  We consider 2-player games played on a finite state space for infinite
rounds. The games are concurrent: in each round, the two players choose their
moves simultaneously; the current state and the moves determine the successor.
We consider omega-regular winning conditions given as parity objectives. We
consider the qualitative analysis problems: the computation of the almost-sure
and limit-sure winning set of states, where player 1 can ensure to win with
probability 1 and with probability arbitrarily close to 1, respectively. In
general the almost-sure and limit-sure winning strategies require both
infinite-memory and infinite-precision. We study the bounded-rationality
problem for qualitative analysis of concurrent parity games, where the strategy
set player 1 is restricted to bounded-resource strategies. In terms of
precision, strategies can be deterministic, uniform, finite-precision or
infinite-precision; and in terms of memory, strategies can be memoryless,
finite-memory or infinite-memory. We present a precise and complete
characterization of the qualitative winning sets for all combinations of
classes of strategies. In particular, we show that uniform memoryless
strategies are as powerful as finite-precision infinite-memory strategies, and
infinite-precision memoryless strategies are as powerful as infinite-precision
finite-memory strategies. We show that the winning sets can be computed in
O(n^{2d+3}) time, where n is the size of the game and 2d is the number of
priorities, and our algorithms are symbolic. The membership problem of whether
a state belongs to a winning set can be decided in NP cap coNP. While this
complexity is the same as for the simpler class of turn-based games, where in
each state only one of the players has a choice of moves, our algorithms, that
are obtained by characterization of the winning sets as mu-calculus formulas,
are considerably more involved.
