  Deployed supervised machine learning models make predictions that interact
with and influence the world. This phenomenon is called performative prediction
by Perdomo et al. (ICML 2020). It is an ongoing challenge to understand the
influence of such predictions as well as design tools so as to control that
influence. We propose a theoretical framework where the response of a target
population to the deployed classifier is modeled as a function of the
classifier and the current state (distribution) of the population. We show
necessary and sufficient conditions for convergence to an equilibrium of two
retraining algorithms, repeated risk minimization and a lazier variant.
Furthermore, convergence is near an optimal classifier. We thus generalize
results of Perdomo et al., whose performativity framework does not assume any
dependence on the state of the target population. A particular phenomenon
captured by our model is that of distinct groups that acquire information and
resources at different rates to be able to respond to the latest deployed
classifier. We study this phenomenon theoretically and empirically.
