Parity Games

Years and Authors of Summarized Original Work

• 1991; Emerson, Jutla

• 1991; Mostowski

Problem Definition

A parity game is an infinite duration game, played by players odd and even, denoted by $\square$ and $\diamond$, respectively on a directed, finite graph. Throughout this note, we let $\bigcirc$ denote an arbitrary player and we write $\bar{\bigcirc }$ for $\bigcirc$’s opponent; i.e. $\bar{\diamond } = \square$ and $\bar{\square } = \diamond$.

Definition 1 (Parity game)

A parity game is a tuple $(V,E,\varOmega ,(V _{\diamond },V _{\square }))$, where

• V is a set of vertices, partitioned in a set $V _{\diamond }$ of vertices owned by player $\diamond$, and a set of vertices $V _{\square }$ owned by player $\square$,

• $E \subseteq V \times V$ is a total edge relation,

• $\varOmega :V \rightarrow \mathbb{N}$ is a priority function that assigns priorities to vertices.

The graph (V, E) underlying a parity game is often referred to as the arena. Parity games...