[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85749-en":3,"doc-seo-85749-105":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},85749,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Threshold Dynamics and Correlated Prophet Inequalities","Prophet inequalities serve as a key analytical tool for online algorithms, yet most results rely on independence among input random variables, restricting real-world applicability. This work studies prophet inequalities under two correlation models generated by a latent world state Z. For the common-base model, it analyzes single-threshold algorithms that always accept the final item, yielding optimal deterministic guarantees of 0.381 and randomized guarantees of 0.4. A new differential-equation framework unifies known single-threshold results, while the common-scale model proves strong impossibility: no algorithm exceeds a competitive ratio of 1/n.","arXiv :2607 .09887v 1 [ cs .GT] 10 Jul 2026  \nThreshold Dynamics and Correlated Prophet Inequalities  \nJos´e Correa ∗ Maximilian Fichtl † Reda Jlibene ‡ Rida Laraki § Vasilis Livanos ¶ Kevin Schewior ‖ Victor Verdugo ∗∗  \nAbstract  \nIn recent years, prophet inequalities have become a central tool for analyzing the performance of online algorithms. However, most existing results assume that input random variables are independent, which significantly limits their applicability. Motivated by this gap, we study prophet inequalities under two elementary correlation models induced by a latent state of the world variable Z. In the common-base model, the algorithm observes the sequence Z + X1 ,..., Z + Xn , a special case of linear correlations. We analyze single-threshold algorithms with the constraint that they always accept the final item, thereby guaranteeing a reward of at least Z. When Z is chosen adversarially, we characterize the optimal deterministic algorithm of this form, achieving a competitive ratio of 0 .381. We then show that randomizing the threshold improves the guarantee to 0 .4, and prove this is optimal via a balanced-prices lower bound. By a minimax argument, the same ratio is achievable when Z is random.  \nWe depart from standard techniques by establishing a stronger lower bound of 0 .41 and an upper bound of 0 .475, ruling out the possibility that this class of algorithms attains the 1/2 ratio known for independent inputs. The core technical contribution is a new analytical framework that captures the reward dynamics of single-threshold algorithms. We introduce a differential equation characterizing the expected reward of a threshold in the worst-case instance, parameterized by the distribution of the maximum. This equation admits a closed-form solution and unifies all known single-threshold prophet inequalities, yielding instance-optimal guaranteesand a simple threshold-optimality condition applicable to the common-base model.  \nFinally, we study the common-scale model, where inputs take the form Z · X1 ,..., Z · Xn. We show that even this minimal multiplicative correlation yields strong impossibility results: no algorithm can achieve a competitive ratio exceeding 1/n. This is proved via a new notion of multiplicative-invariant stopping times, which also rules out competitive guarantees for secretary-type objectives.  \n∗ Department of Industrial Engineering, Universidad de Chile ([correa@uchile.cl](correa@uchile.cl) )  \n†TNG Technology Consulting ([mfichtl@gmail.com](mfichtl@gmail.com) )  \n‡Moroccan Center for Game Theory, Universit´e Mohammed VI Polytechnique ([reda.jlibene@um6p.ma](reda.jlibene@um6p.ma) )  \n§Moroccan Center for Game Theory, Universit´e Mohammed VI Polytechnique ([rida.laraki@um6p.ma](rida.laraki@um6p.ma) )  \n¶ Center for Mathematical Modeling & Department of Computer Science, University of Southern California ([vas.livanos@gmail.com](vas.livanos@gmail.com) )  \n‖Department of Mathematics and Computer Science, University of Cologne ([k.schewior@uni-koeln.de](k.schewior@uni-koeln.de) )  \n∗∗ Institute for Mathematical and Computational Engineering, and Department of Industrial and Systems Engineering, Pontificia Universidad Cat´olica de Chile ([victor.verdugo@uc.cl](victor.verdugo@uc.cl) )  \n1 Introduction  \nProphet inequalities are one of the most important modern paradigms in online algorithms. In the classic model [Correa et al. , 2018 , Hill and Kertz, 1982 , Kertz, 1986 , Samuel-Cahn, 1984], there aren options, modeled by non-negative random variables Y1 ,..., Yn, whose realizations are presented to the decision maker one after the other. At time step i, the decision maker can either choose to stop, obtaining reward Yi, or to continue to get a potentially higher reward in the future. The decision maker’s goal is to maximize the expected value of their choice. Application areas of this problem include online resource allocation, pricing, matching markets, e-commerce, and hiring processes, whi","cbCaitDqPCKY4HDj","https://ap.wps.com/l/cbCaitDqPCKY4HDj","pdf",548577,1,35,"English","en",105,"# Abstract\n## Introduction\n## Correlation models via a latent state Z\n## Single-threshold algorithms under the common-base model\n## Differential-equation framework for reward dynamics\n## Common-scale model and impossibility results\n## Related work and classic prophet inequality setting","[{\"question\":\"What correlation structure does the paper study for prophet inequalities?\",\"answer\":\"It considers two correlation models induced by a latent world variable Z: a common-base model (linear correlations) and a common-scale model (multiplicative correlations).\"},{\"question\":\"What guarantee do single-threshold algorithms achieve in the common-base model?\",\"answer\":\"When Z is chosen adversarially, the optimal deterministic single-threshold strategy has competitive ratio 0.381. Randomizing the threshold improves the guarantee to 0.4, and a lower bound proves this is optimal.\"},{\"question\":\"What impossibility result is shown for the common-scale correlation model?\",\"answer\":\"Even with minimal multiplicative correlation, no algorithm can achieve a competitive ratio exceeding 1/n; the paper also rules out competitive guarantees for secretary-type objectives under this notion.\"}]",1784206003,88,{"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},"threshold-dynamics-and-correlated-prophet-inequalities","",{"@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/threshold-dynamics-and-correlated-prophet-inequalities/85749/",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 correlation structure does the paper study for prophet inequalities?","Question",{"text":75,"@type":76},"It considers two correlation models induced by a latent world variable Z: a common-base model (linear correlations) and a common-scale model (multiplicative correlations).","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What guarantee do single-threshold algorithms achieve in the common-base model?",{"text":80,"@type":76},"When Z is chosen adversarially, the optimal deterministic single-threshold strategy has competitive ratio 0.381. Randomizing the threshold improves the guarantee to 0.4, and a lower bound proves this is optimal.",{"name":82,"@type":73,"acceptedAnswer":83},"What impossibility result is shown for the common-scale correlation model?",{"text":84,"@type":76},"Even with minimal multiplicative correlation, no algorithm can achieve a competitive ratio exceeding 1/n; the paper also rules out competitive guarantees for secretary-type objectives under this notion.","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"]