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A Mean Field Game of Sequential Testing

Steven Campbell, Yuchong Zhang

Abstract

We introduce a mean field game for a family of filtering problems related to the classic sequential testing of the drift of a Brownian motion. To the best of our knowledge this work presents the first treatment of mean field filtering games with stopping and an unobserved common noise in the literature. We show that the game is well-posed, characterize the solution, and establish the existence of an equilibrium under certain assumptions. We also perform numerical studies for several examples of interest.

A Mean Field Game of Sequential Testing

Abstract

We introduce a mean field game for a family of filtering problems related to the classic sequential testing of the drift of a Brownian motion. To the best of our knowledge this work presents the first treatment of mean field filtering games with stopping and an unobserved common noise in the literature. We show that the game is well-posed, characterize the solution, and establish the existence of an equilibrium under certain assumptions. We also perform numerical studies for several examples of interest.
Paper Structure (32 sections, 38 theorems, 213 equations, 3 figures)

This paper contains 32 sections, 38 theorems, 213 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Literature
  3. Notation
  4. Sequential Testing Game
  5. Signal process
  6. Bayes risk
  7. Mean field equilibrium
  8. Analysis of the Single-Agent Problem
  9. Posterior probability and the log-likelihood ratio
  10. Value function and optimal stopping time
  11. Structure of the continuation region
  12. Regularity of the stopping boundaries
  13. Free-boundary problem and boundary integral equations
  14. Existence of a Mean Field Equilibrium
  15. Preemption Games with the Classic Loss
  16. ...and 17 more sections

Key Result

Lemma 2.1

If the family of stopping times $(\tau^i)_{i\in I}$ is essentially pairwise conditionally independent given $\theta$, then That is, the fraction of agents that have stopped by time $t$ equals the conditional probability that a randomly chosen representative agent will stop by time $t$.

Figures (3)

  • Figure 1: $\Pi$ process trajectories in the non-interactive setting $(\lambda_1=0)$ (left panel) and equilibrium stopping boundaries (right panel).
  • Figure 2: Equilibrium stopping time CDFs (left panel) and problem values (right panel).
  • Figure 3: Asymmetric classic problem continuation regions (left panel) and conditional stopping time distributions (right panel) in equilibrium. In the latter plot, the line styles distinguish between the choices of $\lambda_1$.

Theorems & Definitions (81)

  • Lemma 2.1
  • Example 2.2
  • Remark 2.3
  • Example 2.4: Cross-entropy loss
  • Example 2.5: $L_1$ and $L_2$ losses
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9
  • Lemma 3.1
  • ...and 71 more