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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!

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I gave you an extremely vague lead in my other comment. Thinking about it a bit harder, I beieve you are right with having a look into the matching literature which I know a bit less.

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.

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I can't suggest any research papers, but I can suggest one way of studying this problem: using an agent-based model. For example, this is a simple model of an auction on NetLogo: https://ccl.northwestern.edu/netlogo/models/BiddingMarket. Maybe you can tweak the code to make the model run a simulation of your problem.

If this wasn't what you were looking for, maybe you can try doing a literature search with "agent-based model" in the keyword list. This problem seems like something that can be studied using these models.

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