Simple Nash Equilibria for Qualitative Multiplayer Games

We investigate memory requirements for Nash and subgame-perfect equilibria in turn-based deterministic games with $\omega$-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 B\"uchi as well as safety or CoB\"uchi objectives, and that memoryless equilibria need not exist for any other class of Muller objectives (with respect to the Mostowski hierarchy).