{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T20:05:19Z","timestamp":1778270719685,"version":"3.51.4"},"reference-count":50,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Management Science"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:p>Real-time decisions are usually irrevocable in many contexts of online decision making. One common practice is delaying real-time decisions so that the decision maker can gather more information to make better decisions. For example, in online retailing, there is typically a time delay between when an online order is received and when it gets picked and assembled for shipping. However, decisions cannot be delayed forever. In this paper, we study this fundamental trade-off and aim to theoretically characterize the benefits of delaying real-time decisions. We provide a theoretical foundation for a broad family of online decision-making problems by proving that the gap between our proposed online algorithm (called \u201cdelayed Bayesian prophet\u201d) and the offline optimal hindsight policy decays exponentially fast in the length of delay. We also conduct extensive numerical experiments on the benefits of delay, using both synthetic data and publicly available real data. Both our theoretical and empirical results demonstrate an important managerial insight: a little delay is all we need. Finally, we extend our analysis and results to the setting where the arrival distribution is independent but nonidentical, the setting where the arrival distribution is unknown, and the setting where decisions are made in batches.<\/jats:p>\n                  <jats:p>This paper was accepted by Jeannette Song, operations management.<\/jats:p>\n                  <jats:p>Funding: The second author was partially funded by a grant from Amazon.com Inc., awarded through collaboration with the Columbia Center of AI Technology.<\/jats:p>\n                  <jats:p>Supplemental Material: The online appendix and data files are available at https:\/\/doi.org\/10.1287\/mnsc.2023.00549 .<\/jats:p>","DOI":"10.1287\/mnsc.2023.00549","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T15:05:16Z","timestamp":1754492716000},"page":"2826-2841","source":"Crossref","is-referenced-by-count":5,"title":["The Benefits of Delay to Online Decision Making"],"prefix":"10.1287","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0390-7631","authenticated-orcid":false,"given":"Yaqi","family":"Xie","sequence":"first","affiliation":[{"name":"Booth School of Business, University of Chicago, Chicago, Illinois 60637"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2420-4468","authenticated-orcid":false,"given":"Will","family":"Ma","sequence":"additional","affiliation":[{"name":"Graduate School of Business, Columbia University, New York, New York 10027"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8160-6877","authenticated-orcid":false,"given":"Linwei","family":"Xin","sequence":"additional","affiliation":[{"name":"Booth School of Business, University of Chicago, Chicago, Illinois 60637"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"109","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1287\/educ.2019.0199"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1287\/msom.2014.0505"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1086\/704761"},{"key":"B4","doi-asserted-by":"crossref","unstructured":"Amil A, Makhdoumi A, Wei Y (2025) Multi-item order fulfillment revisited: LP formulation and prophet inequality.\n                      Management Sci.\n                      ePub ahead of print April 8, https:\/\/doi.org\/10.1287\/mnsc.2023.00357.","DOI":"10.1287\/mnsc.2023.00357"},{"key":"B5","doi-asserted-by":"crossref","unstructured":"Aminian MR, Ma W, Xin L (2023) Real-time personalized order holding. 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