{"id":392660,"date":"2016-09-21T00:00:23","date_gmt":"2016-09-21T07:00:23","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&#038;p=392660"},"modified":"2018-10-16T20:12:22","modified_gmt":"2018-10-17T03:12:22","slug":"convex-program-duality-fisher-markets-nash-social-welfare","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/convex-program-duality-fisher-markets-nash-social-welfare\/","title":{"rendered":"Convex Program Duality, Fisher Markets, and Nash Social Welfare"},"content":{"rendered":"<p>We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show\u00a0several closely related new results. In particular, we obtain:<\/p>\n<ul>\n<li>A new integer program for the NSW maximization problem whose fractional relaxation has a\u00a0bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap.<\/li>\n<li>An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality\u00a0gap of the above relaxation is at most 2. The approximation factor shown by [7] was 2e\u00a01\/e \u2248 2.89.<\/li>\n<li>A lower bound of e\u00a01\/e \u2248 1.44 on the integrality gap of this relaxation.<\/li>\n<li>New convex programs for natural generalizations of linear Fisher markets and proofs that these\u00a0markets admit rational equilibria.<\/li>\n<\/ul>\n<p>These results were obtained by establishing connections between previously known disparate results,\u00a0and they help uncover their mathematical underpinnings. We show a formal connection between the\u00a0convex programs of Eisenberg and Gale and that of Shmyrev, namely that their duals are equivalent up\u00a0to a change of variables. Both programs capture equilibria of linear Fisher markets. By adding suitable\u00a0constraints to Shmyrev\u2019s program, we obtain a convex program that captures equilibria of the spendingrestricted\u00a0market model defined by [7] in the context of the NSW maximization problem. Further, adding\u00a0certain integral constraints to this program we get the integer program for the NSW mentioned above.<\/p>\n<p>The basic tool we use is convex programming duality. In the special case of convex programs with\u00a0linear constraints (but convex objectives), we show a particularly simple way of obtaining dual programs,\u00a0putting it almost at par with linear program duality. This simple way of finding duals has been used\u00a0subsequently for many other applications.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show\u00a0several closely related new results. In particular, we obtain: A new integer program for the NSW maximization problem whose fractional relaxation has a\u00a0bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap. An improved, and tight, [&hellip;]<\/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":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"","msr_other_contributors":"","msr_speaker":"","msr_award":"","msr_affiliation":"","msr_institution":"","msr_host":"","msr_version":"","msr_duration":"","msr_original_fields_of_study":"","msr_release_tracker_id":"","msr_s2_match_type":"","msr_citation_count_updated":"","msr_published_date":"2016-09-21","msr_highlight_text":"","msr_notes":"","msr_longbiography":"","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1609.06654","msr_external_url":"","msr_secondary_video_url":"","msr_conference_url":"","msr_journal_url":"","msr_s2_pdf_url":"","msr_year":0,"msr_citation_count":0,"msr_influential_citations":0,"msr_reference_count":0,"msr_s2_match_confidence":0,"msr_microsoftintellectualproperty":true,"msr_s2_open_access":false,"msr_s2_author_ids":[],"msr_pub_ids":[],"msr_hide_image_in_river":0,"footnotes":""},"msr-research-highlight":[],"research-area":[13548],"msr-publication-type":[193715],"msr-publisher":[],"msr-focus-area":[],"msr-locale":[268875],"msr-post-option":[],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-392660","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-economics","msr-locale-en_us"],"msr_publishername":"","msr_edition":"","msr_affiliation":"","msr_published_date":"2016-09-21","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"392663","msr_publicationurl":"https:\/\/arxiv.org\/abs\/1609.06654","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"1609.06654","viewUrl":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-content\/uploads\/2017\/06\/1609.06654.pdf","id":392663,"label_id":0},{"type":"url","title":"https:\/\/arxiv.org\/abs\/1609.06654","viewUrl":false,"id":false,"label_id":0}],"msr_related_uploader":"","msr_citation_count":0,"msr_citation_count_updated":"","msr_s2_paper_id":"","msr_influential_citations":0,"msr_reference_count":0,"msr_arxiv_id":"","msr_s2_author_ids":[],"msr_s2_open_access":false,"msr_s2_pdf_url":null,"msr_attachments":[{"id":0,"url":"https:\/\/arxiv.org\/abs\/1609.06654"}],"msr-author-ordering":[{"type":"text","value":"Richard Cole","user_id":0,"rest_url":false},{"type":"user_nicename","value":"nikdev","user_id":33100,"rest_url":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/wp-json\/microsoft-research\/v1\/researchers?person=nikdev"},{"type":"text","value":"Vasilis Gkatzelis","user_id":0,"rest_url":false},{"type":"text","value":"Kamal Jain","user_id":0,"rest_url":false},{"type":"text","value":"Tung Mai","user_id":0,"rest_url":false},{"type":"text","value":"Vijay V. 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