  In a mean-payoff parity game, one of the two players aims both to achieve a
qualitative parity objective and to minimize a quantitative long-term average
of payoffs (aka. mean payoff). The game is zero-sum and hence the aim of the
other player is to either foil the parity objective or to maximize the mean
payoff.
  Our main technical result is a pseudo-quasi-polynomial algorithm for solving
mean-payoff parity games. All algorithms for the problem that have been
developed for over a decade have a pseudo-polynomial and an exponential factors
in their running times; in the running time of our algorithm the latter is
replaced with a quasi-polynomial one. By the results of Chatterjee and Doyen
(2012) and of Schewe, Weinert, and Zimmermann (2018), our main technical result
implies that there are pseudo-quasi-polynomial algorithms for solving parity
energy games and for solving parity games with weights.
  Our main conceptual contributions are the definitions of strategy
decompositions for both players, and a notion of progress measures for
mean-payoff parity games that generalizes both parity and energy progress
measures. The former provides normal forms for and succinct representations of
winning strategies, and the latter enables the application to mean-payoff
parity games of the order-theoretic machinery that underpins a recent
quasi-polynomial algorithm for solving parity games.

Computer Science and Game Theory;Data Structures and Algorithms;Logic in Computer Science