  We consider the dynamics of Q-learning in two-player two-action games with a
Boltzmann exploration mechanism. For any non-zero exploration rate the dynamics
is dissipative, which guarantees that agent strategies converge to rest points
that are generally different from the game's Nash Equlibria (NE). We provide a
comprehensive characterization of the rest point structure for different games,
and examine the sensitivity of this structure with respect to the noise due to
exploration. Our results indicate that for a class of games with multiple NE
the asymptotic behavior of learning dynamics can undergo drastic changes at
critical exploration rates. Furthermore, we demonstrate that for certain games
with a single NE, it is possible to have additional rest points (not
corresponding to any NE) that persist for a finite range of the exploration
rates and disappear when the exploration rates of both players tend to zero.
