Problem 3057. Chess performance

Created by Jean-Marie Sainthillier

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A player's rating depends on the ratings of their opponents, and the results scored against them. The difference in rating between two players determines an estimate for the expected score between them.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Supposing Player A was expected to score Ea points (but actually scored Sa).</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The formula for updating his rating is :</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=\"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>This update can be performed after each game or each tournament, or after any suitable rating period.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Suppose Player A has a rating</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Ra</w:t></w:r><w:r><w:t> of 1613, and plays in a five-round tournament. He (or she) loses to a player rated 1609, draws with a player rated 1477, defeats a player rated 1388, defeats a player rated 1586, and loses to a player rated 1720. The player's actual score</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Sa</w:t></w:r><w:r><w:t> is (0 + 0.5 + 1 + 1 + 0) = 2.5. The expected score</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>Ea</w:t></w:r><w:r><w:t> , calculated according to the formula see in Problem 3056, was (0.506 + 0.686 + 0.785 + 0.539 + 0.351) = 2.867. Therefore the player's new rating</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>R'a</w:t></w:r><w:r><w:t> is (1613 + 32×(2.5 − 2.867)) = 1601. We assume that the</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:b/></w:rPr><w:t>K</w:t></w:r><w:r><w:t> factor is always 32.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>I give you rating of Player A, ratings of their opponents and results.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Compute the new rating (K = 32).</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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"}]}

After Problems <a href = "http://www.mathworks.com/matlabcentral/cody/problems/3054-chess-elo-rating-system/">3054</a> and <a href = "http://www.mathworks.com/matlabcentral/cody/problems/3056-chess-probability/">3056</a>In <a href = "http://en.wikipedia.org/wiki/Elo_rating_system">Chess</a>, performance isn't measured absolutely; it is inferred from wins (1), losses (0), and draws (0.5) against other players. A player's rating depends on the ratings of their opponents, and the results scored against them. The difference in rating between two players determines an estimate for the expected score between them.Supposing Player A was expected to score Ea points (but actually scored Sa).The formula for updating his rating is :<img src = "http://upload.wikimedia.org/math/2/3/f/23fbcb658ac1e2565003c2190f28a21e.png"><ul><li></li><li></li></ul>This update can be performed after each game or each tournament, or after any suitable rating period.Suppose Player A has a rating Ra of 1613, and plays in a five-round tournament. He (or she) loses to a player rated 1609, draws with a player rated 1477, defeats a player rated 1388, defeats a player rated 1586, and loses to a player rated 1720. The player's actual score Sa is (0 + 0.5 + 1 + 1 + 0) = 2.5. The expected score Ea , calculated according to the formula see in Problem 3056, was (0.506 + 0.686 + 0.785 + 0.539 + 0.351) = 2.867. Therefore the player's new rating R'a is (1613 + 32×(2.5 − 2.867)) = 1601. We assume that the K factor is always 32.I give you rating of Player A, ratings of their opponents and results.Compute the new rating (K = 32).

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Andrew Newell on 2 Mar 2015

In the second test, there are four players and five game results.

Andrew Newell on 2 Mar 2015

A delightful series of problems. I especially like the tests with real-world examples.

Jean-Marie Sainthillier on 2 Mar 2015

Fixed. Thank you Andrew.

Rafael S.T. Vieira on 4 Sep 2020

Hum, real life doesn't round up scores. And usually our rating is updated after each match (as it is in cody).

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Problem 3057. Chess performance

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