{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T10:11:28Z","timestamp":1765966288284,"version":"3.48.0"},"reference-count":31,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T00:00:00Z","timestamp":1765929600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council (NSTC) of Taiwan","award":["110-2221-E-005-053"],"award-info":[{"award-number":["110-2221-E-005-053"]}]},{"name":"National Science and Technology Council (NSTC) of Taiwan","award":["111-2221-E-005-084"],"award-info":[{"award-number":["111-2221-E-005-084"]}]},{"name":"National Science and Technology Council (NSTC) of Taiwan","award":["112-2221-E-005-076"],"award-info":[{"award-number":["112-2221-E-005-076"]}]},{"name":"National Science and Technology Council (NSTC) of Taiwan","award":["113-2221-E-005-072"],"award-info":[{"award-number":["113-2221-E-005-072"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this paper, a fast distributed algorithm is proposed for solving the winners and price determination problems in a uniform price auction in which each bidder bids for multiple units out of a lot of k identical items with a per-unit price. In a conventional setting, all bidders disclose their bidding information to an auctioneer and let the auctioneer allocate the items and determine the uniform price, i.e., the least winning price. In our setting, all bidders do not need to disclose their bidding information to the auctioneer. The bidders and the auctioneer collaboratively compute by the distributed algorithm to determine in a small number of steps the units allocated and the uniform price. The number of steps is independent of the number of bidders. At the end of the computing process, each bidder can only know the units allocated to him\/her and the uniform price. The auctioneer can only know the units being allocated to the bidders and the uniform price. Therefore, neither the bidders nor the auctioneer are able to know the per-unit bidding prices of the bidders except the uniform price. Moreover, the auctioneer is not able to know the bidding units of the losing bidders. Bidders\u2019 per-unit bidding prices are protected, and the bidding units of the losing bidders are protected. Bidding information privacy is preserved.<\/jats:p>","DOI":"10.3390\/computation13120294","type":"journal-article","created":{"date-parts":[[2025,12,17]],"date-time":"2025-12-17T09:43:46Z","timestamp":1765964626000},"page":"294","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Fast Distributed Algorithm for Uniform Price Auction with Bidding Information Protection"],"prefix":"10.3390","volume":"13","author":[{"given":"John","family":"Sum","sequence":"first","affiliation":[{"name":"Institute of Technology Management, National Chung Hsing University, Taichung 402, Taiwan"}]},{"given":"Chi-Sing","family":"Leung","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, City University of Hong Kong, Kowloon Tong 999077, Hong Kong"}]},{"given":"Janet C. C.","family":"Chang","sequence":"additional","affiliation":[{"name":"Institute of Technology Management, National Chung Hsing University, Taichung 402, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,17]]},"reference":[{"key":"ref_1","unstructured":"Krishna, V. (2009). Auction Theory, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1111\/joes.12482","article-title":"A review of multiunit auctions with homogeneous goods","volume":"36","author":"Khezr","year":"2022","journal-title":"J. Econ. Surv."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lazzaro, J., Ryckebusch, S., Mahowald, M., and Mead, C. (1988, January 1). Winner-Take-All networks of O(N) complexity. Proceedings of the Advances in Neural Information Processing Systems, Cambridge, MA, USA.","DOI":"10.21236\/ADA451466"},{"key":"ref_4","unstructured":"Touretzky, D.S. (December, January 28). On the K-Winners-Take-All Network. 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