Complexity appeal and challenges of combinatorial games
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Complexity, Appeal and Challenges of Combinatorial Games. Aviezri S. Fraenkel. Outline. Introduction Play game and ponder Some definitions, notations, and properties. Introduction. We’re not familiar to this field Games Playgames Chess, checker, poker, go, … Mathgames

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Complexity, Appeal and Challenges of Combinatorial Games

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Complexity appeal and challenges of combinatorial games

Complexity, Appeal and Challenges of Combinatorial Games

Aviezri S. Fraenkel


Outline

Outline

  • Introduction

  • Play game and ponder

  • Some definitions, notations, and properties


Introduction

Introduction

  • We’re not familiar to this field

  • Games

    • Playgames

      • Chess, checker, poker, go, …

    • Mathgames

      • Nim, Nimania, Welter’s game, …


Complexity appeal and challenges of combinatorial games

Nim

  • Given a finite number of tokens, arranged in piles. A move consists of selecting a pile and removing from it a positive number of tokens, possibly the entire pile

  • The player making the last move wins, the opponent loses


Match 1

Match 1

  • ccli vs shou


Complexity appeal and challenges of combinatorial games

Nim

  • Winning strategy: the XOR of the binary representation of the pile sizes is computed. If the XOR is nonzero, the next player can win

  • Input size:

    • ni is the size of the ith pile

  • Strategy computation: linear in input size

  • Nim is one of the simplest games


Nimania

Nimania

  • There are some numbers. At the kth stage, a player choose to remove a 1, or break a number m into (k+1) copies of (m-1)

  • At the first, there is only one number n

  • The player making the last move wins, the opponent loses


Match 2

Match 2

  • kero vs pheno

  • n = 3


Nimania1

Nimania


Nimania2

Nimania

  • If n = 4, the loser can delay the winner so that play lasts over 244 moves

  • Player I can win for every n 1

  • For n  4, the smallest number of move is

  • For n  4, Player I has a robust winning strategy


Definition 4 1

Definition 4.1

  • A game is impartial if the options (moves) of all positions are the same for both players. Otherwise the game is partizan

  • The game graph of a game  is a digraph G=(V,E), in which every vertex uV represents a game position, and there is a directed edge (u,v)E iff there is a move from u to v in 


Welter s game

Welter’s Game

  • Let’s play on the table, not on the screen 

  • Try to find the winning strategy

  • Nim and Nimania are disjoint sum

  • Welter’s game is not disjoint sum


Domineering

Domineering

  • A chessboard or other doubly ruled board is tiled with dominoes. Every dominoe covers two adjacent squares

  • Left tiles vertically, Right horizontally

  • The player first unable to move loses

  • Domineering is partizan


Grundy s game

Grundy’s Game

  • Given a finite number of piles of finitely many tokens, select a pile and split it into two nonempty piles ofdifferent sizes

  • The player first unable to move loses

  • Find the winning strategy if it is not necessary to split into different sizes


Poset game

Poset Game

  • There are games played on partially ordered set

  • Example

    • Chomp


Chomp

Chomp

  • Two players alternatively move on a given mn matrix of 1’s. For a technical reason there is a 0 at the origin.

  • A move consists of pointing to some 1, say at location (i, j), and removing the entire north-east sector

  • The player removing the last 1 wins


Match

Match ?


Chomp1

Chomp

  • Player I can win

    • A neat proof

    • Could you give me a contructive, polynomial strategy?


Notations

Notations

  • A P-position in a game is any position u from which the previous player can force a win

  • P is the set of all P-position of a game

  • N-position, tie position, D-position, N, D

  • F(u) is the set of all (immediate) follows of position u


Some properties

Some Properties

  • Normal play of a game is when the player making the last move in a game wins; misère play, when the player making the last move loses


Superset game

Superset Game

  • Put

  • A move consists of pointing at an as yet unremoved subset and removing it, together with all sets containing it


Superset game1

Superset Game


Superset game2

Superset Game

  • For normal play

  • Conjecture:

  • If the conjecture is true, then

    • Why?

    • Strategy


Von neumann s hackendot

Von Neumann’s Hackendot

  • It is played on a forest

  • A player points to an as yet unremoved vertex, and remove the unique path from that vertex to the root of the tree the vertex belongs to


Von neumann s hackendot1

Von Neumann’s Hackendot


Definition 8

Definition 8

  • A subset T of combinatorial games with a polynomial strategy has the following properties. For normal play of every GT, and every position u of G

    • The P-, N-, D-, or tie-label of u can be computed in polynomial time

    • The next optimal move can be computed in polynomial time


Definition 81

Definition 8

  • The winner can consummate a win in at most an exponential number of moves

  • The subset T is closed under summation, i.e., G1, G2T implies G1+G2T

  • A game in some such T is called polynomial or efficient or tractable


  • Moore s nim k

    Moore’s Nimk

    • It’s our last game!

    • It is a variation of Nim in which up to k piles can be reduced.

    • Thus Nim is Nim1

    • How to modify the XOR strategy?


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