  Consider a scenario where a player chooses an action in each round $t$ out of
$T$ rounds and observes the incurred cost after a delay of $d_{t}$ rounds. The
cost functions and the delay sequence are chosen by an adversary. We show that
in a non-cooperative game, the expected weighted ergodic distribution of play
converges to the set of coarse correlated equilibria if players use algorithms
that have "no weighted-regret" in the above scenario, even if they have linear
regret due to too large delays. For a two-player zero-sum game, we show that no
weighted-regret is sufficient for the weighted ergodic average of play to
converge to the set of Nash equilibria. We prove that the FKM algorithm with
$n$ dimensions achieves an expected regret of
$O\left(nT^{\frac{3}{4}}+\sqrt{n}T^{\frac{1}{3}}D^{\frac{1}{3}}\right)$ and the
EXP3 algorithm with $K$ arms achieves an expected regret of $O\left(\sqrt{\log
K\left(KT+D\right)}\right)$ even when $D=\sum_{t=1}^{T}d_{t}$ and $T$ are
unknown. These bounds use a novel doubling trick that, under mild assumptions,
provably retains the regret bound for when $D$ and $T$ are known. Using these
bounds, we show that FKM and EXP3 have no weighted-regret even for
$d_{t}=O\left(t\log t\right)$. Therefore, algorithms with no weighted-regret
can be used to approximate a CCE of a finite or convex unknown game that can
only be simulated with bandit feedback, even if the simulation involves
significant delays.
