# Finite-time influence systems and the Wisdom of Crowd effect

## TL;DR

This paper investigates how influence systems affect the wisdom of crowds in finite time, introducing new properties and characterizations that distinguish finite-time from infinite-time wisdom preservation.

## Contribution

It introduces and characterizes finite-time wisdom preservation properties of influence systems, especially the French-DeGroot model, highlighting differences from infinite-time behavior.

## Key findings

- Finite-time wisdom preservation is distinct from infinite-time preservation.
- Counterexamples show some conjectures do not hold.
- Complete characterization of equal-neighbor influence systems.

## Abstract

Recent contributions have studied how an influence system may affect the wisdom of crowd phenomenon. In the so-called naive learning setting, a crowd of individuals holds opinions that are statistically independent estimates of an unknown parameter; the crowd is wise when the average opinion converges to the true parameter in the limit of infinitely many individuals. Unfortunately, even starting from wise initial opinions, a crowd subject to certain influence systems may lose its wisdom. It is of great interest to characterize when an influence system preserves the crowd wisdom effect. In this paper we introduce and characterize numerous wisdom preservation properties of the basic French-DeGroot influence system model. Instead of requiring complete convergence to consensus as in the previous naive learning model by Golub and Jackson, we study finite-time executions of the French-DeGroot influence process and establish in this novel context the notion of prominent families (as a group of individuals with outsize influence). Surprisingly, finite-time wisdom preservation of the influence system is strictly distinct from its infinite-time version. We provide a comprehensive treatment of various finite-time wisdom preservation notions, counterexamples to meaningful conjectures, and a complete characterization of equal-neighbor influence systems.