When Predictions Move Markets: A Phase Transition in Learning Under Strategic Response

This paper establishes a sharp phase transition in learning under performative feedback. When strategic response strength λ is stochastic, we prove the minimax regret is Θ((1+λ)σ√T) with matching upper and lower bounds. When λ is chosen adversarially after observing each prediction, regret becomes Ω(T), making learning impossible. This identifies a fundamental boundary in performative prediction: the standard assumption of fixed response is load-bearing, not merely convenient. We prove the upper bound via an Explore-Then-Commit algorithm and the lower bound via Le Cam's method. The impossibility result is robust to delays, Lipschitz constraints, and nonlinear response. Experiments confirm the √T vs T separation across 100 independent trials.