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'''[[Main Page|Home]] * [[Evaluation]] * Pawn Advantage, Win Percentage and Elo'''
An examination by [[Sune Fischer]] and [[Pradu Kannan]] in December 2007 on the approximate relations between win percentage, pawn advantage, and [https://en.wikipedia.org/wiki/Elo_rating_system Elo rating] advantage for computer chess appeaered with following findings.
=Relationship=
It was found that the the approximate relationship between the winning [https://en.wikipedia.org/wiki/Probability probability] W and the pawn advantage P is
[[FILE:PawnWinELOFormula1.jpg|none|border|text-bottom]]
The inverse relationship can be given as
[[FILE:PawnWinELOFormula2.jpg|none|border|text-bottom]]
From the above, the relationship between the equivalent Elo rating advantage R and the pawn advantage P can be given as
[[FILE:PawnWinELOFormula3.jpg|none|border|text-bottom]]
=Data Acquisition=
Data was taken from a collection of 405,460 computer games in [[Portable Game Notation|PGN format]]. Whenever exactly 5 plys in a game had gone by without captures, the game result was accumulated twice in a table indexed by the material configuration. The data was accumulated twice because it was assumed that material values were equal for both colors. So if there was data for a KPK material configuration, the data was also tallied for the KKP. Only data pertaining to the material configuration was taken. This was considered reasonable because the material configuration is the most important quantity that affects the result of a game.
=Data Reduction and Modeling=
For each material configuration, a pawn value was computed using conventional pawn-normalized material ratios that are close to those used in strong chess programs (P=1, N=4, B=4.1, R=6, Q=12). The relationship between Win Percentage and Pawn Advantage was assumed to follow a [https://en.wikipedia.org/wiki/Logistic_regression logistic model] <ref>[http://mathworld.wolfram.com/LogisticEquation.html logistic model from Wolfram MathWorld]</ref> with its [https://en.wikipedia.org/wiki/Sigmoid_function sigmoid curve], namely,
[[FILE:PawnWinELOFormula4.jpg|none|border|text-bottom]]
where K is an unknown non-zero constant. When applying the condition that the win probability is 0.5 if there is no pawn advantage, the solution to the above seperable differential equation becomes
[[FILE:PawnWinELOFormula5.jpg|none|border|text-bottom]]
For K=4, the proposed logistic model and the data is plotted here for comparison:
[[FILE:wp2pa.PNG|none|border|text-bottom]]
=See also=
* [[Bishop versus Knight#WinningPercantages|Bishop versus Knight - Winning Percentages]]
* [[Match Statistics]]
* [[Material]]
* [[Playing Strength]]
=Publications=
* [[Shogo Takeuchi]], [[Tomoyuki Kaneko]], [[Kazunori Yamaguchi]], [[Satoru Kawai]] ('''2007'''). ''Visualization and Adjustment of Evaluation Functions Based on Evaluation Values and Win Probability''. [http://www.informatik.uni-trier.de/~ley/db/conf/aaai/aaai2007.html AAAI 2007], [https://www.aaai.org/Papers/AAAI/2007/AAAI07-136.pdf pdf]
* [[Kenneth Wingate Regan|Kenneth W. Regan]], [[Tamal T. Biswas]], [[Jason Zhou]] ('''2014'''). ''Human and Computer Preferences at Chess''. [http://www.cse.buffalo.edu/~regan/papers/pdf/RBZ14aaai.pdf pdf]
* [[Tamal T. Biswas]], [[Kenneth Wingate Regan|Kenneth W. Regan]] ('''2015'''). ''Measuring Level-K Reasoning, Satisficing, and Human Error in Game-Play Data''. [[IEEE]] [http://www.icmla-conference.org/icmla15/ ICMLA 2015], [http://www.cse.buffalo.edu/~regan/papers/pdf/BiRe15_ICMLA2015.pdf pdf preprint]
* [[Shogo Takeuchi]], [[Tomoyuki Kaneko]] ('''2015'''). ''[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7336038 Estimating Ratings of Computer Players by the Evaluation Scores and Principal Variations in Shogi]''. [http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7335993 ACIT-CSI]
=Postingss=
* [https://www.stmintz.com/ccc/index.php?id=52542 Elo performance?] by [[Stefan Meyer-Kahlen]], [[CCC]], May 22, 1999 » [[Match Statistics]], [[Playing Strength]]
* [https://www.stmintz.com/ccc/index.php?id=261636 likelihood instead of pawnunits? + chess knowledge] by [[Ingo Lindam]], [[CCC]], October 25, 2002
* [http://www.talkchess.com/forum/viewtopic.php?t=30695 Winning percentage and centipawns] by [[Luca Hemmerich]], [[CCC]], November 18, 2009
* [http://www.talkchess.com/forum/viewtopic.php?t=43323 Pawn Advantage, Win Percentage, and Elo] by [[Adam Hair]], [[CCC]], April 15, 2012
* [http://www.talkchess.com/forum/viewtopic.php?t=44670 normal vs logistic curve for Elo model] by [[Daniel Shawul]], [[CCC]], August 02, 2012
* [http://www.talkchess.com/forum/viewtopic.php?t=50266 Houdini, much weaker engines, and Arpad Elo] by [[Kai Laskos]], [[CCC]], November 29, 2013 » [[Houdini]], [[Match Statistics]] <ref>[https://en.wikipedia.org/wiki/Arpad_Elo Arpad Elo - Wikipedia]</ref>
* [https://chesscomputer.tumblr.com/post/98632536555/using-the-stockfish-position-evaluation-score-to/embed Using the Stockfish position evaluation score to predict victory probability] by unavoidablegrain, [https://en.wikipedia.org/wiki/Tumblr Tumblr], September 28, 2014 » [[Stockfish]]
* [http://www.talkchess.com/forum/viewtopic.php?t=65764 Statistical interpretation of search and eval scores] by [[J. Wesley Cleveland]], [[CCC]], November 18, 2017 » [[Match Statistics]], [[Score]]
=External Links=
* [https://en.wikipedia.org/wiki/Elo_rating_system Elo rating system from Wikipedia]
* [http://www.ratingtheory.com/index.htm Back to Basics in Rating Theory] by [http://www.ratingtheory.com/author.htm Royal Clifford Jones, Jr.]
