|
|
| Line 1: |
Line 1: |
| In [[combinatorial game theory]], '''Maker-Breaker games''' are a subclass of [[positional game]]s.<ref>J. Beck: ''Combinatorial Games: Tic-Tac-Toe Theory'', Cambridge University Press, 2008.</ref>
| | Hello! <br>I'm Arabic male :). <br>I really like Computer programming!<br><br>Feel free to visit my web blog: [http://fungonline.com/profile/124586/jumaestas Info size bike mountain bike sizing.] |
| | |
| It is a two-person [[perfect information|game with complete information]] played on a [[hypergraph]] ''(V,H)'' where ''V'' is an arbitrary set (called the board of the game) and ''H'' is a family of subsets of ''V'', called the ''winning sets''. The two players alternately occupy previously unoccupied elements of ''V''.
| |
| | |
| The first player, ''Maker'', has to occupy a winning set to win; and the second player, ''Breaker'', has to stop Maker from doing so; if Breaker successfully prevents maker from occupying a winning set to the end of the game, then Breaker wins. Thus, in a Maker–Breaker positional game, Maker wins if he occupies all elements of some winning set and Breaker wins if he prevents Maker from doing so. There can be no draw in a Maker-Breaker positional game: one player always wins.
| |
| | |
| The definition of Maker-Breaker game has a subtlety when <math>|V|=\infty</math> and <math>|H|=\infty</math>. In this case we say that Breaker has a winning strategy if, for all ''j'' > 0, Breaker can prevent Maker from completely occupying a winning set by turn ''j''.
| |
| | |
| When [[tictactoe]] is played as a Maker–Breaker positional game, Maker has a winning strategy (Maker does not need to block Breaker from obtaining a winning line)
| |
| .<ref name="kruczek2010">{{cite journal| last=Kruczek| first=Klay|coauthor=Eric Sundberg|title=Potential-based strategies for tic-tac-toe on the integer latticed with numerous directions| journal=[[The Electronic Journal of Combinatorics]]| year=2010| volume=17| pages=R5}}</ref>
| |
| | |
| ==Maker-Breaker games on graphs==
| |
| There has been quite some research done on playing Maker-Breaker games when the board of the game is the edge-set of a [[Graph (mathematics)|graph]] <math>G=(V,E)</math> (usually taken as the complete graph) and the family of winning sets is <math>\mathcal{F}=\{F\subset E\vert G[F]\hbox{ has property }\mathcal{P}\}</math>, where <math>\mathcal{P}</math> is some [[graph property]] (usually taken to be monotone increasing) such as connectivity (see, e.g.,<ref name="ChvatalErdos1978">{{cite journal|last=Chvatal|coauthor=Erdos|title=Biased positional games| journal=[[Annals of discrete mathematics]]| year=1978| volume=2| pages=221–229}}</ref>).
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| [[Category:Combinatorial game theory]]
| |
Hello!
I'm Arabic male :).
I really like Computer programming!
Feel free to visit my web blog: Info size bike mountain bike sizing.