The Reflexivity Threshold: A Phase Transition for Multi-Agent Performative Prediction

Many learning systems are performative: once deployed, their predictions change the data distribution on which future models are trained. In multi-agent settings, this feedback is routed through a network of interacting learners, so stability depends not only on the strength of performativity but also on who affects whom. We study repeated retraining in multi-agent performative prediction and identify a spectral threshold that separates stable learning from worst-case learning obstruction. Prior work often collapses cross-agent feedback into a scalar contraction constant, missing this network structure. We instead introduce the reflexivity matrix $Γ$, which records how strongly one agent's deployment can move another agent's best response, defined from per-agent gradient sensitivities and curvature parameters. The threshold occurs when the spectral radius of this matrix crosses one. Below the threshold, repeated retraining converges geometrically to a unique performatively stable equilibrium, with cumulative excess loss bounded uniformly in the time horizon. Above the threshold, we construct noisy supercritical instances in which every local-information learner incurs expected cumulative excess loss growing linearly for horizons up to order $1/(ρ(Γ)-1)$. Together, these results characterize the threshold behavior through upper and lower bounds and recover the single-agent scalar threshold from prior work as a special case.