Quantitative extensions of parity games have recently attracted significant interest. These extensions include parity games with energy and payoff conditions as well as finitary parity games and their generalization to parity games with costs. Finitary parity games enjoy a special status among these extensions, as they offer a native combination of the qualitative and quantitative aspects in infinite games: The quantitative aspect of finitary parity games is a quality measure for the qualitative aspect, as it measures the limit superior of the time it takes to answer an odd color by a larger even one. Finitary parity games have been extended to parity games with costs, where each transition is labeled with a nonnegative weight that reflects the costs incurred by taking it. We lift this restriction and consider parity games with costs with arbitrary integer weights. We show that solving such games is in NP $\cap$ coNP, the signature complexity for games of this type. We also show that the protagonist has finite-state winning strategies, and provide tight pseudo-polynomial bounds for the memory he needs to win the game. Naturally, the antagonist may need infinite memory to win. Moreover, we present tight bounds on the quality of winning strategies for the protagonist. Furthermore, we investigate the problem of determining, for a given threshold $b$, whether the protagonist has a strategy of quality at most $b$ and show this problem to be EXPTIME-complete. The protagonist inherits the necessity of exponential memory for implementing such strategies from the special case of finitary parity games.

@article{paperbot765,
  title = {Parity Games with Weights},
  author = {Sven Schewe and Alexander Weinert and Martin Zimmermann},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 15, Issue 3},
  year = {2019},
  doi = {10.23638/lmcs-15(3:20)2019}
}