  We study two-player general sum repeated finite games where the rewards of
each player are generated from an unknown distribution. Our aim is to find the
egalitarian bargaining solution (EBS) for the repeated game, which can lead to
much higher rewards than the maximin value of both players. Our most important
contribution is the derivation of an algorithm that achieves simultaneously,
for both players, a high-probability regret bound of order
$\mathcal{O}(\sqrt[3]{\ln T}\cdot T^{2/3})$ after any $T$ rounds of play. We
demonstrate that our upper bound is nearly optimal by proving a lower bound of
$\Omega(T^{2/3})$ for any algorithm.
