A positional ${\mathbf{\Pi }}_3^0$-complete objective

We study zero-sum turn-based games on graphs. In this note, we show the existence of a game objective that is ${\mathbf{\Pi }}_3^0$-complete for the Borel hierarchy and that is positional, i.e., for which positional strategies suffice for the first player to win over arenas of arbitrary cardinality. To the best of our knowledge, this is the first known such objective; all previously known positional objectives are in ${\mathbf{\Sigma }}_3^0$. The objective in question is a qualitative variant of the well-studied total-payoff objective, where the goal is to maximise the sum of weights.