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Incomplete Information Robustness

Stephen Morris, Takashi Ui

TL;DR

It is shown that, for every belief-invariant correlating device that maximizes the expected value of a generalized potential function, there exists a BIBCE in which every player chooses an action from a subset of actions prescribed by the device, and that the set of such BIBCEs is robust, which can differ from the set from potential maximizing BNE.

Abstract

Consider an analyst who models a strategic situation using an incomplete information game. The true game may involve correlated, duplicated belief hierarchies, but the analyst lacks knowledge of the correlation structure and can only approximate each belief hierarchy. To make predictions in this setting, the analyst uses belief-invariant Bayes correlated equilibria (BIBCE) and seeks to determine which one is justifiable. We address this question by introducing the notion of robustness: a BIBCE is robust if, for every nearby incomplete information game, there exists a BIBCE close to it. Our main result provides a sufficient condition for robustness using a generalized potential function. In a supermodular potential game, a robust BIBCE is a Bayes Nash equilibrium, whereas this need not hold in other classes of games.

Incomplete Information Robustness

TL;DR

It is shown that, for every belief-invariant correlating device that maximizes the expected value of a generalized potential function, there exists a BIBCE in which every player chooses an action from a subset of actions prescribed by the device, and that the set of such BIBCEs is robust, which can differ from the set from potential maximizing BNE.

Abstract

Consider an analyst who models a strategic situation using an incomplete information game. The true game may involve correlated, duplicated belief hierarchies, but the analyst lacks knowledge of the correlation structure and can only approximate each belief hierarchy. To make predictions in this setting, the analyst uses belief-invariant Bayes correlated equilibria (BIBCE) and seeks to determine which one is justifiable. We address this question by introducing the notion of robustness: a BIBCE is robust if, for every nearby incomplete information game, there exists a BIBCE close to it. Our main result provides a sufficient condition for robustness using a generalized potential function. In a supermodular potential game, a robust BIBCE is a Bayes Nash equilibrium, whereas this need not hold in other classes of games.

Paper Structure

This paper contains 31 sections, 22 theorems, 105 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Incomplete information games
  4. Belief-invariant Bayes correlated equilibrium
  5. Elaborations
  6. Non-redundant games
  7. Robustness
  8. Generalized potentials
  9. $\mathcal{A}$-Decision rule
  10. Generalized potentials and BIBCE
  11. Main results
  12. Robustness of GP-maximizing BIBCE
  13. Proof of Theorem \ref{['main theorem']}
  14. Proof outline for Lemma \ref{['main theorem lemma']}
  15. Supermodular games
  16. ...and 16 more sections

Key Result

Lemma 1

The set of all BIBCE of $(T,\Theta,\pi,u)$ is a nonempty convex compact subset of $\Sigma$ containing all BNE, where a convex combination $\alpha\sigma+(1-\alpha)\sigma'$ for $\sigma,\sigma'\in\Sigma^{BI}$ and $\alpha\in(0,1)$ is given by $(\alpha\sigma+(1-\alpha)\sigma')(a|t,\theta)=\alpha\sigma(a|

Theorems & Definitions (40)

  • Lemma 1
  • Definition 1
  • Lemma 2
  • Definition 2
  • Lemma 3
  • Lemma 4
  • Definition 3
  • Remark 1
  • Definition 4
  • Remark 2
  • ...and 30 more