Positional games

Positional game

A positional game is a kind of a combinatorial game for two players. It is described by: * – a finite set of elements. Often is called the board and its elements are called positions. * – a family of subsets of . These subsets are usually called the winning-sets. * A criterion for winning the game. During the game, players alternately claim previously-unclaimed positions, until one of the players wins. If all positions in are taken while no player wins, the game is considered a draw. The classic example of a positional game is Tic-tac-toe. In it, contains the 9 squares of the game-board, contains the 8 lines that determine a victory (3 horizontal, 3 vertical and 2 diagonal), and the winning criterion is: the first player who holds an entire winning-set wins. Other examples of positional games are Hex and the Shannon switching game. For every positional game there are exactly three options: either the first player has a winning strategy, or the second player has a winning strategy, or both players have strategies to enforce a draw. The main question of interest in the study of these games is which of these three options holds in any particular game. A positional game is finite, deterministic and has perfect information; therefore, in theory it is possible to create the full game tree and determine which of these three options holds. In practice, however, the game-tree might be enormous. Therefore, positional games are usually analyzed via more sophisticated combinatorial techniques. (Wikipedia).

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Spanning tree | Topological space | Strong positional game | Clique game | Discrepancy game | Finite set | Position (poker) | Hamiltonian path | Sim (pencil game) | Topological game | Arithmetic progression | Biased positional game | Shannon switching game | Banach–Mazur game | Game tree | Perfect information | Hex (board game) | Complete graph | Family of sets | Hypergraph | Arithmetic progression game | Tic-tac-toe | Avoider-Enforcer game