Revisiting the convergence theorem for competitive bidding in common value auctions

In common value auctions, the value of the item for sale is identical among bidders, but bidders have different information (noisy signal) about the item’s value. Wilson (Rev Econ Stud 4:511–518, 1977) and Milgrom (Econometrica 47:679–688, 1979) proved the convergence theorem of competitive bidding that the winning bid converges to the true value almost surely or in probability respectively. In particular, Milgrom provided a necessary and sufficient condition for convergence when the common value is a random variable that is nowhere dense. A counterexample is given for which Milgrom’s condition is not necessary when the common value is a continuous random variable. We provide a sufficient condition for the convergence theorem in a wallet game which is a special case of a common value auction.