{"id":184521,"date":"2004-04-19T00:00:00","date_gmt":"2009-10-31T13:50:53","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/stable-marriage-of-poisson-and-lebesgue\/"},"modified":"2016-12-27T11:02:30","modified_gmt":"2016-12-27T19:02:30","slug":"stable-marriage-of-poisson-and-lebesgue","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/stable-marriage-of-poisson-and-lebesgue\/","title":{"rendered":"A Stable Marriage of Poisson and Lebesgue"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Given a point process M of intensity one in the plane, the well-known Voronoi tesselation assigns a polygon (of different area) to each point of M. The geometry of &#8220;fair&#8221; allocations (assigning unit area to each point of M) is richer and more mysterious: see http:\/\/www.math.ubc.ca\/~holroyd\/stable.html<\/p>\n<p>There is a unique &#8220;fair&#8221; allocation that is &#8220;stable&#8221; in the sense of the Gale-Shapley stable marriage problem, every point of M is assigned a bounded region with finitely many components, but obtaining any(!) tail estimate for the diameter of these regions is open. These allocations arose from the continuum version of the &#8220;extra head&#8221; problem, studied by Thorisson and Liggett. The original problem is to find in a sequence of i.i.d. coins with heads probability <i>p<\/i>, one coin that landed heads so that all other coins are still i.i.d. with heads probability <i>p<\/i>. [This is possible without randomization only when <i>1\/p<\/i> is an integer].<\/p>\n<p>Talk based on joint works with C. Hoffman and A. Holroyd.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Given a point process M of intensity one in the plane, the well-known Voronoi tesselation assigns a polygon (of different area) to each point of M. The geometry of &#8220;fair&#8221; allocations (assigning unit area to each point of M) is richer and more mysterious: see http:\/\/www.math.ubc.ca\/~holroyd\/stable.html There is a unique &#8220;fair&#8221; allocation that is &#8220;stable&#8221; [&hellip;]<\/p>\n","protected":false},"featured_media":290063,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-184521","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/MPrB1HiAlJw","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184521","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":1,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184521\/revisions"}],"predecessor-version":[{"id":341774,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184521\/revisions\/341774"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/290063"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184521"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184521"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184521"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184521"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184521"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184521"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184521"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184521"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184521"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184521"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}