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We study the bidding behavior of spiteful agents who, contrary to the common assumption of self-interest, maximize a convex combination of their own profit and their competitors' losses. The motivation for this assumption stems from inherent spitefulness or, for example, from competitive scenarios such as in closed markets where the loss of a competitor will likely result in future gains for oneself. We derive symmetric Bayes Nash equilibria for spiteful agents in 1st-price and 2nd-price sealedbid auctions. In 1st-price auctions, bidders become "more truthful" the more spiteful they are. Surprisingly, the equilibrium strategy in 2nd-price auctions does not depend on the number of bidders. Based on these equilibria, we compare the revenue in both auction types. It turns out that expected revenue in 2nd-price auctions is higher than expected revenue in 1st-price auctions in the case of even the most modestly spiteful agents, provided they still care at least at little for their own profit. In other words, revenue equivalence only holds for auctions in which all agents are either self-interested or completely malicious. We furthermore investigate the impact of common knowledge on spiteful bidding. Divulging the bidders' valuations reduces revenue in 2nd-price auctions, whereas it has the opposite effect in 1st-price auctions.