Separating the Revenue-Optimal Buy-\(k\) and Buy-(\(k+1\)) Auction (Continued)

This week I have spent a lot of time trying to prove that there exists a revenue-optimal buy-\(k\) incentive-compatible mechanism for a particular distribution that is of a certain "nice" form. With this nice form, I would be able to prove a strict gap in revenue between the optimal buy-\(k\) and buy-\((k+1)\) mechanism.

I have attempted to break down the proof in several smaller claims which I believe together will help prove the overall result. In particular, I have so far managed to show that "nice" mechanism I have in mind is buy-\(k\) incentive-compatible and achieves a certain revenue. I have also shown several other properties about what all revenue-optimal mechanisms should look like for this distribution. Unfortunately, I thought I had a proof for one key critical piece which would tie in all the claims together but it broke down beyond buy-one case.

With the program ending soon, I hope to spend the next week or two finishing up this proof and beginning to prepare our overall paper for submission to a conference.