A positional $\mathbf{\Pi}^0_3$-complete objective

TL;DR

This paper introduces a new positional game objective that is 3-complete in the Borel hierarchy, demonstrating the existence of complex yet positional winning strategies in graph games.

Contribution

It is the first known 3-complete objective, expanding understanding of positional strategies beyond previously known 3 objectives.

Findings

Existence of a 3-complete positional objective.

This objective is a qualitative variant of total-payoff.

Positional strategies suffice for the first player in this context.

Abstract

We study zero-sum turn-based games on graphs. In this note, we show the existence of a game objective that is $\mathbf{\Pi}^0_3$-complete for the Borel hierarchy and that is positional, i.e., for which positional strategies suffice for the first player to win over arenas of arbitrary cardinality. To the best of our knowledge, this is the first known such objective; all previously known positional objectives are in $\mathbf{\Sigma}^0_3$. The objective in question is a qualitative variant of the well-studied total-payoff objective, where the goal is to maximise the sum of weights.