Models of Combinatorial Games and Some Applications: a Survey
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Impartial Games
Combinatorial Games MSRI Publications Volume 29, 1995 Impartial Games RICHARD K. GUY In memory of Jack Kenyon, 1935-08-26 to 1994-09-19 Abstract. We give examples and some general results about impartial games, those in which both players are allowed the same moves at any given time. 1. Introduction We continue our introduction to combinatorial games with a survey of im- partial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81{116, and in ONAG [Conway 1976], particu- larly pp. 112{130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996], pp. ??{?? in this volume. An impartial game is one in which the set of Left options is the same as the set of Right options. We've noticed in the preceding article the impartial games = 0=0; 0 0 = 1= and 0; 0; = 2: {|} ∗ { | } ∗ ∗ { ∗| ∗} ∗ that were born on days 0, 1, and 2, respectively, so it should come as no surprise that on day n the game n = 0; 1; 2;:::; (n 1) 0; 1; 2;:::; (n 1) ∗ {∗ ∗ ∗ ∗ − |∗ ∗ ∗ ∗ − } is born. In fact any game of the type a; b; c;::: a; b; c;::: {∗ ∗ ∗ |∗ ∗ ∗ } has value m,wherem =mex a;b;c;::: , the least nonnegative integer not in ∗ { } the set a;b;c;::: . To see this, notice that any option, a say, for which a>m, { } ∗ This is a slightly revised reprint of the article of the same name that appeared in Combi- natorial Games, Proceedings of Symposia in Applied Mathematics, Vol. 43, 1991. Permission for use courtesy of the American Mathematical Society. -
Algorithmic Combinatorial Game Theory∗
Playing Games with Algorithms: Algorithmic Combinatorial Game Theory∗ Erik D. Demaine† Robert A. Hearn‡ Abstract Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer. 1 Introduction Many classic games are known to be computationally intractable (assuming P 6= NP): one-player puzzles are often NP-complete (as in Minesweeper) or PSPACE-complete (as in Rush Hour), and two-player games are often PSPACE-complete (as in Othello) or EXPTIME-complete (as in Check- ers, Chess, and Go). Surprisingly, many seemingly simple puzzles and games are also hard. Other results are positive, proving that some games can be played optimally in polynomial time. In some cases, particularly with one-player puzzles, the computationally tractable games are still interesting for humans to play. We begin by reviewing some basics of Combinatorial Game Theory in Section 2, which gives tools for designing algorithms, followed by reviewing the relatively new theory of Constraint Logic in Section 3, which gives tools for proving hardness. -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #34: Partisan Games (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 9 December 2016 Partisan Games • Conway's Game Theory • Hackenbush and Domineering • Four Types of Games and an Order • Some Games are Numbers • Values of Numbers • Single-Stalk Hackenbush • Some Domineering Examples Conway’s Game Theory • J. H. Conway introduced his combinatorial game theory in his 1976 book On Numbers and Games or ONAG. Researchers in the area are sometimes called onagers. • Another resource is the book Winning Ways by Berlekamp, Conway, and Guy. Conway’s Game Theory • Games, like everything else in the theory, are defined recursively. A game consists of a set of left options, each a game, and a set of right options, each a game. • The base of the recursion is the zero game, with no options for either player. • Last time we saw non-partisan games, where each player had the same options from each position. Today we look at partisan games. Hackenbush • Hackenbush is a game where the position is a diagram with red and blue edges, connected in at least one place to the “ground”. • A move is to delete an edge, a blue one for Left and a red one for Right. • Edges disconnected from the ground disappear. As usual, a player who cannot move loses. Hackenbush • From this first position, Right is going to win, because Left cannot prevent him from killing both the ground supports. It doesn’t matter who moves first. -
ES.268 Dynamic Programming with Impartial Games, Course Notes 3
