Conditional combinatorial games

Martin Muller

報告人 :張歐丞

Outline

1. Introduction

2. Combinatorial game theory

3. Conditional combinatorial games

4. Application to Go

1.Introduction

Classical combinatorial game studies games that can be represented as sums of independent subgames.

In many games true independence of subgames is a rare event.

Describe weak dependencies between games in a concise

2.Combinatorial game theory

Combinatorial game theory breaks up game positions into pieces and analyzes an overrall game in terms of these smaller local subgames.

A combinatorial game is defined recursively by G={GL|GR}, where GL and GR are sets of combinatorial games representing the move options for the players Left (L) and Right (R) in G.

2.Combinatorial game theoryexample

Nim game: At each move,a player removes an arbitrary

number of tokens from a single heap, and whoever runs out of moves first loses.

3.Conditional combinatorial games

Only a restricted type of dependency between CCG is allowed.

The idea of CCG is to develop a framework for describing loosely coupled games, which takes the dependencies between games into account.

Example:Prime-Nim

1. As in normal Nim, players remove a number of tokens from a single heap at each move.

2. Additionally, after each move the total number of tokens in the whole game must be a prime number.

Let *n denote a heap of n pebbles.

P is the predicate.

*n={*0P,*1P,…,*(n-1)P|*0P,*1P,…,*(n-1)P}

In first exp:

CCG sum=*5+*3+*4

in here *3 are {*2Prime(5+2+4),*1Prime(5+1+4),*0Prime(5+0+4)}

{*2true,*1false,*0false}{*2}

Isolated view of a CCG

a CCG is a function that maps tuples consisting of a local game state and the boolean values

describes the behavior of a CCG under all possible combinations of context condition values

Embedded view of a CCG

the current value of all context conditions and therefore the set of currently legal moves are defined at each move of the global game as functions of the global game state.

Differences CCG & Combinatorial games

Each CCG is played in the context of a sum of other CCG, which are part of the overall game position.

Most properties of independent combinatorial games are lost in CCG.

A few properties are still valid even in CCG,