  We analyze a stochastic approximation algorithm for decision-dependent
problems, wherein the data distribution used by the algorithm evolves along the
iterate sequence. The primary examples of such problems appear in performative
prediction and its multiplayer extensions. We show that under mild assumptions,
the deviation between the average iterate of the algorithm and the solution is
asymptotically normal, with a covariance that clearly decouples the effects of
the gradient noise and the distributional shift. Moreover, building on the work
of H\'ajek and Le Cam, we show that the asymptotic performance of the algorithm
with averaging is locally minimax optimal.
