Games with $\ddot{I}‰$-Automatic Preference Relations

Abstract

This paper investigates Nash equilibria (NEs) in multi-player turn-based games on graphs, where player preferences are modeled as $\ddot{I}‰$-automatic relations via deterministic parity automata. Unlike much of the existing literature, which focuses on specific reward functions, our results apply to any preference relation definable by an $\ddot{I}‰$-automatic relation. We analyze the computational complexity of determining the existence of an NE (possibly under some constraints), verifying whether a given strategy profile forms an NE, and checking whether a specific outcome can be realized by an NE. When a (constrained) NE exists, we show that there always exists one with finite-memory strategies. Finally, we explore fundamental properties of $\ddot{I}‰$-automatic relations and their implications for the existence of equilibria.

Extended version of a MFCS 2025 paper