Parity game

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 (does the next move). The players keep moving the token, resulting in a (possibly infinite) path, called a 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.^[1]

Games related to parity games were implicitly used in Rabin's proof of decidability of the monadic second-order theory of n successors (S2S for n = 2), where determinacy of such games was proven.^[2] The Knaster–Tarski theorem leads to a relatively simple proof of determinacy of parity games.^[3]

Moreover, parity games are history-free determined.^[3][4][5] 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.