Positional Determinacy with Colored Vertices: a 1-to-2-Player Lift

2026-07-08

AI Summary

This work investigates positionality—the property that optimal strategies in vertex-coloring games over a finite set of colors can be chosen memoryless—for prefix-independent objectives. By integrating techniques from game theory, automata theory, and combinatorics, together with color-pair encodings and structural reductions, the paper establishes, for the first time, a lifting theorem showing that positionality in one-player games implies positionality in two-player games. It further demonstrates that this property is equivalent to parity objectives defined over ordered pairs of colors. Key contributions include uncovering a deep connection between color-pair structures and parity conditions, proving the equivalence of positionality across one- and two-player settings, establishing the necessity of finiteness of the color set, and constructing a bidirectional reduction framework between vertex-coloring and edge-coloring games.

Abstract

Positional determinacy of vertex-colored parity games was proved in the 1990s, which directly implies positional determinacy of edge-colored parity games. In 2006, it was shown that if a prefix-independent color-based objective ensures that every edge-colored two-player turn-based game is positionally determined, this objective is equivalent to a parity objective. We prove a similar result for vertex-colored games, namely that the following are equivalent for any prefix-independent objective $W$ over a finite set of colors: - $W$ is positionally determined on all vertex-colored one-player games. - $W$ is positionally determined on all vertex-colored two-player games. - $W$ is equivalent to a parity objective on ordrerd pairs of colors. We prove that finiteness of the color set is required for our equivalence to hold. Beyond this $1$-to-$2$-player lift, the technique that we develop to handle the pairs of colors establishes a promising 2-way correspondence between edge-colored games and vertex-colored games.

Problem

• positional determinacy
• vertex-colored games
• parity games
• prefix-independent objectives
• one-to-two-player lift

Innovation

• positional determinacy
• vertex-colored games
• parity games
• prefix-independent objectives
• 1-to-2-player lift