Payoff Continuity in Games of Incomplete Information Across Models of Knowledge

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.