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

TL;DR

The paper identifies a spectral threshold based on a reflexivity matrix that determines stability in multi-agent performative prediction systems. When the matrix's spectral radius is below one, learning converges quickly to a stable equilibrium; above one, learning becomes unstable with linearly growing loss. This framework captures networked feedback effects missed by prior scalar models.

Why It Matters

Understanding stability in multi-agent learning systems is critical for real-world applications like algorithmic trading, recommendation systems, and autonomous coordination, where predictions influence data and feedback loops are complex. The threshold enables early detection of instability and guides robust system design.

Key Findings

• A reflexivity matrix Γ captures cross-agent influence through gradient sensitivities and curvature, encoding how one agent's predictions shift another's optimal response.
• The spectral radius of Γ acts as a phase transition: below one, repeated retraining converges geometrically to a unique stable equilibrium with bounded cumulative loss.
• Above the threshold, even local-information learners face expected cumulative excess loss growing linearly over time, demonstrating worst-case instability.