page_title: Parity game
hierarchical_section_title: Parity game
caption_attribution_description: English: The image shows a simple example parity game with winning regions and winning strategies.
context_page_description: A parity game is played on a colored directed graph, where each node has been colored by a priority – one of finitely many natural numbers. Two players, 0 and 1, move a token along the edges of the graph. The owner of the node that the token falls on selects the successor node, resulting in a path, called the play.
  The winner of a finite play is the player whose opponent is unable to move. The winner of an infinite play is determined by the priorities appearing in the play. Typically, player 0 wins an infinite play if the largest priority that occurs infinitely often in the play is even. Player 1 wins otherwise. This explains the word "parity" in the title.
  Parity games lie in the third level of the Borel hierarchy, and are consequently determined.
  Games related to parity games were implicitly used in Rabin's
  proof of decidability of second-order theory of n successors, where determinacy of such games was
  proven. The Knaster–Tarski theorem leads to a relatively simple proof of determinacy of parity games.
  Moreover, parity games are history-free determined.
context_section_description: A parity game is played on a colored directed graph, where each node has been colored by a priority – one of (usually) finitely many natural numbers.  Two players, 0 and 1, move a (single, shared) token along the edges of the graph. The owner of the node that the token falls on selects the successor node, resulting in a (possibly infinite) path, called the play.
  The winner of a finite play is the player whose opponent is unable to move. The winner of an infinite play is determined by the priorities appearing in the play. Typically, player 0 wins an infinite play if the largest priority that occurs infinitely often in the play is even. Player 1 wins otherwise. This explains the word "parity" in the title.
  Parity games lie in the third level of the Borel hierarchy, and are consequently determined.
  Games related to parity games were implicitly used in Rabin's
  proof of decidability of second-order theory of n successors, where determinacy of such games was
  proven. The Knaster–Tarski theorem leads to a relatively simple proof of determinacy of parity games.
  Moreover, parity games are history-free determined. This means that if a player has a winning strategy then that player has a winning strategy that depends only on the current board position, and not on the history of the play.
caption: A parity game. Circular nodes belong to player 0, rectangular nodes belong to player 1. On the left side is the winning region of player 0, on the right side is the winning region of player 1.
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