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.
