  Many projects (both practical and academic) have designed algorithms to match
users to content they will enjoy under the assumption that user's preferences
and opinions do not change with the content they see. Evidence suggests that
individuals' preferences are directly shaped by what content they see --
radicalization, rabbit holes, polarization, and boredom are all example
phenomena of preferences affected by content. Polarization in particular can
occur even in ecosystems with "mass media," where no personalization takes
place, as recently explored in a natural model of preference dynamics
by~\citet{hkazla2019geometric} and~\citet{gaitonde2021polarization}. If all
users' preferences are drawn towards content they already like, or are repelled
from content they already dislike, uniform consumption of media leads to a
population of heterogeneous preferences converging towards only two poles.
  In this work, we explore whether some phenomenon akin to polarization occurs
when users receive \emph{personalized} content recommendations. We use a
similar model of preference dynamics, where an individual's preferences move
towards content the consume and enjoy, and away from content they consume and
dislike. We show that standard user reward maximization is an almost trivial
goal in such an environment (a large class of simple algorithms will achieve
only constant regret). A more interesting objective, then, is to understand
under what conditions a recommendation algorithm can ensure stationarity of
user's preferences. We show how to design a content recommendations which can
achieve approximate stationarity, under mild conditions on the set of available
content, when a user's preferences are known, and how one can learn enough
about a user's preferences to implement such a strategy even when user
preferences are initially unknown.
