Positional game

Article summary

A positional game in game theory is a kind of a combinatorial game for two players. It is described by: * X a finite set of elements. Often X is called the board and its elements are called positions. * \mathcal{F} a family of subsets of X . 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 X 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, X contains the 9 squares of the game-board, \mathcal{F} 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.…

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