The Targeted-Loss Exposure Frontier in Auctions

Pay-to-bid auctions charge participants before allocation and therefore make losing-side payments vulnerable to seller intervention. This paper introduces targeted-loss exposure, a stress test that records a designated bidder's payment when one rival's equilibrium report is fixed at an arbitrarily high level while her type distribution is preserved. Under nonnegative payments and an ordering in which a winner pays at least as much as a loser, revenue equivalence yields a sharp two-bidder bound attained by the silent war of attrition. For any number of bidders, the same format attains the upper bound among payment-ordered rank-local rules. The proof formulates payment location as a linear program and constructs a dual probability measure. With more than two bidders, complementary slackness determines the optimal loser-payment schedule almost everywhere. Winner-pay auctions have zero exposure, standard all-pay auctions lie strictly below the frontier, and frontier exposure decreases as the number of bidders rises. A loser-only rule shows that removing payment ordering can make exposure unbounded. A common-shock extension also shows that anticipating intervention changes bids but preserves the ordering of standard formats. The results separate the total interim payment fixed by revenue equivalence from the winning or losing state to which that payment is attached.