{"id":1063,"date":"2010-02-28T02:08:34","date_gmt":"2010-02-28T06:08:34","guid":{"rendered":"http:\/\/mat.tepper.cmu.edu\/blog\/?p=1063"},"modified":"2010-02-28T02:08:34","modified_gmt":"2010-02-28T06:08:34","slug":"canada-owns-40-of-the-olympic-podium","status":"publish","type":"post","link":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/2010\/02\/28\/canada-owns-40-of-the-olympic-podium\/","title":{"rendered":"Canada owns 40% of the Olympic podium"},"content":{"rendered":"

In this year’s Olympics, much has been made of the Canadian efforts to “own the podium<\/a>“.\u00a0 Canada has spent $118 million in training its athletes, far more than the US has spent ($55 million over four years).\u00a0 Since it seems that, despite a late rush, the Canadian goal of winning more medals than any other country will not be met, the Own the Podium effort appears to be a failure.\u00a0 But perhaps operations research can come to the rescue here.<\/p>\n

The problem is, perhaps, in defining the goal.\u00a0 By defining the goal in terms of overall medals, the Canadians were perhaps too modest.\u00a0 If they had simply strived for excellence and defined their goal in terms of “Most gold medals”, then they would have succeeded:\u00a0 they have 13 gold compared to the Germany’s 10 gold with two events to go.<\/p>\n

It does seem kind of strange to define winning as “Most Medals”:\u00a0 a bronze is not the same as a gold!\u00a0 But it also seems pretty strange to only count gold:\u00a0 the others seem to have some value.<\/p>\n

Rather than look at any particular weighting of the medals, perhaps we should look at any<\/em> reasonable weighting and see who wins.\u00a0 If we give weights wg, ws, and wb to each of gold, silver, and bronze, and let ng, ns, and nb be the number of such medals won, then the score of a country is wg*ng+ws*ns+wb*nb.\u00a0 The stated Canadian goal had (wg,ws,wb)= (1,1,1).\u00a0 Counting gold only has (wg,ws,wb) = (1,0,0).\u00a0 What other weights would be reasonable?<\/p>\n

Clearly, gold is at least as valuable as silver which is at least as valuable as bronze, so we want wg>=ws>=wb.\u00a0 Also, we can normalize so that wb+ws+wb=1 (since, for instance, (2,2,2) is the same as (1,1,1) which is the same as (1\/3, 1\/3, 1\/3)). \u00a0 With these requirements, there are only three teams that might be considered ahead at this point.\u00a0 Consider the leading countries (from nbcolympics.com<\/a>:<\/p>\n\n\n\n\n\n\n\n\n\n
Country<\/th>\nMedalists<\/th>\n\"GOLD\"<\/th>\n\"SILVER\"<\/th>\n\"BRONZE\"<\/th>\nTotal<\/th>\n<\/tr>\n<\/thead>\n
\"CAN\"<\/td>\nUnited States<\/a><\/td>\nSee Names<\/a><\/td>\n9<\/td>\n14<\/td>\n13<\/td>\n36<\/strong><\/td>\n<\/tr>\n
\"CAN\"<\/td>\nGermany<\/a><\/td>\nSee Names<\/a><\/td>\n10<\/td>\n12<\/td>\n7<\/td>\n29<\/strong><\/td>\n<\/tr>\n
\"CAN\"<\/td>\nCanada<\/a><\/td>\nSee Names<\/a><\/td>\n13<\/td>\n7<\/td>\n5<\/td>\n25<\/strong><\/td>\n<\/tr>\n
\"CAN\"<\/td>\nNorway<\/a><\/td>\nSee Names<\/a><\/td>\n8<\/td>\n8<\/td>\n6<\/td>\n22<\/strong><\/td>\n<\/tr>\n
\"CAN\"<\/td>\nAustria<\/a><\/td>\nSee Names<\/a><\/td>\n4<\/td>\n6<\/td>\n6<\/td>\n16<\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Norway, Austria, and every other country (other than Germany and Canada) is dominated by the USA, so cannot be the winner, no matter the weight.<\/p>\n

There are many weights other than (1,0,0) for which Canada is the winner.\u00a0 For instance (.68, .16, .16) is also a win for Canada.\u00a0 Even (.64, .32,.04) results in Canada in first.<\/p>\n

Going through the grid of possible values, it seems that Canada is currently in first in about 40% of the cases;\u00a0 the USA is in first in the remaining 60% of the weights.\u00a0 Germany is never in first, being dominated by the combination of 74% USA and 26% Canada.\u00a0 So perhaps it is fair to say that Canada owns 40% of the podium, trailing the USA with 60%.<\/p>\n

If Canada were to beat the USA in hockey on Sunday, they would go up to 45%.\u00a0 This assumes no further medals in the men’s 50km cross country.\u00a0 But if a Canadian could also win the cross country, then the fraction of weights for which Canada wins goes up to 54.7%.\u00a0 There is still a chance for Canada to “Own the Podium!”.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this year’s Olympics, much has been made of the Canadian efforts to “own the podium“.\u00a0 Canada has spent $118 million in training its athletes, far more than the US has spent ($55 million over four years).\u00a0 Since it seems that, despite a late rush, the Canadian goal of winning more medals than any other … Continue reading “Canada owns 40% of the Olympic podium”<\/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],"tags":[],"class_list":["post-1063","post","type-post","status-publish","format-standard","hentry","category-sports"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1063","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=1063"}],"version-history":[{"count":0,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/posts\/1063\/revisions"}],"wp:attachment":[{"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=1063"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=1063"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.tepper.cmu.edu\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=1063"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}