Learning, Misspecification, and Cognitive Arbitrage in Linear-Quadratic Network Games

TL;DR

This paper analyzes how agents with misspecified models behave in linear-quadratic network games, introducing a framework for influencing their equilibrium outcomes through strategic observation distortions.

Contribution

It introduces the concept of cognitive arbitrage, a novel approach to steer agent behavior by shaping their conjectures via minimal observation modifications.

Findings

BNE can significantly diverge from Nash equilibrium under misspecification.

A closed-form Stackelberg solution for cognitive arbitrage is derived.

A convergent learning algorithm for optimal BNE is proposed.

Abstract

We study strategic interaction in linear-quadratic network games where agents act on subjective, misspecified models of their environment. Agents observe noisy aggregate signals generated by local network externalities and interpret them through simplified conjectures, such as constant or mean-field representations. We characterize the long-run behavior using the Berk-Nash equilibrium (BNE) concept, establishing conditions under which BNE diverges from the Nash equilibrium of the perfectly specified game. We quantify this divergence using a Value of Misspecification (VoM) metric. Building on this framework, we introduce "cognitive arbitrage" -- a design paradigm where a system designer strategically shapes agents' conjectures via minimal observation distortions to steer equilibrium outcomes. We formulate the cognitive arbitrage problem as a Stackelberg optimization with closed-form solutions and prove the convergence of a two-time-scale learning algorithm to the optimal BNE. Our results provide a principled framework for influencing behavior in networked systems with bounded rationality, offering a new perspective on mechanism design that operates on agents' representations rather than their incentives.