But the key observation is that *some* equilibrium makes this mechanism optimal. Indeed, recall that an optimal allocation rule gives the good to whoever has the highest ironed virtual value (IVV) if that value is positive. But there could be many such rules!

When both bidders have types in a region where the IVV is flat, then how we allocate among those bidders is revenue-neutral. So use the following bidding rule: - In a flat-IVV region: bidder 1 [resp. 2] bids the supremum [infimum] type in the region. - Elsewhere: bid your type.