Positionality in $\Sigma_0^2$ and a completeness result

TL;DR

This paper characterizes positional strategies in infinite games with prefix-independent objectives in a_0^2, linking them to automata theory, and applies this to mean-payoff games and a completeness result.

Contribution

It provides a new characterization of positional objectives in a_0^2 using automata theory and proves their closure properties and applications to mean-payoff games.

Findings

Mean-payoff objective is positional over arbitrary game graphs.

Objectives with certain automata recognition are exactly those with positional strategies.

A completeness result for objectives equivalent over finite graphs and positional over all graphs.

Abstract

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\"chi 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.