  Numerous online services are data-driven: the behavior of users affects the
system's parameters, and the system's parameters affect the users' experience
of the service, which in turn affects the way users may interact with the
system. For example, people may choose to use a service only for tasks that
already works well, or they may choose to switch to a different service. These
adaptations influence the ability of a system to learn about a population of
users and tasks in order to improve its performance broadly. In this work, we
analyze a class of such dynamics -- where users allocate their participation
amongst services to reduce the individual risk they experience, and services
update their model parameters to reduce the service's risk on their current
user population. We refer to these dynamics as \emph{risk-reducing}, which
cover a broad class of common model updates including gradient descent and
multiplicative weights. For this general class of dynamics, we show that
asymptotically stable equilibria are always segmented, with sub-populations
allocated to a single learner. Under mild assumptions, the utilitarian social
optimum is a stable equilibrium. In contrast to previous work, which shows that
repeated risk minimization can result in (Hashimoto et al., 2018; Miller et
al., 2021), we find that repeated myopic updates with multiple learners lead to
better outcomes. We illustrate the phenomena via a simulated example
initialized from real data.
