Discontinuous Selection and Cyclic Dynamics in Performative Learning

Adaptive learning systems often select a model or policy using the state created by previous deployments. Even when either fixed policy would make the environment converge, threshold selection can cause persistent oscillation because each choice may activate the other. In contractive affine models, we prove that if the deployment score evolves autonomously and each fixed policy’s equilibrium selects the other policy, every trajectory approaches an alternating cycle. Writing $\rho$ for state persistence and $\beta$ for the policy-induced shift, the cycle has amplitude $\beta/(1+\rho)$, which also sets the sharp transition scale for linearly smoothing the threshold or selecting no action within a deadband. A tanh rule shares this threshold but develops its cycle continuously from zero, while hysteresis requires the larger width $\beta/(1-\rho)$ to stop repeated switching. In higher dimensions, fixed states and alternating cycles depend on different conditions, allowing two stable states to coexist with a stable cycle. We also derive an exact test for finite periodic switching patterns in one-dimensional systems with unequal responses. Across 600 synthetic systems, all 300 systems predicted to admit an alternating cycle reached period two from 20 initial states, while none of the remaining 300 did. Of those 300 systems, 285 reached longer finite periods and 15 remained unresolved within the simulation horizon. Therefore, fixed policy stability cannot determine whether adaptive deployment converges, cycles, or depends on initialization, while selector design provides a direct means of controlling these outcomes.