{"id":1681,"date":"2012-08-12T17:52:44","date_gmt":"2012-08-12T21:52:44","guid":{"rendered":"http:\/\/mat.tepper.cmu.edu\/blog\/?p=1681"},"modified":"2012-08-12T17:52:44","modified_gmt":"2012-08-12T21:52:44","slug":"owning-the-podium-summer-2012-edition","status":"publish","type":"post","link":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/2012\/08\/12\/owning-the-podium-summer-2012-edition\/","title":{"rendered":"Owning the Podium: Summer 2012 edition"},"content":{"rendered":"

During the last winter Olympics, I had what I thought was a pretty good idea<\/a>. \u00a0There are many ways to rank countries during the Olympics: \u00a0you can rank them by total number of medals, or you can rank them by number of gold medals, or by some point scheme (5 for gold, 3 for silver, 1 for bronze) and so on. \u00a0Point schemes seem to make sense, but then people argue about points. \u00a0Is a gold worth 5 bronzes or 4? Is 2 silvers more than, less than, or the same as a gold?<\/p>\n

So my idea was to rank countries by the fraction of reasonable weights that result in them having the highest point count. \u00a0Not every point scheme is reasonable: \u00a0only a bronze lover (pyropusaphile?)\u00a0would score bronze higher than gold. \u00a0So we need gold >= silver >= bronze. \u00a0And it seems unreasonable to have a negative weight on a medal. \u00a0Finally, the weights can be scaled so that the total weight is one.<\/p>\n

\"\"In the Winter 2010 Olympics, Canada was edged out by the United States in the Trick Medal Championship (TMC) by a narrow margin. \u00a0Canada had 14 gold, 7 silver, and 5 bronze; \u00a0the US went 9, 15, 13. \u00a0If you put enough weight on gold, then Canada wins. \u00a0But only 45.25% of the reasonable weights put enough weight on Gold for Canada to win; \u00a0the US wins for the remaining 54.75% of the weights.<\/p>\n

The summer Olympics are now over, with the final medal count being:<\/p>\n

US: 46,29,29<\/p>\n

China: 38,27,22<\/p>\n

Russia: 24,25,33<\/p>\n

Great Britain: 29, 17,19<\/p>\n

with no other country winning at least 20 medals of a single type.<\/p>\n

So the coveted TMC Award goes to …. the United States in a rout! \u00a0In fact, the US wins for every reasonable weighting. \u00a0Russia could win with a lot of weight on bronze medals, but not if the weight on gold and silver is at least that of bronze.<\/p>\n

A necessary and sufficient condition to win for any reasonable weight is to have<\/p>\n

    \n
  1. more gold than anyone else,<\/li>\n
  2. more gold+silver than anyone else, and<\/li>\n
  3. more gold+silver+bronze than anyone else.<\/li>\n<\/ol>\n

    Equality in any of these can lead to weights where the country ties for the win.<\/p>\n

    Here, the US meets that condition. \u00a0Of course, it helps that there are a zillion medals in swimming (where the US does well) and only one in, say, team handball (described here as water polo without the water, which is only marginally informative). \u00a0But a win is a win: \u00a0if any representative of the US Olympic teams would like to drop by my office, I will be glad to give them the TMC trophy (which I will be recycling from my stash of high-school curling trophies I have dragged around the world).<\/p>\n

    P.S. The Wall Street Journal Numbers Guy has a similar discussion<\/a> though, sadly, it does not include the above approach.<\/p>\n","protected":false},"excerpt":{"rendered":"

    During the last winter Olympics, I had what I thought was a pretty good idea. \u00a0There are many ways to rank countries during the Olympics: \u00a0you can rank them by total number of medals, or you can rank them by number of gold medals, or by some point scheme (5 for gold, 3 for silver, … Continue reading “Owning the Podium: Summer 2012 edition”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[51,57],"tags":[],"class_list":["post-1681","post","type-post","status-publish","format-standard","hentry","category-sports","category-voting"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1681","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=1681"}],"version-history":[{"count":0,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1681\/revisions"}],"wp:attachment":[{"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=1681"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=1681"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=1681"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}