{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,16]],"date-time":"2026-03-16T15:23:23Z","timestamp":1773674603529,"version":"3.50.1"},"reference-count":20,"publisher":"Association for Computing Machinery (ACM)","issue":"2","license":[{"start":{"date-parts":[[2007,5,1]],"date-time":"2007-05-01T00:00:00Z","timestamp":1177977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Internet Technol."],"published-print":{"date-parts":[[2007,5]]},"abstract":"<jats:p>\n            This article considers a canonical auction protocol that forms the basis of nearly all current online auctions. Such\n            <jats:italic>discrete bid<\/jats:italic>\n            auctions require that the bidders submit bids at predetermined discrete bid levels, and thus, there exists a minimal increment by which the bid price may be raised. In contrast, the academic literature of optimal auction design deals almost solely with continuous bid auctions. As a result, there is little practical guidance as to how an auctioneer, seeking to maximize its revenue, should determine the number and value of these discrete bid levels, and it is this omission that is addressed here. To this end, a model of an ascending price English auction with discrete bid levels is considered. An expression for the expected revenue of this auction is derived and used to determine numerical and analytical solutions for the optimal bid levels in the case of uniform and exponential bidder's valuation distributions. Finally, in order to develop an intuitive understanding of how these optimal bid levels are distributed, the limiting case where the number of discrete bid levels is large is considered, and an analytical expression for their distribution is derived.\n          <\/jats:p>","DOI":"10.1145\/1239971.1239976","type":"journal-article","created":{"date-parts":[[2007,6,6]],"date-time":"2007-06-06T14:37:11Z","timestamp":1181140631000},"page":"12","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":30,"title":["Optimal design of english auctions with discrete bid levels"],"prefix":"10.1145","volume":"7","author":[{"given":"Esther","family":"David","sequence":"first","affiliation":[{"name":"University of Southampton, Southampton, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alex","family":"Rogers","sequence":"additional","affiliation":[{"name":"University of Southampton, Southampton, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas R.","family":"Jennings","sequence":"additional","affiliation":[{"name":"University of Southampton, Southampton, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeremy","family":"Schiff","sequence":"additional","affiliation":[{"name":"Bar-Ilan University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sarit","family":"Kraus","sequence":"additional","affiliation":[{"name":"Bar-Ilan University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael H.","family":"Rothkopf","sequence":"additional","affiliation":[{"name":"Rutgers Business School and RUTCOR: the Rutgers Center for Operations Research"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2007,5]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/857166.857167"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.2307\/1593721"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.49.1.85.12754"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1111\/j.1540-5915.2002.tb01656.x"},{"key":"e_1_2_1_5_1","first-page":"233","article-title":"Auctions with severely bounded communication","volume":"28","author":"Blumrosen L.","year":"2007","unstructured":"Blumrosen , L. , Nisan , N. , and Segal , I. 2007 . Auctions with severely bounded communication . J. AI Resear. 28 , 233 -- 266 . Blumrosen, L., Nisan, N., and Segal, I. 2007. Auctions with severely bounded communication. J. AI Resear. 28, 233--266.","journal-title":"J. AI Resear."},{"key":"e_1_2_1_6_1","doi-asserted-by":"crossref","unstructured":"Cassidy R. 1967. Auctions and Auctioneering. University of California Press.  Cassidy R. 1967. Auctions and Auctioneering. University of California Press.","DOI":"10.1525\/9780520322257"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/0165-1765(89)90019-0"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/1064009.1064020"},{"key":"e_1_2_1_9_1","unstructured":"Hageman L. A. and Young D. M. 1981. Applied Iterative Methods. Academic Press.  Hageman L. A. and Young D. M. 1981. Applied Iterative Methods. Academic Press."},{"key":"e_1_2_1_10_1","volume-title":"Proceedings of IJCAI-05 Workshop on Game Theoretic and Decision Theoretic Agents. 98--107","author":"Jiang A. X.","unstructured":"Jiang , A. X. and Leyton-Brown , K . 2005. Estimating bidders' valuation distributions in online auctions . In Proceedings of IJCAI-05 Workshop on Game Theoretic and Decision Theoretic Agents. 98--107 . Jiang, A. X. and Leyton-Brown, K. 2005. Estimating bidders' valuation distributions in online auctions. In Proceedings of IJCAI-05 Workshop on Game Theoretic and Decision Theoretic Agents. 98--107."},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/AAMAS.2004.31"},{"key":"e_1_2_1_12_1","first-page":"585","article-title":"Equilibrium in auctions with entry","volume":"84","author":"Levin D.","year":"1994","unstructured":"Levin , D. and Smith , J. L. 1994 . Equilibrium in auctions with entry . Amer. Econom. Rev. 84 , 3, 585 -- 599 . Levin, D. and Smith, J. L. 1994. Equilibrium in auctions with entry. Amer. Econom. Rev. 84, 3, 585--99.","journal-title":"Amer. Econom. Rev."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1111\/1467-6451.00122"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1287\/moor.6.1.58"},{"key":"e_1_2_1_15_1","volume-title":"Numerical Recipes: The Art of Scientific Computing","author":"Press W. H.","year":"1992","unstructured":"Press , W. H. , Flannery , B. P. , Teukolsky , S. A. , and Vetterling , W. T . 1992 . Numerical Recipes: The Art of Scientific Computing . Cambridge University Press . Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. 1992. Numerical Recipes: The Art of Scientific Computing. Cambridge University Press."},{"key":"e_1_2_1_16_1","first-page":"381","article-title":"Optimal auctions","volume":"71","author":"Riley J. G.","year":"1981","unstructured":"Riley , J. G. and Samuelson , W. F. 1981 . Optimal auctions . Amer. Econom. Rev. 71 , 381 -- 392 . Riley, J. G. and Samuelson, W. F. 1981. Optimal auctions. Amer. Econom. Rev. 71, 381--392.","journal-title":"Amer. Econom. Rev."},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1145\/1064009.1064020"},{"key":"e_1_2_1_18_1","doi-asserted-by":"crossref","DOI":"10.1145\/1255438.1255441","article-title":"The effects of proxy bidding and minimum bid increments within eBay auctions. ACM","author":"Rogers A.","year":"2007","unstructured":"Rogers , A. , David , E. , Schiff , J. , Kraus , S. , and Jennings , N. R. 2007 . The effects of proxy bidding and minimum bid increments within eBay auctions. ACM Trans. Web. http:\/\/eprints.ecs.soton.ac.uk\/12716\/ Rogers, A., David, E., Schiff, J., Kraus, S., and Jennings, N. R. 2007. The effects of proxy bidding and minimum bid increments within eBay auctions. ACM Trans. Web. http:\/\/eprints.ecs.soton.ac.uk\/12716\/","journal-title":"Trans. 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