ES.268 , Lecture 3 , Feb 16, 2010 http://erikdemaine.org/papers/AlgGameTheory_GONC3 Playing Games with Algorithms: { most games are hard to play well: { Chess is EXPTIME-complete: { n × n board, arbitrary position { need exponential (cn) time to find a winning move (if there is one) { also: as hard as all games (problems) that need exponential time { Checkers is EXPTIME-complete: ) Chess & Checkers are the \same" computationally: solving one solves the other (PSPACE-complete if draw after poly. moves) { Shogi (Japanese chess) is EXPTIME-complete { Japanese Go is EXPTIME-complete { U. S. Go might be harder { Othello is PSPACE-complete: { conjecture requires exponential time, but not sure (implied by P 6= NP) { can solve some games fast: in \polynomial time" (mostly 1D) Kayles: [Dudeney 1908] (n bowling pins) { move = hit one or two adjacent pins { last player to move wins (normal play) Let's play! 1 First-player win: SYMMETRY STRATEGY { move to split into two equal halves (1 pin if odd, 2 if even) { whatever opponent does, do same in other half (Kn + Kn = 0 ::: just like Nim) Impartial game, so Sprague-Grundy Theory says Kayles ≡ Nim somehow { followers(Kn) = fKi + Kn−i−1;Ki + Kn−i−2 j i = 0; 1; :::;n − 2g ) nimber(Kn) = mexfnimber(Ki + Kn−i−1); nimber(Ki + Kn−i−2) j i = 0; 1; :::;n − 2g { nimber(x + y) = nimber(x) ⊕ nimber(y) ) nimber(Kn) = mexfnimber(Ki) ⊕ nimber(Kn−i−1); nimber(Ki) ⊕ nimber(Kn−i−2) j i = 0; 1; :::n − 2g RECURRENCE! | write what you want in terms of smaller things Howe do w compute it? nimber(K0) = 0 (BASE CASE) nimber(K1) = mexfnimber(K0) ⊕ nimber(K0)g 0 ⊕ 0 = 0 = 1 nimber(K2) = mexfnimber(K0) ⊕ nimber(K1); 0 ⊕ 1 = 1 nimber(K0) ⊕ nimber(K0)g 0 ⊕ 0 = 0 = 2 so e.g. -
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game
Combinatorial Game Theory, Well-Tempered Scoring Games, and a Knot Game Will Johnson June 9, 2011 Contents 1 To Knot or Not to Knot 4 1.1 Some facts from Knot Theory . 7 1.2 Sums of Knots . 18 I Combinatorial Game Theory 23 2 Introduction 24 2.1 Combinatorial Game Theory in general . 24 2.1.1 Bibliography . 27 2.2 Additive CGT specifically . 28 2.3 Counting moves in Hackenbush . 34 3 Games 40 3.1 Nonstandard Definitions . 40 3.2 The conventional formalism . 45 3.3 Relations on Games . 54 3.4 Simplifying Games . 62 3.5 Some Examples . 66 4 Surreal Numbers 68 4.1 Surreal Numbers . 68 4.2 Short Surreal Numbers . 70 4.3 Numbers and Hackenbush . 77 4.4 The interplay of numbers and non-numbers . 79 4.5 Mean Value . 83 1 5 Games near 0 85 5.1 Infinitesimal and all-small games . 85 5.2 Nimbers and Sprague-Grundy Theory . 90 6 Norton Multiplication and Overheating 97 6.1 Even, Odd, and Well-Tempered Games . 97 6.2 Norton Multiplication . 105 6.3 Even and Odd revisited . 114 7 Bending the Rules 119 7.1 Adapting the theory . 119 7.2 Dots-and-Boxes . 121 7.3 Go . 132 7.4 Changing the theory . 139 7.5 Highlights from Winning Ways Part 2 . 144 7.5.1 Unions of partizan games . 144 7.5.2 Loopy games . 145 7.5.3 Mis`eregames . 146 7.6 Mis`ereIndistinguishability Quotients . 147 7.7 Indistinguishability in General . 148 II Well-tempered Scoring Games 155 8 Introduction 156 8.1 Boolean games . -
Combinatorial Game Theory
Combinatorial Game Theory Aaron N. Siegel Graduate Studies MR1EXLIQEXMGW Volume 146 %QIVMGER1EXLIQEXMGEP7SGMIX] Combinatorial Game Theory https://doi.org/10.1090//gsm/146 Combinatorial Game Theory Aaron N. Siegel Graduate Studies in Mathematics Volume 146 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 91A46. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-146 Library of Congress Cataloging-in-Publication Data Siegel, Aaron N., 1977– Combinatorial game theory / Aaron N. Siegel. pages cm. — (Graduate studies in mathematics ; volume 146) Includes bibliographical references and index. ISBN 978-0-8218-5190-6 (alk. paper) 1. Game theory. 2. Combinatorial analysis. I. Title. QA269.S5735 2013 519.3—dc23 2012043675 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
Crash Course on Combinatorial Game Theory
CRASH COURSE ON COMBINATORIAL GAME THEORY ALFONSO GRACIA{SAZ Disclaimer: These notes are a sketch of the lectures delivered at Mathcamp 2009 containing only the definitions and main results. The lectures also included moti- vation, exercises, additional explanations, and an opportunity for the students to discover the patterns by themselves, which is always more enjoyable than merely reading these notes. 