# Multi-learner risk reduction under endogenous participation dynamics

Prediction systems face exogenous and endogenous distribution shift -- the world constantly changes, and the predictions the system makes change the environment in which it operates. For example, a music recommender observes exogeneous changes in the user distribution as different communities have increased access to high speed internet. If users under the age of 18 enjoy their recommendations, the proportion of the user base comprised of those under 18 may endogeneously increase. Most of the study of endogenous shifts has focused on the single decision-maker setting, where there is one learner that users either choose to use or not. This paper studies participation dynamics between sub-populations and possibly many learners. We study the behavior of systems with \emph{risk-reducing} learners and sub-populations. A risk-reducing learner updates their decision upon observing a mixture distribution of the sub-populations $\mathcal{D}$ in such a way that it decreases the risk of the learner on that mixture. A risk reducing sub-population updates its apportionment amongst learners in a way which reduces its overall loss. Previous work on the single learner case shows that myopic risk minimization can result in high overall loss~\citep{perdomo2020performative, miller2021outside} and representation disparity~\citep{hashimoto2018fairness, zhang2019group}. Our work analyzes the outcomes of multiple myopic learners and market forces, often leading to better global loss and less representation disparity.

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