Half-positional $ω$-regular languages

In the context of two-player games over graphs, a language $L$ is called half-positional if, in all games using $L$ as winning objective, the protagonist can play optimally using positional strategies, that is, strategies that do not depend on the history of the play. In this work, we describe the class of parity automata recognising half-positional languages, providing a complete characterisation of half-positionality for $\omega$-regular languages. As corollaries, we establish decidability of half-positionality in polynomial time, finite-to-infinite and 1-to-2-players lifts, and show the closure under union of prefix-independent half-positional objectives, answering a conjecture by Kopczy\'nski.