# Beliefs in Decision-Making Cascades

## TL;DR

This paper investigates decision-making cascades with agents having mismatched beliefs and varying noise levels, revealing counterintuitive optimal beliefs, and explores predecessor selection and augmented intelligence design.

## Contribution

It introduces a recursive belief update method, analyzes the effect of belief deviations, and examines predecessor selection and augmented intelligence in social learning.

## Key findings

- Counterintuitive optimal beliefs deviate from the true prior.
- Agents with varying noise levels influence decision regions.
- Suboptimal predecessor choices highlight the need for a social planner.

## Abstract

This work explores a social learning problem with agents having nonidentical noise variances and mismatched beliefs. We consider an $N$-agent binary hypothesis test in which each agent sequentially makes a decision based not only on a private observation, but also on preceding agents' decisions. In addition, the agents have their own beliefs instead of the true prior, and have nonidentical noise variances in the private signal. We focus on the Bayes risk of the last agent, where preceding agents are selfish. We first derive the optimal decision rule by recursive belief update and conclude, counterintuitively, that beliefs deviating from the true prior could be optimal in this setting. The effect of nonidentical noise levels in the two-agent case is also considered and analytical properties of the optimal belief curves are given. Next, we consider a predecessor selection problem wherein the subsequent agent of a certain belief chooses a predecessor from a set of candidates with varying beliefs. We characterize the decision region for choosing such a predecessor and argue that a subsequent agent with beliefs varying from the true prior often ends up selecting a suboptimal predecessor, indicating the need for a social planner. Lastly, we discuss an augmented intelligence design problem that uses a model of human behavior from cumulative prospect theory and investigate its near-optimality and suboptimality.