Extending Finite Memory Determinacy to Multiplayer Games

We provide several techniques to extend finite memory determinacy from some restricted class of games played on a finite graph to a larger class. As a particular example, we study energy parity games. First we show that under some general conditions the finite memory determinacy of a class of two-player win/lose games played on finite graphs implies the existence of a Nash equilibrium built from finite memory strategies for the corresponding class of multi-player multi-outcome games. This generalizes a previous result by Brihaye, De Pril and Schewe. Then we investigate adding additional constraints to the winning conditions, in a way that generalizes the move from parity to bounded energy parity games. We prove that under some conditions, this preserves finite memory determinacy. We show that bounded energy parity games and unbounded energy parity games are equivalent, and thus obtain a new proof of finite memory determinacy for energy parity games. Our proof yields significantly improved bounds on the memory required compared to the original one by Chatterjee and Doyen. We then apply our main theorem to show that multi-player multi-outcome energy parity games have finite memory Nash equilibria. Our proofs are generally constructive, that is, provide upper bounds for the memory required, as well as algorithms to compute the relevant winning strategies.