AbstractIn auctions with externalities, it is well‐known that the core can be empty, which is undesirable both in terms of stability and “fairness.” Nevertheless, some auction outcome must be chosen. We separate deviations into two types: deviations by paying more and deviations by refusing to pay. In high‐stakes auctions where bidders also care about their reputation, the latter are unlikely to occur, or else can be prevented by legal interventions. In contrast, the former is more undesirable in the sense that the seller and the bidders experience justified envy. We show that the core is nonempty if bidders cannot refuse to pay.