Maybe he’ll be a social scientist some day

2yo is a little confused about “because”

2yo keeps invading our personal space while shouting "Cuz I a doctor!"

On the latter, it's useful to think about the "must-sell" case, where a 2PA with no reserve is optimal in the symmetric regular case whatever the type distribution. This argument says it's still optimal in the symmetric 2-player case, but now the equilibrium depends on details.

Second, we've lost some simplicity: Whereas the auction format only amounts to setting a reserve price, the desired equilibrium now depends in a detailed way on the type distributions.

This example is a bit silly, though, because it loses two of the main desiderata of 2PAs: First, we've lost dominance: A bidder's willingness to bid as described in the bunching regions relies on the other bidder's complementary weird behavior there.

This is an equilibrium for the same reason asymmetric bidding can happen in the complete-info case: I'm indifferent to my bid within a bunching region because it's not pivotal to the trade decision or the price of trade. And binding IR and revenue equivalence—> this is optimal.

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.

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!

How can this be? If nobody plays a weakly dominated strategy, then everyone with type above the reserve price is bidding their value, and so I'm not successfully ironing—meaning the induced allocation rule is suboptimal in the irregular case.