{"id":318989,"date":"2016-11-14T14:43:37","date_gmt":"2016-11-14T22:43:37","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=318989"},"modified":"2018-10-16T20:14:27","modified_gmt":"2018-10-17T03:14:27","slug":"unifying-hierarchy-valuations-complements-substitutes","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/unifying-hierarchy-valuations-complements-substitutes\/","title":{"rendered":"A Unifying Hierarchy of Valuations with Complements and Substitutes"},"content":{"rendered":"

We introduce a new hierarchy over monotone set functions, that we refer to as MPH<\/em> (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m<\/em> (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1<\/em>, captures all monotone submodular functions, and more generally, the class of functions known as X OS<\/em>. Every monotone function that has a positive hypergraph representation of rank k<\/em> (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k<\/em>. Every monotone function that has supermodular degree k<\/em> (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k + 1)<\/em>. In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k<\/em>. One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH<\/em> hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k + 1<\/em> if all players hold valuation functions in MPH-k<\/em>. The other is an upper bound of 2k<\/em> on the price of anarchy of simultaneous first price auctions. Being in MPH-k<\/em> can be shown to involve two requirements \u2013 one is monotonicity and the other is a certain requirement that we refer to as P LE<\/em> (Positive Lower Envelope). Removing the monotonicity requirement, one obtains the PLE<\/em> hierarchy over all non-negative set functions (whether monotone or not), which can be fertile ground for further research.<\/p>\n","protected":false},"excerpt":{"rendered":"

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561,13548,13546],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-318989","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-research-area-economics","msr-research-area-computational-sciences-mathematics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"Twenty-Ninth AAAI Conference on Artificial Intelligence (AAAI) 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