Download 2.Combinatorial game theory

Conditional combinatorial
games
Martin Muller
報告人:張歐丞
Outline
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3.
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Introduction
Combinatorial game theory
Conditional combinatorial games
Application to Go
1.Introduction
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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
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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 theory
example
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
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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
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As in normal Nim, players remove a
number of tokens from a single heap at
each move.
Additionally, after each move the total
number of tokens in the whole game must
be a prime number.
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Let *n denote a heap of n pebbles.
P is the predicate.
 *n={*0P,*1P,…,*(n-1)P|*0P,*1P,…,*(n-1)P}
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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
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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
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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
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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,