Home > Combinatorial game theory

Contents

1 Formal definitions

2 Finite nonloopy games

3 Nimbers

4 Domineering

5 See also

Combinatorial game theory (CGT) is a mathematical theory of games, which while part of game theory in a broad sense has its own tradition going back to the solution of Nim. It deals abstractly with a very large range of games for two players (only) that can be reduced to tree-like structures, with a characteristic ending rule: the player left with no (legal) play loses. On that rather slender basis has been constructed a theory that can be applied to some traditional games (most notably go), as well as a large number of new games the investigation of which it has stimulated. The founders of the general theory were Elwyn Berlekamp, John Conway and Richard Guy , in collaborative work during the 1960s that took some time fully to be published.

For a pedagogical discussion, see Combinatorial game theory (pedagogy). For its history, see Combinatorial game theory (history).

1 Formal definitions

A structure is called a collection of games if

and

where is the power set of

and

The elements of are called games and the convention here is that they would be denoted by the upper case Latin letters G,H,K,... .

Define the binary relation, R (for reachable) between and itself by

iff .

is called loopy if where is the transitive closure of R. Otherwise, it's called nonloopy.

If there exists an element 0 of , with , then we call it the zero element. The zero element, if it exists, is unique.

2 Finite nonloopy games

If is finite and nonloopy, then it contains a zero element.

Let be the smallest collection of games containing 0 and such that

Then all finite nonloopy games are isomorphic to a subcollection of . We can work solely with .

Define a binary operator

recursiveSee: Recursion Recursive function Recursive set Recursively enumerable set Recursively enumerable language Primitive recursive function.ly by

and .

This definition of addition of games is well-definedIn mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way. The concept of well-definednes and unique; and it is commutative.

The setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now of second-player-win games, P is defined recursively. The negative of a game is defined recursively as follows:

This definition is well-defined and unique.

The relation is defined by iff . It is an equivalence relationIn mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive symmetric and transitive i. if the relation is written as ~ it holds for all a b and c in X that # (Reflexivity) a ~ a # (Symmetry) if a ~ b then b ~ a # (Trans; and it respects the addition and negative operations. Therefore, the operations + and - can be defined on the quotient set defined by the equivalence relation . Finally one can show that the addition is an abelian group operation.

3 Nimbers

An impartial game is one where .

The set of nimbers is defined as the smallest subcollection containing 0 and containing for every G in the subcollection.

Nimbers are the combinatorial game theoretic analogue of the ordinal numbers. A function from the ordinals to nimbers is defined. The Sprague-Grundy theorem states that every impartial game is -equivalent to a nimber.

4 Domineering

An example of a partial game is Domineering. In this game, a grid is drawn, on which Left can play vertical dominoes and Right can play horizontal dominoes. Various interesting Games, such as hot game s, appear in Domineering, due to the fact that there is sometimes an incentive to move, and sometimes not.

5 See also

Game theory *

<Hide>

Topics: Combinatorial Game Theory, Formal Definitions, Finite Nonloopy...

Combinatorial chemistryCombinatorial chemistry involves the synthesis of, the computer simulation of, or both, a large number of different molecules and the selective testing of these to find ones with desired characteristics. Although combinatorial chemistry has only really be	Combinatorial logicThis article is not about combinatory logic, a topic in mathematical logic. In digital circuit theory, combinatorial logic (also called combinational logic is a type of logic circuit whose output is a function of only the present input. This is in contras	Combinatorial speciesThis article is about a concept in combinatorial mathematics. Another article treats the concept of species in biology. In combinatorial mathematics, the theory of combinatorial species is an abstract, systematic method for analysing discrete structures i
Combinatorial searchCombinatorial search is a branch of computer science that sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Combinatorial search algorithms solve instances of problems that are believed to	Combinatorial enumerationCombinatorics Combinatorial enumeration is a subfield of enumeration that deals with the counting of objects that have symmetries which are combinatorial in nature. See combinatorics.	Combinatorial game theoryCombinatorial game theory (CGT is a mathematical theory of games, which while part of game theory in a broad sense has its own tradition going back to the solution of Nim. It deals abstractly with a very large range of games for two players (only) that ca
Combinatorial game theory (history)	Combinatorial game theory (pedagogy)	Combinatorial topology
Combinatorial optimization	Combinatoriality	Combinatorial proof
Combinatorial class

Non User