Combinations of Qualitative Winning for Stochastic Parity Games

We study Markov decision processes and turn-based stochastic games with parity conditions. There are three qualitative winning criteria, namely, sure winning, which requires all paths must satisfy the condition, almost-sure winning, which requires the condition is satisfied with probability~1, and limit-sure winning, which requires the condition is satisfied with probability arbitrarily close to~1. We study the combination of these criteria for parity conditions, e.g., there are two parity conditions one of which must be won surely, and the other almost-surely. The problem has been studied recently by Berthon et.~al for MDPs with combination of sure and almost-sure winning, under infinite-memory strategies, and the problem has been established to be in NP cap coNP. Even in MDPs there is a difference between finite-memory and infinite-memory strategies. Our main results for combination of sure and almost-sure winning are as follows: (a)~we show that for MDPs with finite-memory strategies the problem lie in NP cap coNP; (b)~we show that for turn-based stochastic games the problem is coNP-complete, both for finite-memory and infinite-memory strategies; and (c)~we present algorithmic results for the finite-memory case, both for MDPs and turn-based stochastic games, by reduction to non-stochastic parity games. In addition we show that all the above results also carry over to combination of sure and limit-sure winning, and results for all other combinations can be derived from existing results in the literature. Thus we present a complete picture for the study of combinations of qualitative winning criteria for parity conditions in MDPs and turn-based stochastic games.