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, 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.

Solving a parity game played on a finite graph means deciding, for a given starting position, which of the two players has a winning strategy. It has been shown that this problem is in NP and Co-NP, as well as UP and co-UP. It remains an open question whether this decision problem is solvable in PTime.

Given that parity games are history-free determined, solving a given parity game is equivalent to solving the following simple looking graph-theoretic problem. Given a finite colored directed bipartite graph with n vertices ${\displaystyle V=V_{0}\cup V_{1}}$, and V colored with colors from 1 to m, is there a choice function selecting a single out-going edge from each vertex of ${\displaystyle V_{0}}$, such that the resulting subgraph has the property that in each cycle the largest occurring color is even.

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