Sequential elimination in multi-stage all-pay auctions

I study a multi-stage all-pay auction in which the lowest bidder in each stage is eliminated. Elimination continues until only two bidders remain, one of whom wins the auction. I analyse optimal bidding behaviour and the seller's expected revenue when bidders have independent and private values. In contrast to typical bidding strategies, the optimal bid in each stage is strictly decreasing in the number of bidders. For a fixed number of bidders, however, bids increase as bidders progress through the stages of the auction. Despite independent values, this multi-stage auction yields less expected revenue to the seller than its single-stage counterpart when there are more than three bidders.