1. The games we will analyze We will study combinatorial games. These are games: • with two players, who take turns moving, • with complete information (for example, there are no cards hidden in your hand that only you can see), • deterministic (for example, no dice or flipped coins), • guaranteed to finish in a finite amount of time, • without loops, ties, or draws, • where the first player who cannot move loses. In this course, we will specifically concentrate on impartial combinatorial games. Those are combinatorial games in which the legal moves for both players are the same. 2. Nim Rules of nim: • We play with various piles of beans. • On their turn, a player removes one or more beans from the same pile. • Final position: no beans left. Notation: • A P-position is a \previous player win", e.g. (0,0,0), (1,1,0). • An N-position is a \next player win", e.g. (1,0,0), (2,1,0). We win by always moving onto P-positions. Definition. Given two natural numbers a and b, their nim-sum, written a ⊕ b, is calculated as follows: • write a and b in binary, • \add them without carrying", i.e., for each position digit, 0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1. -
Pumac 2018 Power Round: “Life Is a Game I Lost...”
PUMaC 2018 Power Round Solutions Page 1 PUMaC 2018 Power Round: \Life is a game I lost..." November 18, 2018 \The Game gives you a Purpose. The Real Game is, to Find a Purpose." | Vineet Raj Kapoor Contents 0 Acknowledgements 2 1 Surreal Numbers (93 points) 2 1.1 Defining the Surreal Numbers (41 points) . .2 1.2 General Statements about Surreal Numbers (56 points) . .5 2 Introduction to Combinatorial Game Theory (57 points) 9 2.1 Combinatorial Game Definitions (3 points) . .9 2.2 G~ (16 points) . .9 2.3 G (38 points) . 11 3 Nim and the Sprague-Grundy Theorem (92 points) 15 3.1 Nim (18 points) . 15 3.2 Nim Variants (41 points) . 16 3.3 Sprague-Grundy (33 points) . 17 4 Specific Games & Questions (240 points) 20 4.1 Toads and Frogs (75 points) . 20 4.2 Partizan Splittles (90 points) . 21 4.3 Wythoff (95 points) . 24 PUMaC 2018 Power Round Solutions Page 2 0 Acknowledgements We'd like to take a moment to thank the individuals without whose contributions this Power Round would not have been nearly as successful: • Evan Chen and Bryan Gillespie, whose thorough and thoughtful remarks helped guide this round to completion in the weeks leading up to PUMaC 2018. A good deal of content was drawn from Combinatorial Game Theory by Aaron Siegel. Students looking for a more comprehensive treatment may want to look into Siegel's text. Much of the framework of the Surreal Numbers section was adapted from On Numbers and Games by John Conway. Although we created many of these problems ourselves, several were adapted straight from Conway's book, such as Problem 1.1.9, 1.2.1, 1.2.2, 1.2.3, 1.2.4, and 1.2.5. -
Notes on Impartial Game Theory
IMPARTIAL GAMES KENNETH MAPLES In the theory of zero-sum and general-sum games, we studied games where each player made their moves simultaneously. Equivalently, we can think of the moves of the alternate player as hidden until both have made their decision. In this part of the course, we will instead consider games that have complete information; that is, both players know the entire state of the game. This means that players move sequentially and there are no unknown or random factors in the game. We will first consider the theory of impartial games. An impartial game is one where two players move alternately according to a set of rules. If either player has no available moves when it is her turn, that player loses. We also require that both players have the same set of moves from a position. The theory of game of complete information where the players have different sets of moves available is called the theory of partizan games, which we will cover in the next week of the course. For our analysis the following definition is very convenient. Definition. An impartial game is a set. We will call the elements of a game A the options of the game, where they represent the resulting game after the move. Thus the game fg, denoting the empty set, is a game where the player-to-move loses, and the game ffgg, denoting a set containing the empty set, is a game where the player-to-move wins (as after her move, teh opponent is given a losing game). -
COMPSCI 575/MATH 513 Combinatorics and Graph Theory
COMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #35: Conway’s Number System (from Conway, On Numbers and Games and Berlekamp, Conway, and Guy, Winning Ways) David Mix Barrington 12 December 2016 Conway’s Number System • Review: Games and Numbers • Single-Stalk Hackenbush • What are Real Numbers? • Infinitesimals • Ordinals • The Games “Up” and “Down” • Multiplying Numbers Conway’s Combinatorial Games • Conway recursively defines a game to be a set of left options and a set of right options, each of which is a game. • The base case of the recursion is the zero game with no options for either player (so that the second player wins). • Any game must end in a finite (but possibly unbounded) number of moves, even if it has infinitely many states. The Order on Games • Given any game, there is a winner under optimal play if Left moves first, and a winner under optimal play if Right moves first. • A game where Left wins in both scenarios is called positive, and one where Right wins both is called negative. One where the first player wins is called fuzzy, and one where the second player wins is called a zero game. • Non-partisan games, where both players have the same options, are always zero or fuzzy. The Order on Games • We define a partial order on games denoted by the usual symbols >, ≥, =, ≤, and <. • Any game G has an additive inverse -G made by switching the roles of Left and Right. Using the game sum operation, G + (-G) is always a zero game. • Given two games G and H, we say that G > H if G - H is positive, and that G < H if G - H is negative. -
Combinatorial Game Complexity: an Introduction with Poset Games
Combinatorial Game Complexity: An Introduction with Poset Games Stephen A. Fenner∗ John Rogersy Abstract Poset games have been the object of mathematical study for over a century, but little has been written on the computational complexity of determining important properties of these games. In this introduction we develop the fundamentals of combinatorial game theory and focus for the most part on poset games, of which Nim is perhaps the best- known example. We present the complexity results known to date, some discovered very recently. 1 Introduction Combinatorial games have long been studied (see [5, 1], for example) but the record of results on the complexity of questions arising from these games is rather spotty. Our goal in this introduction is to present several results— some old, some new—addressing the complexity of the fundamental problem given an instance of a combinatorial game: Determine which player has a winning strategy. A secondary, related problem is Find a winning strategy for one or the other player, or just find a winning first move, if there is one. arXiv:1505.07416v2 [cs.CC] 24 Jun 2015 The former is a decision problem and the latter a search problem. In some cases, the search problem clearly reduces to the decision problem, i.e., having a solution for the decision problem provides a solution to the search problem. In other cases this is not at all clear, and it may depend on the class of games you are allowed to query. ∗University of South Carolina, Computer Science and Engineering Department. Tech- nical report number CSE-TR-2015-001. -
New Results for Domineering from Combinatorial Game Theory Endgame Databases
New Results for Domineering from Combinatorial Game Theory Endgame Databases Jos W.H.M. Uiterwijka,∗, Michael Bartona aDepartment of Knowledge Engineering (DKE), Maastricht University P.O. Box 616, 6200 MD Maastricht, The Netherlands Abstract We have constructed endgame databases for all single-component positions up to 15 squares for Domineering. These databases have been filled with exact Com- binatorial Game Theory (CGT) values in canonical form. We give an overview of several interesting types and frequencies of CGT values occurring in the databases. The most important findings are as follows. First, as an extension of Conway's [8] famous Bridge Splitting Theorem for Domineering, we state and prove another theorem, dubbed the Bridge Destroy- ing Theorem for Domineering. Together these two theorems prove very powerful in determining the CGT values of large single-component positions as the sum of the values of smaller fragments, but also to compose larger single-component positions with specified values from smaller fragments. Using the theorems, we then prove that for any dyadic rational number there exist single-component Domineering positions with that value. Second, we investigate Domineering positions with infinitesimal CGT values, in particular ups and downs, tinies and minies, and nimbers. In the databases we find many positions with single or double up and down values, but no ups and downs with higher multitudes. However, we prove that such single-component ups and downs easily can be constructed, in particular that for any multiple up n·" or down n·# there are Domineering positions of size 1+5n with these values. Further, we find Domineering positions with 11 different tinies and minies values.