[论文解读] Learning, Misspecification, and Cognitive Arbitrage in Linear-Quadratic Network Games

Quanyan Zhu, Zhengye Han|arXiv (Cornell University)|Mar 17, 2026

Game Theory and Applications被引用 0

一句话总结

该论文研究带有错误设定代理模型的线性-二次网络博弈中的 Berk-Nash 均衡，通过最小扭曲观测实现认知套利设计，并推导出带有两时间尺度收敛的 Stackelberg 最优扭曲策略。

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.

研究动机与目标

• Understand how subjective misspecifications affect equilibrium outcomes in linear-quadratic network games.
• Characterize Berk-Nash equilibrium and its divergence from Nash equilibrium under misspecification.
• Quantify the Value of Misspecification (VoM) and identify conditions for stability.
• Propose and solve a cognitive arbitrage design problem to influence equilibrium through minimal observation distortions.
• Establish convergence of a two-time-scale learning algorithm to the optimal Berk-Nash equilibrium.

提出的方法

• Model a directed network with local externalities and Gaussian observation noise.
• Define subjective conjectures with simple regressors (constant, aggregate, mean-field, feature-based).
• Derive the Berk-Nash equilibrium as a fixed-point of optimality and consistency conditions (best-response under subjective distributions).
• Quantify the divergence from Nash equilibrium via Value of Misspecification (VoM).
• Formulate cognitive arbitrage as a Stackelberg-like design: minimize aggregate cost subject to a distortion budget, yielding a closed-form solution for the optimal distortion δ* and induced BN equilibrium.
• Prove almost-sure convergence of a two-time-scale learning dynamics to the unique BNE.

Figure 1: Network mismatch: True dense interactions (gray) vs. sparse subjective attention (blue). Agents ignore long-range dependencies, creating persistent model misspecification.

实验结果

研究问题

• RQ1 How does misspecification in agents' conjectures affect the resulting equilibrium in linear-quadratic network games?
• RQ2 What is the relationship between Berk-Nash equilibrium and the true Nash equilibrium under various conjecture classes?
• RQ3 How large can the misspecification be before it substantially worsens (or improves) system performance, as captured by VoM?
• RQ4 Can a system designer optimally influence equilibrium outcomes by minimally distorting observations (cognitive arbitrage), and how can this be computed and implemented?
• RQ5 Do learning dynamics converge to the Berk-Nash equilibrium, and under what conditions?

主要发现

Agent	δ*	x*	θ*
0	0.39	1.03	0.00
1	-0.10	-0.49	0.35
2	0.65	1.58	-1.13
3	0.30	1.00	-0.19
4	0.39	0.82	0.24
5	0.41	1.37	-0.08
6	-0.16	-0.62	0.77
7	0.43	1.52	-0.16
8	0.10	0.09	0.15
9	0.86	2.88	-0.71
10	0.55	1.58	-0.28
11	0.23	1.09	0.55

• BNE actions coincide with NE under constant conjectures, yielding no distortion.
• Global mean-field conjectures can lead to BNE actions equivalent to a scalar aggregation of local influences in symmetric or large-population limits.
• Local mean-field conjectures cause a sparsified perception of the network, modifying equilibrium via a reweighted interaction matrix.
• VoM provides a bound on how misspecification distorts aggregate costs; the bound scales with network distortion and system parameters.
• Cognitive arbitrage enables optimal distortion δ* to steer the BN equilibrium, with a unique solution given by a closed-form (involving a QCQP with a complementary slackness condition).
• Two-time-scale learning experiments validate rapid convergence of agent-level variables to BN equilibrium while the designer’s distortion evolves slowly, supporting the Stackelberg approximation.

Figure 2: Convergence diagnostics plotted on a log scale. The rapid decay of action and conjecture updates relative to the distortion updates ( $\|\Delta\delta\|$ ) empirically validates the two-time-scale separation principle.