A positional game is a played on a finite board V, where a
family of subsets (a hypergraph) H, whose members are usually
called winning sets, is specified. The game is played by two
players, taking turns in claiming previously unoccupied elements
of V, and ends whenever there are no unoccupied elements. In
general, there are two additional parameters, p and q, the first
player takes p elements in his turn, while the second one claims
There are several types of positional games. In the so called
Strong Game, a player completing a winning set A of H first
wins, otherwise the game ends in a draw. In a Weak Game, the first
player (Maker) wins if he completes a winning set by the end of
the game, otherwise the game is won by the second player
(Breaker). In the Avoider-Enforcer version, the first player
(Avoider) aims to avoid occupying a winning set completely, while
the second player (Enforcer) tries to force Avoider to do just so.
There are also hybrid versions, where, for example, the first
player acts both as Breaker and Avoider.
There is an amazing variety of recreational and mathematical games
that can be casted into the above described framework. Examples
include Tic-Tac-Toe and its multi-dimensional generalizations, the
game of Hex played and studied by John Nash, and various
achievement games played on the edges of the complete graph K^n,
where for example Maker tries to create a Hamilton cycle, while
Breaker aims to prevent Maker from fulfilling his goal.
In this survey-type talk I will introduce the subject of positional
games, and will define and discuss a variety of types of positional
games. I will indicate some typical approaches and tools available.
Some recent results will be discussed too.
I will stress a perhaps surprising yet quite ubiquitous role of
probabilistic intuition in analyzing these deterministic games.
No experience (theoretical at least) with positional games will be