Maker-Breaker game

Maker-Breaker games are a subclass of positional games^[1].

It is a two-person 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 $|V|=\infty$ and $|H|=\infty$. 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.

It is interesting to note that 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) ^[2].

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 G = (V,E) (usually taken as the complete graph) and the family of winning sets is $\mathcal{F}=\{F\subset E\vert G[F]\hbox{ has property }\mathcal{P}\}$, where $\mathcal{P}$ is some graph property (usually taken to be monotone increasing) such as connectivity (see, e.g., ^[3]).

References

1. ^ J. Beck: Combinatorial Games: Tic-Tac-Toe Theory, Cambridge University Press, 2008.
2. ^ Kruczek, Klay; Eric Sundberg (2010). "Potential-based strategies for tic-tac-toe on the integer latticed with numerous directions". The Electronic Journal of Combinatorics 17: R5. 
3. ^ Chvatal; Erdos (1978). "Biased positional games". Annals of discrete mathematics 2: 221–229.