  We consider two-player games played on weighted directed graphs with
mean-payoff and total-payoff objectives, two classical quantitative objectives.
While for single-dimensional games the complexity and memory bounds for both
objectives coincide, we show that in contrast to multi-dimensional mean-payoff
games that are known to be coNP-complete, multi-dimensional total-payoff games
are undecidable. We introduce conservative approximations of these objectives,
where the payoff is considered over a local finite window sliding along a play,
instead of the whole play. For single dimension, we show that (i) if the window
size is polynomial, deciding the winner takes polynomial time, and (ii) the
existence of a bounded window can be decided in NP $\cap$ coNP, and is at least
as hard as solving mean-payoff games. For multiple dimensions, we show that (i)
the problem with fixed window size is EXPTIME-complete, and (ii) there is no
primitive-recursive algorithm to decide the existence of a bounded window.
