Classical objectives in two-player zero-sum games played on graphs often deal with limit behaviors of infinite plays: e.g., mean-payoff and total-payoff in the quantitative setting, or parity in the qualitative one (a canonical way to encode omega-regular properties). Those objectives offer powerful abstraction mechanisms and often yield nice properties such as memoryless determinacy. However, their very nature provides no guarantee on time bounds within which something good can be witnessed. In this work, we consider two approaches toward inclusion of time bounds in parity games. The first one, parity-response games, is based on the notion of finitary parity games [CHH09] and parity games with costs [FZ14,WZ16]. The second one, window parity games, is inspired by window mean-payoff games [CDRR15]. We compare the two approaches and show that while they prove to be equivalent in some contexts, window parity games offer a more tractable alternative when the time bound is given as a parameter (P-c. vs. PSPACE-c.). In particular, it provides a conservative approximation of parity games computable in polynomial time. Furthermore, we extend both approaches to the multi-dimension setting. We give the full picture for both types of games with regard to complexity and memory bounds.
[CHH09] K. Chatterjee, T.A. Henzinger, F. Horn (2009): Finitary winning in omega-regular games. ACM Trans. Comput. Log. 11(1). [FZ14] N. Fijalkow, M. Zimmermann (2014): Parity and Streett Games with Costs. LMCS 10(2). [WZ16] A. Weinert, M. Zimmermann (2016): Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs. Proc. of CSL, LIPIcs, Schloss Dagstuhl - LZI. To appear. [CDRR15] K. Chatterjee, L. Doyen, M. Randour, J.-F. Raskin (2015): Looking at mean-payoff and total-payoff through windows. Information and Computation 242, pp. 25-52.