Payoff continuity in incomplete information games

An incomplete information game is defined by a probability distribution mu over a type space and payoff functions u. Probability distribution mu' is strategically close to mu if, for any bounded payoff functions u and any equilibrium of the game (mu, u), there exists an approximate equilibrium of the game (mu', u) under which all players get approximately the same payoffs. This note shows that two probability distributions are strategically close if and only if (1) they assign similar ex ante probability to all events; and (2) with high ex ante probability, it is approximate common knowledge that they assign similar conditional probabilities to all events. (C) 1998 Academic Press.