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We study the existence of positional strategies for the protagonist ininfinite duration games over arbitrary game graphs. We prove thatprefix-independent objectives in {\Sigma}_0^2 which are positional and admit a(strongly) neutral letter are exactly those that are recognised byhistory-deterministic monotone co-B\"uchi automata over countable ordinals.This generalises a criterion proposed by [Kopczy\'nski, ICALP 2006] and givesan alternative proof of closure under union for these objectives, which wasknown from [Ohlmann, TheoretiCS 2023].  We then give two applications of our result. First, we prove that themean-payoff objective is positional over arbitrary game graphs. Second, weestablish the following completeness result: for any objective W which isprefix-independent, admits a (weakly) neutral letter, and is positional overfinite game graphs, there is an objective W' which is equivalent to W overfinite game graphs and positional over arbitrary game graphs.

Positionality in {\Sigma}_0^2 and a completeness result