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...

We study all-pay auctions (or wars of attrition), where the highest bidder wins an object, but all bidders pay their bids. We consider such auctions when two bidders alternate in raising their bids and where all aspects of the auction are common knowledge including bidders' valuations. We...