Problem 3061. Mirror, mirror on the wall, who is fairest of them all?

Created by Jean-Marie Sainthillier

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The Elo system is employed to rank coeds by their attractiveness. The equations driving the algorithm are shown briefly (Ea and Eb).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>You should know these equations now (See problem</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www.mathworks.com/matlabcentral/cody/problems/3056-chess-probability/\"><w:r><w:t>3056</w:t></w:r></w:hyperlink><w:r><w:t>) :</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:customXml w:element=\"image\"><w:customXmlPr><w:attr w:name=\"height\" w:val=\"-1\"/><w:attr w:name=\"width\" w:val=\"-1\"/><w:attr w:name=\"relationshipId\" w:val=\"rId1\"/></w:customXmlPr></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Ea is the expected probability that Girl A will win the match against Girl B.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Ra is the rating of Girl A, which changes after tournament, according to the formula (Ra )n = (Ra )n-1 + 32 (W - Ea ) where W = {1,0.5,0}.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Now imagine a single round-robin tournament where each girl plays (is compared with) every other girl once. A judge (me for the problem) gives a note :</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>1 if girl A is more attractive than girl B</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>0 if girl B is more attractive than girl A</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr></w:pPr><w:r><w:t>0.5 if same attractiveness</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>I give you the tournament results (2 on the main diagonal).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Find the final rating of Snow White (she is unique).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Consider that all girl begin the tournament with a rating of 1000.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>You can observe that the total number of attractiveness (ELO) points remains constant.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"},{"partUri":"/media/image1.png","contentType":"image/png","content":"data:image/png;base64,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"}]}

The Elo rating system was featured in the movie The Social Network during the <a href = "https://www.youtube.com/watch?v=BzZRr4KV59I/">algorithm scene</a> where Mark Zuckerberg released Facemash.In the scene Eduardo Saverin writes mathematical formulas for the Elo rating system on Zuckerberg's dorm room window. 
The Elo system is employed to rank coeds by their attractiveness. 
The equations driving the algorithm are shown briefly (Ea and Eb).You should know these equations now (See problem <a href = "http://www.mathworks.com/matlabcentral/cody/problems/3056-chess-probability/">3056</a>) :<img src = "http://upload.wikimedia.org/math/b/0/3/b0366725c224ee55eab6e2371dc6a0ef.png"><ul><li></li></ul>Ea is the expected probability that Girl A will win the match against Girl B.Ra is the rating of Girl A, which changes after tournament, according to the formula (Ra )n = (Ra )n-1 + 32 (W - Ea ) where W = {1,0.5,0}.Now imagine a single round-robin tournament where each girl plays (is compared with) every other girl once.
A judge (me for the problem) gives a note :<ul><li>1 if girl A is more attractive than girl B</li><li>0 if girl B is more attractive than girl A</li><li>0.5 if same attractiveness</li></ul>I give you the tournament results (2 on the main diagonal).Find the final rating of Snow White (she is unique).Consider that all girl begin the tournament with a rating of 1000.You can observe that the total number of attractiveness (ELO) points remains constant.

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Last Solution submitted on Dec 23, 2020

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Rafael S.T. Vieira on 4 Sep 2020

The score should be updated only after all matches. I initially thought that it should be updated after each tournament round (that's how it is done while playing on the net, but I guess each tournament has its rules).

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Problem 3061. Mirror, mirror on the wall, who is fairest of them all?

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