{
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  "slug": "positional-game",
  "title": "Positional game",
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  "lead": "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.…",
  "body": "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. In practice, however, the game-tree might be enormous. Therefore, positional games are usually analyzed via more sophisticated combinatorial techniques. ==Variants== There are many variants of positional games, differing in their rules and their winning criteria. Biased positional game: Each positional game has a biased variant, in which the first player can take p elements at a time and the second player can take q elements at a time (in the unbiased variant, p=q=1). ==See also== Topological game, a generalization of a positional game to infinite sets Banach–Mazur game, a game played on a topological space by choosing among certain subsets, with winning conditions resembling those of a maker-breaker game ==References==",
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    "title": "Positional game",
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