Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Aggregative Efficiency of Bayesian Learning in Networks

Krishna Dasaratha, Kevin He

TL;DR

This work considers sequential social learning with rational agents and Gaussian signals and asks how the efficiency of signal aggregation changes with the network, leading to a detailed ranking of networks for social learning based on their aggregation efficiency index.

Abstract

When individuals in a social network learn about an unknown state from private signals and neighbors' actions, the network structure often causes information loss. We consider rational agents and Gaussian signals in the canonical sequential social-learning problem and ask how the network changes the efficiency of signal aggregation. Rational actions in our model are log-linear functions of observations and admit a signal-counting interpretation of accuracy. Networks where agents observe multiple neighbors but not their common predecessors confound information, and even a small amount of confounding can lead to much lower accuracy. In a class of networks where agents move in generations and observe the previous generations, we quantify the information loss with an aggregative efficiency index. Aggregative efficiency is a simple function of network parameters: increasing in observations and decreasing in confounding. Later generations contribute little additional information, even when generations are arbitrarily large and agents observe arbitrarily far into the past.

Aggregative Efficiency of Bayesian Learning in Networks

TL;DR

This work considers sequential social learning with rational agents and Gaussian signals and asks how the efficiency of signal aggregation changes with the network, leading to a detailed ranking of networks for social learning based on their aggregation efficiency index.

Abstract

When individuals in a social network learn about an unknown state from private signals and neighbors' actions, the network structure often causes information loss. We consider rational agents and Gaussian signals in the canonical sequential social-learning problem and ask how the network changes the efficiency of signal aggregation. Rational actions in our model are log-linear functions of observations and admit a signal-counting interpretation of accuracy. Networks where agents observe multiple neighbors but not their common predecessors confound information, and even a small amount of confounding can lead to much lower accuracy. In a class of networks where agents move in generations and observe the previous generations, we quantify the information loss with an aggregative efficiency index. Aggregative efficiency is a simple function of network parameters: increasing in observations and decreasing in confounding. Later generations contribute little additional information, even when generations are arbitrarily large and agents observe arbitrarily far into the past.

Paper Structure

This paper contains 38 sections, 38 theorems, 160 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model
  3. Basic Results about Beliefs and Learning
  4. Optimal Actions Are Log-Linear
  5. Measure of Accuracy
  6. Examples of Information Confounding in Networks
  7. Long-Run Learning and Quantifying Information Loss
  8. Generations Networks
  9. Symmetric Generations Networks
  10. Multi-Generation Observation Networks
  11. Random Regular Networks
  12. Finite-Population Learning
  13. Organizational Applications
  14. Value of Mentorship
  15. Information Silos
  16. ...and 23 more sections

Key Result

Proposition 1

For each agent $i$ with $N(i)=\{j(1),...,j(d_{i})\},$ there exist constants $(\beta_{i,j(k)})_{k=1}^{d_{i}}$ so that $i$'s rational log-strategy is given by The vector of coefficients $\vec{\beta}_{i}$ is given by where $\textsc{Cov}[\ell_{j(1)},...,\ell_{j(d_{i})}\mid\omega=1]^{-1}$ is the inverse of the conditional covariance matrix. These coefficients do not depend on the signal variance $\si

Figures (7)

  • Figure 1: Left: The shield network with four agents. Right: The diamond network with four agents.
  • Figure 2: A three-generation network with $K_{1}$ agents in generation 1, $K_{2}$ agents in generation 2, and one agent in generation 3.
  • Figure 3: A generations network with $K=3$ agents per generation and the observation sets $\Psi_{1}=\{1,2\},$$\Psi_{2}=\{2,3\},$ and $\Psi_{3}=\{1,3\}$.
  • Figure 4: Number of signals aggregated by social learning in maximal generations networks with different generation sizes, $K\in\{2,3,4,5\}.$
  • Figure 5: Signal aggregation by generation in networks with $K=100$ agents per generation. In both plots, the solid lines show the largest one-generation increase in $r_i$ across agents in all 100 positions, averaged across 1000 repetitions of the simulation. The shaded areas show the range of this increase across the 1000 repetitions. In each repetition, the observation sets for the 100 positions are randomly and independently drawn, but kept fixed for all generations. Subfigure (a): All positions have $d$ neighbors each. Subfigure (b): Positions 1 through 50 have $d_1$ neighbors each and positions 51 through 100 have $d_2$ neighbors each.
  • ...and 2 more figures

Theorems & Definitions (80)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Example 1: The Shield and the Diamond
  • Example 2
  • Definition 2
  • Proposition 3
  • Definition 3
  • Definition 4
  • Theorem 1
  • ...and 70 more