  A satisfactory multiagent learning algorithm should, {\em at a minimum},
learn to play optimally against stationary opponents and converge to a Nash
equilibrium in self-play. The algorithm that has come closest, WoLF-IGA, has
been proven to have these two properties in 2-player 2-action repeated
games--assuming that the opponent's (mixed) strategy is observable. In this
paper we present AWESOME, the first algorithm that is guaranteed to have these
two properties in {\em all} repeated (finite) games. It requires only that the
other players' actual actions (not their strategies) can be observed at each
step. It also learns to play optimally against opponents that {\em eventually
become} stationary. The basic idea behind AWESOME ({\em Adapt When Everybody is
Stationary, Otherwise Move to Equilibrium}) is to try to adapt to the others'
strategies when they appear stationary, but otherwise to retreat to a
precomputed equilibrium strategy. The techniques used to prove the properties
of AWESOME are fundamentally different from those used for previous algorithms,
and may help in analyzing other multiagent learning algorithms also.
