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Observational Learning with a Budget

Shuo Wu, Pawan Poojary, Randall Berry

TL;DR

This work studies budgeted information design in Bayesian observational learning with sequential agents facing a binary world state. It develops a Markovian framework for the cascade process, yielding explicit cascade-probability expressions under irrational and rational cascade constants, and analyzes how to allocate a fixed budget to improve private signal qualities $p_1$ and $p_2$. The authors prove continuity and monotonicity properties of the probability of a correct cascade $\mathbb{P}_{cc}$ with respect to signal-quality improvements, and derive two optimal allocation strategies, including scenarios where it is best to devote the entire budget to one signal or to symmetrically balance improvements. The appendix provides rigorous proofs of lemmas, propositions, and the main theorem, including the behavior when $p_1=p_2$, which exhibits special discontinuities. Overall, the results inform information-design decisions that maximize social learning accuracy under budget constraints in sequential decision settings.

Abstract

We consider a model of Bayesian observational learning in which a sequence of agents receives a private signal about an underlying binary state of the world. Each agent makes a decision based on its own signal and its observations of previous agents. A central planner seeks to improve the accuracy of these signals by allocating a limited budget to enhance signal quality across agents. We formulate and analyze the budget allocation problem and propose two optimal allocation strategies. At least one of these strategies is shown to maximize the probability of achieving a correct information cascade.

Observational Learning with a Budget

TL;DR

This work studies budgeted information design in Bayesian observational learning with sequential agents facing a binary world state. It develops a Markovian framework for the cascade process, yielding explicit cascade-probability expressions under irrational and rational cascade constants, and analyzes how to allocate a fixed budget to improve private signal qualities and . The authors prove continuity and monotonicity properties of the probability of a correct cascade with respect to signal-quality improvements, and derive two optimal allocation strategies, including scenarios where it is best to devote the entire budget to one signal or to symmetrically balance improvements. The appendix provides rigorous proofs of lemmas, propositions, and the main theorem, including the behavior when , which exhibits special discontinuities. Overall, the results inform information-design decisions that maximize social learning accuracy under budget constraints in sequential decision settings.

Abstract

We consider a model of Bayesian observational learning in which a sequence of agents receives a private signal about an underlying binary state of the world. Each agent makes a decision based on its own signal and its observations of previous agents. A central planner seeks to improve the accuracy of these signals by allocating a limited budget to enhance signal quality across agents. We formulate and analyze the budget allocation problem and propose two optimal allocation strategies. At least one of these strategies is shown to maximize the probability of achieving a correct information cascade.
Paper Structure (21 sections, 17 theorems, 47 equations, 3 figures)

This paper contains 21 sections, 17 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model
  4. Optimal decision and cascades
  5. Optimal Budget Allocation Problem
  6. Markovian Analysis of Cascade Probability
  7. Markovian Reformulation
  8. Irrational Cascade Constant
  9. Rational Cascade Constant
  10. Single-side Signal Quality Improvement
  11. Double-side Signal Quality Improvement
  12. Appendix
  13. Proof of Lemma \ref{['lemma: action region']}
  14. Proof of Lemma \ref{['lemma: irrational a never reach b_r-1']}
  15. Proof of Proposition \ref{['coro: prob of CC']}
  16. ...and 6 more sections

Key Result

Lemma 1

Agent $n$ cascades to a Yes (No) action if and only if $l_{n-1} < \frac{1-p_1}{p_2}$$( l_{n-1} \!>\!\! \frac{p_1}{1-p_2} )$ and otherwise follows its private signal $S_n$.

Figures (3)

  • Figure 1: The channel through which agents receive private signals.
  • Figure 2: Example transition diagram for $\{h_n\}$ when $V=G$.
  • Figure 3: Comparison of $\mathbb{P}_{\text{cc}}(p_1,p_2)$ using the irrational and rational formulas with increasing $p_2$.

Theorems & Definitions (34)

  • Definition 1
  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 1
  • ...and 24 more