On the Impact of Performative Risk Minimization for Binary Random Variables

Performativity, the phenomenon where outcomes are influenced by predictions, is particularly prevalent in social contexts where individuals strategically respond to a deployed model. In order to preserve the high accuracy of machine learning models under distribution shifts caused by performativity, Perdomo et al. (2020) introduced the concept of performative risk minimization (PRM). While this framework ensures model accuracy, it overlooks the impact of the PRM on the underlying distributions and the predictions of the model. In this paper, we initiate the analysis of the impact of PRM, by studying performativity for a sequential performative risk minimization problem with binary random variables and linear performative shifts. We formulate two natural measures of impact. In the case of full information, where the distribution dynamics are known, we derive explicit formulas for the PRM solution and our impact measures. In the case of partial information, we provide performative-aware statistical estimators, as well as simulations. Our analysis contrasts PRM to alternatives that do not model data shift and indicates that PRM can have amplified side effects compared to such methods.