Controlled Iterated Function Systems for Minority Dynamics and Diversity-Preserving Ergodic Control

We study \emph{ergodic regulation} of large-agent strategic systems in which agents prefer to be in the minority. The population dynamics are modeled as a controlled iterated function system (IFS) on a Polish space; the control reshapes the transition kernel and hence the stationary law of the ensemble. We formulate a distribution-shaping problem that penalizes collapse into majority states and rewards \emph{diversity} by aligning the controlled invariant measure with a dispersed target in Wasserstein geometry. Our contributions are fourfold. (i) We prove Fréchet differentiability of the stationary law with respect to control and obtain a \emph{linear-response} (resolvent) representation. (ii) We derive an adjoint \emph{Poisson equation} yielding an explicit gradient formula for the Wasserstein objective, providing a measure-valued Pontryagin-type optimality condition suitable for feedback design. (iii) We establish non-asymptotic convergence of projected stochastic gradient with \emph{dependent} samples, with explicit bias--variance tradeoffs induced by finite trajectory length and truncated Poisson solves. (iv) We quantify \emph{robustness} of both the invariant law and the optimal value under kernel misspecification. Numerical studies on minority-game stylizations in low- and moderate-dimensional IFS illustrate suppression of informational cascades, preservation of multi-modality, and favorable sample–compute profiles consistent with theory.