# Multi-item multi-unit matching markets

This is a reference request.

I'm interested in auction design problems of the following form:

- Consider one seller and many buyers.

- The seller seeks to sell multiple items where each item has multiple units.

- Each buyer is interested in getting exactly one unit from only one item.

- The seller wants to maximize the social welfare of all buyers.

Has this problem been studied in the literature? I have tried looking at combinatorial auctions, matching markets, and multi-parameter markets but so far haven't been lucky.

I'd appreciate any hints of direction where to look. Thank you!

Here is a setting that might be of interest: Kelso & Crawford, Econometrica 1982. They study $$m$$ workers (=buyers in your setting) and $$n$$ firms (=items), where a worker can only work at one firm, but a firm can hire many workers. Firms pay salaries to workers (=accept payments for a unit). If that is a setting that appeals to you, go straight to "Matching with contracts" by Hatfield & Milgrom, AER 2005.