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

Abstract We study 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 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 linear-response (resolvent) representation. (ii) We derive an adjoint 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 dependent samples, with explicit bias--variance tradeoffs induced by finite trajectory length and truncated Poisson solves. (iv) We quantify 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.