  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 {\Sigma}_0^2 which are positional and admit a
(strongly) neutral letter are exactly those that are recognised by
history-deterministic monotone co-B\"uchi automata over countable ordinals.
This generalises a criterion proposed by [Kopczy\'nski, 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.
