Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Pettis Integral Approach to Large Population Games

Masaki Miyashita, Takashi Ui

Abstract

The analysis of large population economies with incomplete information often entails the integration of a continuum of random variables. We showcase the usefulness of the integral notion à la Pettis (1938) to study such models. We present several results on Pettis integrals, including convenient sufficient conditions for Pettis integrability and Fubini-like exchangeability formulae, illustrated through a running example. Building on these foundations, we conduct a unified analysis of Bayesian games with arbitrarily many heterogeneous agents. We provide a sufficient condition on payoff structures, under which the equilibrium uniqueness is guaranteed across all signal structures. Our condition is parsimonious, as it turns out necessary when strategic interactions are undirected. We further identify the moment restrictions, imposed on the equilibrium action-state joint distribution, which have crucial implications for information designer's problem of persuading a population of strategically interacting agents. To attain these results, we introduce and develop novel mathematical tools, built on the theory of integral kernels and reproducing kernel Hilbert spaces in functional analysis.

On the Pettis Integral Approach to Large Population Games

Abstract

The analysis of large population economies with incomplete information often entails the integration of a continuum of random variables. We showcase the usefulness of the integral notion à la Pettis (1938) to study such models. We present several results on Pettis integrals, including convenient sufficient conditions for Pettis integrability and Fubini-like exchangeability formulae, illustrated through a running example. Building on these foundations, we conduct a unified analysis of Bayesian games with arbitrarily many heterogeneous agents. We provide a sufficient condition on payoff structures, under which the equilibrium uniqueness is guaranteed across all signal structures. Our condition is parsimonious, as it turns out necessary when strategic interactions are undirected. We further identify the moment restrictions, imposed on the equilibrium action-state joint distribution, which have crucial implications for information designer's problem of persuading a population of strategically interacting agents. To attain these results, we introduce and develop novel mathematical tools, built on the theory of integral kernels and reproducing kernel Hilbert spaces in functional analysis.
Paper Structure (21 sections, 21 theorems, 134 equations, 2 figures)

This paper contains 21 sections, 21 theorems, 134 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Aggregating Random Variables
  3. Large Population Games with Incomplete Information
  4. Pettis Integral
  5. Pettis Integral of Hilbert-valued Processes
  6. Pettis Integral of Stochastic Processes
  7. Pettis Integral of Gaussian Processes
  8. Equilibrium Analysis
  9. Setup
  10. Moment Restrictions
  11. Uniqueness of Equilibrium
  12. Application to Information Design
  13. Designer's Problem
  14. Targeted Disclosure
  15. Symmetric Disclosure
  16. ...and 6 more sections

Key Result

Proposition 1

Consider the following conditions on a process $f:T \to X$. If (and only if) $f$ satisfies (P1), it is weakly measurable. If, in addition, $f$ satisfies (P2), then it is Pettis integrable.

Figures (2)

  • Figure 1: The set of points $(\alpha,\beta)$ satisfying each of (T1), (T2), and (T3) are displayed. The left panel corresponds to the case of small strategic interactions, $r \in (-1,1)$, resulting in an upward-sloping ray that divides (T2) and (T3). In contrast, the right panel considers the case of large strategic substitutes, $r < -1$, resulting in a downward-sloping ray.
  • Figure 2: In the left panel, the dashed line represents $(\alpha,\beta)$ in Example \ref{['ex_market']} for different values of $\lambda$. Full disclosure (resp. partial disclosure) is optimal if and only if $(\alpha,\beta)$ lies below (resp. above) the ray of slope $\frac{1-\gamma}{2}$. The right panel displays the region of $(\lambda,\gamma)$ for which each disclosure policy is optimal.

Theorems & Definitions (47)

  • Example 1
  • Definition 1
  • Proposition 1
  • Remark 1
  • Corollary 1
  • Example 1: Continued
  • Proposition 2
  • Example 1: Continued
  • Proposition 3
  • Corollary 2
  • ...and 37 more