depending on the bet

### Monte-Carlo Tree Search Solver

Monte-Carlo Tree Search Solver Mark H.M. Winands 1, Yngvi Björnsson 2, and Jahn-Takeshi Saito 1 1 Games and AI Group, MICC, Universiteit Maastricht, Maastricht, The Netherlands {m.winands,j.saito}@micc.unimaas.nl

### Generalized Widening

Generalized Widening Tristan Cazenave Abstract. We present a new threat based search algorithm that outperforms other threat based search algorithms and selective knowledge-based for open life and death

### I d Rather Stay Stupid: The Advantage of Having Low Utility

I d Rather Stay Stupid: The Advantage of Having Low Utility Lior Seeman Department of Computer Science Cornell University lseeman@cs.cornell.edu Abstract Motivated by cost of computation in game theory,

### Would You Like To Earn \$1000 s With The Click Of A Button?

Would You Like To Earn \$1000 s With The Click Of A Button? (Follow these easy step by step instructions and you will) This Version of the ebook is for all countries other than the USA. If you need the

### During the course of our research on NBA basketball, we found out a couple of interesting principles.

After mining all the available NBA data for the last 15 years, we found the keys to a successful basketball betting system: If you follow this system exactly, you can expect to hit 90% of your NBA bets.

### Ch. 13.3: More about Probability

Ch. 13.3: More about Probability Complementary Probabilities Given any event, E, of some sample space, U, of a random experiment, we can always talk about the complement, E, of that event: this is the

### Ready, Set, Go! Math Games for Serious Minds

Math Games with Cards and Dice presented at NAGC November, 2013 Ready, Set, Go! Math Games for Serious Minds Rande McCreight Lincoln Public Schools Lincoln, Nebraska Math Games with Cards Close to 20 -

### Monte-Carlo Tree Search (MCTS) for Computer Go

Monte-Carlo Tree Search (MCTS) for Computer Go Bruno Bouzy bruno.bouzy@parisdescartes.fr Université Paris Descartes Séminaire BigMC 5 mai 2011 Outline The game of Go: a 9x9 game The «old» approach (*-2002)

### TeachingEnglish Lesson plans

Topic: Talking About the Future - a one-hour revision lesson Aims To help students talk about the future To review 4 ways to talk about the future To help students choose the most appropriate future tense