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Endogenous Network Structures with Precision and Dimension Choices

Nikhil Kumar

TL;DR

The paper develops a DeGroot-based model of social learning in which agents simultaneously choose private-information precision and the state dimension to learn, endogenously shaping the network over which beliefs are updated. On a fixed network, each agent's optimal precision is sublinear in their stationary influence, and the socially optimal precision exceeds the individual optimum by up to a factor of n^{1/3}, with a welfare gap that grows as Θ(n^2). It then extends to dynamic networks where the network weights are determined by a kernel distance between agents' dimension choices, leading to recommendations that agents learn on dimensions where their influence is most evenly distributed; it further explores multi-dimension learning and multiplexed networks, plus iterative DeGroot learning with memory. The results contribute to understanding endogenous networks, dimensional specialization, and efficiency of social learning, with potential extensions to sequential interactions and AI-human networks.

Abstract

This paper presents a social learning model where the network structure is endogenously determined by signal precision and dimension choices. Agents not only choose the precision of their signals and what dimension of the state to learn about, but these decisions directly determine the underlying network structure on which social learning occurs. We show that under a fixed network structure, the optimal precision choice is sublinear in the agent's stationary influence in the network, and this individually optimal choice is worse than the socially optimal choice by a factor of $n^{1/3}$. Under a dynamic network structure, we specify the network by defining a kernel distance between agents, which then determines how much weight agents place on one another. Agents choose dimensions to learn about such that their choice minimizes the squared sum of influences of all agents: a network with equally distributed influence across agents is ideal.

Endogenous Network Structures with Precision and Dimension Choices

TL;DR

The paper develops a DeGroot-based model of social learning in which agents simultaneously choose private-information precision and the state dimension to learn, endogenously shaping the network over which beliefs are updated. On a fixed network, each agent's optimal precision is sublinear in their stationary influence, and the socially optimal precision exceeds the individual optimum by up to a factor of n^{1/3}, with a welfare gap that grows as Θ(n^2). It then extends to dynamic networks where the network weights are determined by a kernel distance between agents' dimension choices, leading to recommendations that agents learn on dimensions where their influence is most evenly distributed; it further explores multi-dimension learning and multiplexed networks, plus iterative DeGroot learning with memory. The results contribute to understanding endogenous networks, dimensional specialization, and efficiency of social learning, with potential extensions to sequential interactions and AI-human networks.

Abstract

This paper presents a social learning model where the network structure is endogenously determined by signal precision and dimension choices. Agents not only choose the precision of their signals and what dimension of the state to learn about, but these decisions directly determine the underlying network structure on which social learning occurs. We show that under a fixed network structure, the optimal precision choice is sublinear in the agent's stationary influence in the network, and this individually optimal choice is worse than the socially optimal choice by a factor of . Under a dynamic network structure, we specify the network by defining a kernel distance between agents, which then determines how much weight agents place on one another. Agents choose dimensions to learn about such that their choice minimizes the squared sum of influences of all agents: a network with equally distributed influence across agents is ideal.

Paper Structure

This paper contains 24 sections, 4 theorems, 67 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. The Model
  4. Precision Choice on a Fixed Network Structure
  5. Examples Across Standard Network Topologies
  6. Multi-Dimension Choice on a Fixed Network Structure
  7. Multiplexed Network Structures
  8. Varying Precision Choices
  9. Single Precision Parameter
  10. Effort Allocation Across Dimensions
  11. Dynamic Network Structures
  12. Single DeGroot Learning Iteration
  13. Iterative DeGroot Learning
  14. Summary and Conclusions
  15. Derivations and Proofs
  16. ...and 9 more sections

Key Result

Theorem 1

Under normal signals, each agent's optimal choice of precision is increasing in their influence in the network, but at a sublinear rate. Formally, each agent $i$ chooses an individually optimal precision choice of

Figures (6)

  • Figure 1: A general $n$-agent complete network weight matrix parameterized by self‐weights $x_i$ and a corresponding simplified network diagram for $n=8$.
  • Figure 2: Core–periphery network
  • Figure 3: Star network
  • Figure 5: Dimension choices to learn about: agent $n$ picks dimension 2, and agent $1$ picks dimension 3, and all others choose dimension 1. Yellow nodes indicate the agents who choose that corresponding dimension.
  • Figure 6: Different proportions of specialists and generalists in the network. Specialists concentrate effort on one coordinate (variance $1/\tau_i^2$ there, $1/\underline\tau^2$ elsewhere); generalists split effort equally (variance $1/(\underline\tau+\tau_i/m)^2$ on every coordinate).
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1: Optimal Precision Choice Under Fixed Network Structure, One-Dimensional State
  • proof
  • Corollary 1: Socially Optimal Precision Choice
  • proof : Proof Sketch:
  • Claim 1: Social Welfare Gap
  • proof : Proof Sketch:
  • Claim 2
  • Theorem 2: Dimension Choices under Multi-Dimensional State
  • proof : Proof:
  • Corollary 2: Bayesian Updating Under Iterative DeGroot Updating