Rationalizable outcomes of large private-value first-price discrete auctions

We consider discrete versions of first-price auctions. We present a condition on beliefs about players' values such that, with any fixed finite set of possible bids and sufficiently many players, only bidding the bid closest from below to one's true value survives iterative deletion of bids that are dominated, where the dominance is evaluated using beliefs that satisfy the condition. The condition holds in an asymmetric conditionally independent environment so long as the likelihood of each type is bounded from below. In particular, with many players, common knowledge of rationality and that all types are possible in an independent and private values auction implies that players will bid just below their true value.