The Architecture of Illusion: Network Opacity and Strategic Escalation

TL;DR

The paper introduces Connected Minds, a model that fuses Iterative Reasoning (Level-$k$/CH) with network-based observation through a locality parameter $p\in(0,1]$, which governs how much of the population a player sees. It shows that $p$ smoothly interpolates between myopic Level-$k$ behavior ($p\to 0$) and standard Cognitive Hierarchy ($p=1$), while preserving log-concavity of beliefs and producing a Sophisticated Bias that overstates opponents’ depth. A Poisson-shift convergence result shows that, as reasoning depth grows, agents’ perceived population remains biased with an effective mean $\tau/p$, implying topology can dominate cognition and sustain miscalibration and bubbles. The paper then develops a normative mechanism-design framework with a Cognitive Designer who tunes information architecture to trade off escalation and coordination, yielding the Escalation Principle for strategic complements and a Transparency Reversal for coordination games. Overall, network topology acts as a cognitive zoom lens, shaping whether agents behave as local imitators or global optimizers, with practical implications for platform design, policy, and welfare. The analysis highlights identification challenges and proposes experimental strategies to disentangle network opacity from true cognitive ability.”

Abstract

Standard models of bounded rationality typically assume agents either possess accurate knowledge of the population's reasoning abilities (Cognitive Hierarchy) or hold dogmatic, degenerate beliefs (Level-$k$). We introduce the ``Connected Minds'' model, which unifies these frameworks by integrating iterative reasoning with a parameterized network bias. We posit that agents do not observe the global population; rather, they observe a sample biased by their network position, governed by a locality parameter $p$ representing algorithmic ranking, social homophily, or information disclosure. We show that this parameter acts as a continuous bridge: the model collapses to the myopic Level-$k$ recursion as networks become opaque ($p \to 0$) and recovers the standard Cognitive Hierarchy model under full transparency ($p=1$). Theoretically, we establish that network opacity induces a \emph{Sophisticated Bias}, causing agents to systematically overestimate the cognitive depth of their opponents while preserving the log-concavity of belief distributions. This makes $p$ an actionable lever: a planner or platform can tune transparency -- globally or by segment (a personalized $p_k$) -- to shape equilibrium behavior. From a mechanism design perspective, we derive the \emph{Escalation Principle}: in games of strategic complements, restricting information can maximize aggregate effort by trapping agents in echo chambers where they compete against hallucinated, high-sophistication peers. Conversely, we identify a \emph{Transparency Reversal} for coordination games, where maximizing network visibility is required to minimize variance and stabilize outcomes. Our results suggest that network topology functions as a cognitive zoom lens, determining whether agents behave as local imitators or global optimizers.

Figures (6)

• Figure 1: Example with k=5 (reasoning levels) showing how filtering happens. Agents only see neighbors with specific probability weights. Layout optimized to show connection spread clearly as we go from myopic agent (left) to global agent (right).
• Figure 2: The impact of biased network sampling showing the difference between true and perceived distributions of levels for true Geometric (left) and Poisson (right) distributions.
• Figure 3: Aggregate welfare $W(p)$ as a function of network transparency $p$ under three planner objectives. Competition (red) prefers opacity to deter tacit collusion and reduce mean action; beauty-contest stability (green) prefers transparency to minimize quadratic-loss deviations from the endogenous target; and the innovation objective (dark) exhibits an interior optimum, balancing average performance with exploratory heterogeneity while accounting for transparency costs. Vertical dashed lines mark the optimizing $p^*$ for each welfare (here $p^*_{\mathrm{innov}}\approx 0.54$).
• Figure 4: The Topological Microfoundations. (A) The effective transparency $\hat{p}$ decays as structural homophily $\beta$ increases. (B) The Kullback-Leibler divergence between the theoretical belief $g_k(\cdot | \hat{p})$ and the actual empirical neighborhood remains low ($< 0.2$ nats) across the spectrum. This "Variance Absorption" indicates that $p$ is a sufficient statistic for complex topological distortions. (C-D) Adjacency matrices sorted by cognitive level (Spy Plots) reveal the physical transition: from global random mixing (C) to stratified, diagonal clusters (D).
• Figure 5: The Identifiability Crisis. We visualize the log-likelihood surface for three different ground-truth scenarios. In each case, a "Ridge of Indeterminacy" emerges, prohibiting the separate identification of network bias ($p$) and cognitive level ($\tau$).

Theorems & Definitions (16)

• definition 1
• theorem 1
• proposition 1
• proposition 2
• corollary 1
• theorem 2
• theorem 3
• theorem 4
• theorem 6
• Remark 7