Positionality in $Σ_0^2$ and a completeness result

We study the existence of positional strategies for the protagonist in infinite duration games over arbitrary game graphs. We prove that prefix-independent objectives in $Σ_0^2$ which are positional and admit a (strongly) neutral letter are exactly those that are recognised by history-deterministic monotone co-Bchi automata over countable ordinals. This generalises a criterion proposed by [Kopczyński, ICALP 2006] and gives an alternative proof of closure under union for these objectives, which was known from [Ohlmann, TheoretiCS 2023]. We then give two applications of our result. First, we prove that the mean-payoff objective is positional over arbitrary game graphs. Second, we establish the following completeness result: for any objective $W$ which is prefix-independent, admits a (weakly) neutral letter, and is positional over finite game graphs, there is an objective $W'$ which is equivalent to $W$ over finite game graphs and positional over arbitrary game graphs.