  The beyond worst-case synthesis problem was introduced recently by Bruy\`ere
et al. [BFRR14]: it aims at building system controllers that provide strict
worst-case performance guarantees against an antagonistic environment while
ensuring higher expected performance against a stochastic model of the
environment. Our work extends the framework of [BFRR14] and follow-up papers,
which focused on quantitative objectives, by addressing the case of
$\omega$-regular conditions encoded as parity objectives, a natural way to
represent functional requirements of systems.
  We build strategies that satisfy a main parity objective on all plays, while
ensuring a secondary one with sufficient probability. This setting raises new
challenges in comparison to quantitative objectives, as one cannot easily mix
different strategies without endangering the functional properties of the
system. We establish that, for all variants of this problem, deciding the
existence of a strategy lies in ${\sf NP} \cap {\sf coNP}$, the same complexity
class as classical parity games. Hence, our framework provides additional
modeling power while staying in the same complexity class.
  [BFRR14] V\'eronique Bruy\`ere, Emmanuel Filiot, Mickael Randour, and
Jean-Fran\c{c}ois Raskin. Meet your expectations with guarantees: Beyond
worst-case synthesis in quantitative games. In Ernst W. Mayr and Natacha
Portier, editors, 31st International Symposium on Theoretical Aspects of
Computer Science, STACS 2014, March 5-8, 2014, Lyon, France, volume 25 of
LIPIcs, pages 199-213. Schloss Dagstuhl - Leibniz - Zentrum fuer Informatik,
2014.
