  In mean-payoff games, the objective of the protagonist is to ensure that the
limit average of an infinite sequence of numeric weights is nonnegative. In
energy games, the objective is to ensure that the running sum of weights is
always nonnegative. Multi-mean-payoff and multi-energy games replace individual
weights by tuples, and the limit average (resp. running sum) of each coordinate
must be (resp. remain) nonnegative. These games have applications in the
synthesis of resource-bounded processes with multiple resources.
  We prove the finite-memory determinacy of multi-energy games and show the
inter-reducibility of multimean-payoff and multi-energy games for finite-memory
strategies. We also improve the computational complexity for solving both
classes of games with finite-memory strategies: while the previously best known
upper bound was EXPSPACE, and no lower bound was known, we give an optimal
coNP-complete bound. For memoryless strategies, we show that the problem of
deciding the existence of a winning strategy for the protagonist is
NP-complete. Finally we present the first solution of multi-meanpayoff games
with infinite-memory strategies. We show that multi-mean-payoff games with
mean-payoff-sup objectives can be decided in NP and coNP, whereas
multi-mean-payoff games with mean-payoff-inf objectives are coNP-complete.
