Payoff Continuity in Games of Incomplete Information Across Models of Knowledge

TL;DR

This paper establishes a fundamental equivalence between two topological notions of proximity in models of incomplete information, showing they are consistent under a suitable labeling of types, which clarifies how equilibrium payoffs vary with information structures.

Contribution

It proves the open conjecture that the topologies on partition profiles and common priors are equivalent when appropriately labeled, unifying different approaches to information structure proximity.

Findings

Proves the equivalence of two topological notions of proximity in information structures.

Shows that equilibrium payoffs are continuous with respect to these topologies.

Provides a unified framework for analyzing the sensitivity of game equilibria to information changes.

Abstract

Equilibrium predictions in games of incomplete information are sensitive to the assumed information structure. Monderer and Samet (1996) and Kajii and Morris (1998) define topological notions of proximity for common prior information structures such that two information structures are close if and only if (approximate) equilibrium payoffs are close. However, Monderer and Samet (1996) fix a common prior and define their topology on profiles of partitions over a state space, whereas Kajii and Morris (1998) define their topology on common priors over the product of a state space and a type space. We prove the open conjecture that two partition profiles are close in the Monderer and Samet (1996) topology if and only if there exists a labeling of types such that the associated common priors are close in the Kajii and Morris (1998) topology.