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.

翻译：我们研究了在任意博弈图上无限时长博弈中主角位置策略的存在性。我们证明了，在 Σ_0^2 中具有前缀独立性、可位置化且允许（强）中性字母的目标，恰好是那些可由在可数序数上的历史确定性单调 co-Büchi 自动机识别的目标。这推广了 [Kopczyński, ICALP 2006] 提出的一个判据，并为这些目标在并集下的封闭性提供了一个替代证明（该性质已知于 [Ohlmann, TheoretiCS 2023]）。随后，我们给出了我们结果的两个应用。首先，我们证明了平均收益目标在任意博弈图上是可位置化的。其次，我们建立了如下完备性结果：对于任意具有前缀独立性、允许（弱）中性字母且在有限博弈图上可位置化的目标 W，存在一个目标 W'，它在有限博弈图上与 W 等价，并且在任意博弈图上是可位置化的。