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Bayesian games with nested information

Royi Jacobovic, John Yehuda Levy, Eilon Solan

TL;DR

This paper studies Bayesian games with nested information and proves the existence of a Bayesian 0-equilibrium when action spaces are finite, payoffs are bounded, and type spaces are Polish. It introduces two main techniques: a finite- belief-hierarchy approximation to obtain Bayesian ε-equilibria, and a Measurable Choice-based construction to pass to an exact equilibrium while preserving measurability across all players. The results extend prior ε-equilibrium findings by eliminating the need for density absolute continuity in the information structure and by handling arbitrarily many players under nested information. The approach leverages a careful synthesis of δ-approximation, finite information structures, selection theorems (Kuratowski–Ryll-Nardzewski, Meertens), and Aumann integration to build a convergent, measurable equilibrium. The findings have implications for hierarchical information settings and provide a framework for exact equilibrium existence in broader classes of Bayesian games with structured information.

Abstract

A Bayesian game is said to have nested information if the players are ordered, and each player knows the types of all players that follow her in that order. We prove that all multiplayer Bayesian games with finite actions spaces, bounded payoffs,Polish type spaces, and nested information admit a Bayesian equilibrium.

Bayesian games with nested information

TL;DR

This paper studies Bayesian games with nested information and proves the existence of a Bayesian 0-equilibrium when action spaces are finite, payoffs are bounded, and type spaces are Polish. It introduces two main techniques: a finite- belief-hierarchy approximation to obtain Bayesian ε-equilibria, and a Measurable Choice-based construction to pass to an exact equilibrium while preserving measurability across all players. The results extend prior ε-equilibrium findings by eliminating the need for density absolute continuity in the information structure and by handling arbitrarily many players under nested information. The approach leverages a careful synthesis of δ-approximation, finite information structures, selection theorems (Kuratowski–Ryll-Nardzewski, Meertens), and Aumann integration to build a convergent, measurable equilibrium. The findings have implications for hierarchical information settings and provide a framework for exact equilibrium existence in broader classes of Bayesian games with structured information.

Abstract

A Bayesian game is said to have nested information if the players are ordered, and each player knows the types of all players that follow her in that order. We prove that all multiplayer Bayesian games with finite actions spaces, bounded payoffs,Polish type spaces, and nested information admit a Bayesian equilibrium.
Paper Structure (17 sections, 17 theorems, 80 equations)

This paper contains 17 sections, 17 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Literature on Bayesian Equilibrium and on Nested Information
  3. The Model and Main result
  4. The Driving Force Behind our Proofs
  5. Proof of Theorem \ref{['thm:0']}
  6. Existence of a Bayesian $\varepsilon$-equilibrium under A1
  7. $\delta$-approximations
  8. A finite approximation of the information structure
  9. Proof: existence of Bayesian $\varepsilon$-equilibrium under A1
  10. Existence of a Bayesian $0$-equilibrium under A1 and A2
  11. Selectors and Integration
  12. Conditional Expected Payoffs
  13. The Limit of $\frac{1}{k}$-Equilibria: The Operator $\Psi_i$
  14. Equilibrium Characterization via $\Psi_1,\Psi_2,\ldots,\Psi_n$
  15. Selections from $(\Psi_i)_{i \in {\cal N}}$
  16. ...and 2 more sections

Key Result

Theorem 1

$\text{ }$ Every Bayesian game with nested information that satisfies Conditions A1 and A2 admits a Bayesian $0$-equilibrium.

Theorems & Definitions (31)

  • Definition 1: Bayesian game
  • Definition 2: Bayesian $\varepsilon$-equilibrium
  • Remark 1: The relation between Bayesian and Harsanyi equilibria
  • Definition 3: Nested information
  • Remark 2
  • Remark 3: Players possessing the same information
  • Remark 4: ${\mathbb P}$-a.s. versus everywhere in Eq. \ref{['equ:nested']}
  • Remark 5: Nested information and absolute continuity of information
  • Theorem 1: Existence of 0-equilibrium
  • Remark 6: Tightness of the conditions in Theorem 1
  • ...and 21 more