A Strategic Topology on Information Structures

TL;DR

This paper introduces an 'almost common knowledge topology' to measure the closeness of information structures, showing it is the coarsest topology for continuous equilibrium outcomes and that simple structures are dense within it.

Contribution

It defines a new topology on information structures, proves its properties, and demonstrates the density of simple structures, simplifying analysis in information design.

Findings

The 'almost common knowledge topology' is the coarsest topology for equilibrium continuity.

Simple information structures are dense in this topology.

Restricting to simple structures is without loss in information design.

Abstract

Two information structures are said to be close if, with high probability, there is approximate common knowledge that interim beliefs are close under the two information structures. We define an "almost common knowledge topology" reflecting this notion of closeness. We show that it is the coarsest topology generating continuity of equilibrium outcomes. An information structure is said to be simple if each player has a finite set of types and each type has a distinct first-order belief about payoff states. We show that simple information structures are dense in the almost common knowledge topology and thus it is without loss to restrict attention to simple information structures in information design problems.