[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83895-en":3,"doc-seo-83895-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83895,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Multi Choice Min Prophet Inequality","The prophet inequality studies sequential decision-making with unknown stopping outcomes. This work investigates its minimization counterpart, the min prophet (or cost prophet inequality): players observe independent values from known distributions and must irrevocably select one element while minimizing the chosen value relative to the offline minimum. A multi-choice relaxation is proposed, selecting multiple elements and paying their minimum, trading expected selections against competitive ratio. For adversarial order, constant competitiveness needs nearly linear expected choices, while random order achieves constant competitiveness with only O(ln n) expected choices, with tight bounds via M.","arXiv :2607 .05085v 1 [ cs .GT] 6 Jul 2026  \nMulti Choice Min Prophet  \nYossi Azar∗ Itamar Biran∗ Amos Fiat∗  \nJuly 7, 2026  \nAbstract  \nThe prophet inequality is a fundamental problem in optimal stopping theory. Given n independent variables drawn from known distributions, a player observes values sequentially and must decide irrevocably whether to stop and accept the current value or continue. The goal is to select a single element while maximizing the ratio between the value chosen and that of the maximum value in the sequence. In this paper, we study the minimization counterpart, often termed the min prophet or cost prophet inequality. Unlike the maximization setting, where simple threshold algorithms achieve half of the prophet’s value, the minimization setting is significantly harder, with an exponential lower bound even for i.i.d. variables.  \nWe study a multi-choice relaxation in which the algorithm may select multiple variables and gets to choose the best amongst them (the minimum amongst those selected) . Our goal is to minimize the expected number of selections while achieving a constant competitive ratio. For adversarial order, we show that a constant competitive ratio requires a nearly linear number of choices in expectation, ergo,Ω(n/ln n) . In contrast, we show that for the prophet secretary model (random order) one can attain constant competitiveness while requiring only an exponentially smaller expected number of choices i.e. O(ln n) . We give a refined analysis and define M to be the ratio of the minimum expected value of any single variable to the expected minimum value of all variables (the prophet’s value) and present an algorithm that achieves a constant competitive ratio with O(min{ln ln M, ln n}) choices in expectation for the prophet secretary. We show that this is tight up to low order log factors even for the special case of the i.i.d. model. Specifically, the lower bound on the expected number of choices for any constant competitive algorithm is Ω(min{ln ln M/ln ln ln M, ln n/ln ln n}) . We also show that if we insist on a deterministic bound on the number of choices then every constant competitive algorithm requires n choices. This holds even in the i.i.d. setting and shows that to achieve a constant competitive algorithm there is an exponential gap between the lower bound on the deterministic number of choices and the upper bound on the expected number of choices.  \nFinally, we consider a variant where both the algorithm and the adversary choose r values and pay their sum, this is the minimization multi unit version. We extend our techniques to the multi-unit variant for i.i.d. variables, achieving a constant competitive ratio with a small expected number of choices.  \n∗ Blavatnik School of Computer Science, Tel Aviv University, Israel. Emails: [azar@tau.ac.il](azar@tau.ac.il), [itamarbiran@mail.tau.ac.il](itamarbiran@mail.tau.ac.il), [fiat@tau.ac.il](fiat@tau.ac.il).  \n1 Introduction  \nConsider the following problem, there are n rewards X1 ,..., Xn drawn independently from known distributions F1 ,..., Fn. The player observes the values one by one and may stop at any point, collecting the last observed reward. The goal is to maximize the payoff. This problem is well known and is called the prophet inequality [24, 25] . The optimum offline solution (prophet’s expected payoff) is E[max j Xj] . For general variables, the best online algorithms achieve at least half of the optimum prophet’s value [24, 25 , 29] . For random order (also called prophet secretary) the bound is better (between 0 .688 [10] and 0.723 [17]) . For i.i.d. variables, the optimal online algorithm achieves 0 .745 [11, 20 , 22]  \n. This result has been extended in many directions, including multi-choice prophet inequalities [6], multi-unit prophet settings [3], and resource augmentation [8] .  \nIn this work, we study the minimization counterpart, sometimes called the cost prophet inequality [26] . Here the Xi’s represent costs, and","cbCaibuKEQY5Z1vq","https://ap.wps.com/l/cbCaibuKEQY5Z1vq","pdf",556834,1,33,"English","en",105,"# Abstract\n# Introduction\n## Problem setting: prophet inequality and cost (min prophet)\n## Min prophet vs max prophet\n## Multi-choice relaxation and competitiveness definition","[{\"question\":\"What is the min prophet (cost prophet inequality) problem studied in the paper?\",\"answer\":\"Values arrive sequentially and the player must immediately decide whether to stop and select a value or continue. The objective is to minimize the expected selected value relative to the offline optimum minimum.\"},{\"question\":\"How does the paper extend the classic min prophet setting?\",\"answer\":\"It studies a multi-choice relaxation where the algorithm selects multiple variables and the outcome cost is the minimum among those selected, aiming to reduce the expected number of selections while keeping a constant competitive ratio.\"},{\"question\":\"How do the results differ between adversarial order and random order (prophet secretary)?\",\"answer\":\"With adversarial order, achieving constant competitiveness requires nearly linear expected selections, Ω(n/ln n). For random order, constant competitiveness can be achieved with a much smaller expected number of choices, O(ln n), along with matching lower bounds.\"}]",1784191277,83,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"multi-choice-min-prophet-inequality","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/multi-choice-min-prophet-inequality/83895/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the min prophet (cost prophet inequality) problem studied in the paper?","Question",{"text":75,"@type":76},"Values arrive sequentially and the player must immediately decide whether to stop and select a value or continue. The objective is to minimize the expected selected value relative to the offline optimum minimum.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper extend the classic min prophet setting?",{"text":80,"@type":76},"It studies a multi-choice relaxation where the algorithm selects multiple variables and the outcome cost is the minimum among those selected, aiming to reduce the expected number of selections while keeping a constant competitive ratio.",{"name":82,"@type":73,"acceptedAnswer":83},"How do the results differ between adversarial order and random order (prophet secretary)?",{"text":84,"@type":76},"With adversarial order, achieving constant competitiveness requires nearly linear expected selections, Ω(n/ln n). For random order, constant competitiveness can be achieved with a much smaller expected number of choices, O(ln n), along with matching lower bounds.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]