  Multi-dimensional mean-payoff and energy games provide the mathematical
foundation for the quantitative study of reactive systems, and play a central
role in the emerging quantitative theory of verification and synthesis. In this
work, we study the strategy synthesis problem for games with such
multi-dimensional objectives along with a parity condition, a canonical way to
express $\omega$-regular conditions. While in general, the winning strategies
in such games may require infinite memory, for synthesis the most relevant
problem is the construction of a finite-memory winning strategy (if one
exists). Our main contributions are as follows. First, we show a tight
exponential bound (matching upper and lower bounds) on the memory required for
finite-memory winning strategies in both multi-dimensional mean-payoff and
energy games along with parity objectives. This significantly improves the
triple exponential upper bound for multi energy games (without parity) that
could be derived from results in literature for games on VASS (vector addition
systems with states). Second, we present an optimal symbolic and incremental
algorithm to compute a finite-memory winning strategy (if one exists) in such
games. Finally, we give a complete characterization of when finite memory of
strategies can be traded off for randomness. In particular, we show that for
one-dimension mean-payoff parity games, randomized memoryless strategies are as
powerful as their pure finite-memory counterparts.
