Destructive Bidding in Common-Valuation, All-Pay Auctions and Lotteries

In the classic first-price all-pay auction, bidders compete for a common-value prize and every bidder pays her bid. This paper considers the case in which losing bids are paid and additionally asymmetrically reduce the value of the prize for the winning bidder. This set-up provides a better fit for some political contests and military conflicts as well as having applications to lobbying and R&D races. We first solve for the n-player, symmetric, mixed-strategy symmetric equilibrium to the auction and compare the results to the standard all-pay auction model. We then further generalize the model into a two round game in which a first round destructive investment is made followed by an all-pay lottery. Risk-aversion of contestants and effectiveness of the investment are shown to have a non-monotonic effect on the size of the destructive investment.