Simple Nash Equilibria for Qualitative Multiplayer Games

2026-07-08

AI Summary

This study investigates the memory requirements for Nash equilibria and subgame-perfect equilibria in turn-based deterministic multi-player games with ω-regular objectives, including reachability, safety, and specific Muller conditions such as the 0-2 Muller objective. By integrating game theory, automata theory, and formal methods—particularly through an analysis based on the Mostowski hierarchy—the paper establishes that memoryless randomized (i.e., stationary) subgame-perfect equilibria always exist for several classes of objective combinations. The main contributions are threefold: first, it demonstrates the necessity of randomization for equilibrium existence under certain objective combinations; second, it precisely characterizes the boundary conditions under which such equilibria exist for Muller objectives; and third, it provides an effective algorithmic construction for these equilibria.

Abstract

We investigate memory requirements for Nash and subgame-perfect equilibria in turn-based deterministic games with $ω$-regular objectives. We prove that memoryless randomised (i.e., stationary) subgame-perfect equilibria always exist in games with reachability, safety, and 0-2 Muller objectives (i.e., Muller objectives for which accepting sets are either up- or downward closed), and any combination of these objectives. We provide an algorithm to construct such an equilibrium. We also show that randomisation may be required to construct memoryless equilibria in games with reachability or Büchi as well as safety or CoBüchi objectives, and that memoryless equilibria need not exist for any other class of Muller objectives (with respect to the Mostowski hierarchy).

Problem

• Nash equilibrium
• subgame-perfect equilibrium
• memoryless strategies
• omega-regular objectives
• multiplayer games

Innovation

• Nash equilibrium
• subgame-perfect equilibrium
• memoryless strategies
• ω-regular objectives
• Muller objectives