* [http://macechess.blogspot.de/2014/03/pawn-advantage-in-ice.html Pawn Advantage in iCE] by [[Thomas Petzke]], [http://macechess.blogspot.de/ mACE Chess], March 16, 2014 » [[iCE]]
* [https://rjlipton.wordpress.com/2015/10/06/depth-of-satisficing/ Depth of Satisficing] by [[Kenneth Wingate Regan|Ken Regan]], [https://rjlipton.wordpress.com/ Gödel's Lost Letter and P=NP], October 06, 2015 » [[Depth]], [[Match Statistics]], [[Pawn Advantage, Win Percentage, and Elo]], [[Stockfish]], [[Komodo]] <ref>[http://www.talkchess.com/forum/viewtopic.php?t=57890 Regan's latest: Depth of Satisficing] by [[Carl Lumma]], [[CCC]], October 09, 2015</ref>
* [https://rjlipton.wordpress.com/2016/12/08/magnus-and-the-turkey-grinder/ Magnus and the Turkey Grinder] by [[Kenneth Wingate Regan|Ken Regan]], [https://rjlipton.wordpress.com/ Gödel's Lost Letter and P=NP], December 08, 2016 » [[Match Statistics]] <ref>[https://rjlipton.wordpress.com/2016/11/30/when-data-serves-turkey/ When Data Serves Turkey] by [[Kenneth Wingate Regan|Ken Regan]], [https://rjlipton.wordpress.com/ Gödel's Lost Letter and P=NP], November 30, 2016</ref> <ref>[https://en.wikipedia.org/wiki/World_Chess_Championship_2016 World Chess Championship 2016 from Wikipedia]</ref> <ref>[http://www.talkchess.com/forum/viewtopic.php?t=62435 Regan's conundrum] by [[Carl Lumma]], [[CCC]], December 09, 2016</ref>
* [http://wismuth.com/elo/calculator.html Elo Win Probability Calculator] by [[Mathematician#FLabelle|François Labelle]]
=References=
<references />
'''[[Evaluation|Up one Level]]'''
An examination by [[Sune Fischer]] and [[Pradu Kannan]] in December 2007 on the approximate relations between win percentage, pawn advantage, and [https://en.wikipedia.org/wiki/Elo_rating_system Elo rating] advantage for computer chess appeaered with following findings.
=Relationship=
It was found that the the approximate relationship between the winning [https://en.wikipedia.org/wiki/Probability probability] W and the pawn advantage P is
[[FILE:PawnWinELOFormula1.jpg|none|border|text-bottom]]
The inverse relationship can be given as
[[FILE:PawnWinELOFormula2.jpg|none|border|text-bottom]]
From the above, the relationship between the equivalent Elo rating advantage R and the pawn advantage P can be given as
[[FILE:PawnWinELOFormula3.jpg|none|border|text-bottom]]
=Data Acquisition=
Data was taken from a collection of 405,460 computer games in [[Portable Game Notation|PGN format]]. Whenever exactly 5 plys in a game had gone by without captures, the game result was accumulated twice in a table indexed by the material configuration. The data was accumulated twice because it was assumed that material values were equal for both colors. So if there was data for a KPK material configuration, the data was also tallied for the KKP. Only data pertaining to the material configuration was taken. This was considered reasonable because the material configuration is the most important quantity that affects the result of a game.
=Data Reduction and Modeling=
For each material configuration, a pawn value was computed using conventional pawn-normalized material ratios that are close to those used in strong chess programs (P=1, N=4, B=4.1, R=6, Q=12). The relationship between Win Percentage and Pawn Advantage was assumed to follow a [https://en.wikipedia.org/wiki/Logistic_regression logistic model] <ref>[http://mathworld.wolfram.com/LogisticEquation.html logistic model from Wolfram MathWorld]</ref> with its [https://en.wikipedia.org/wiki/Sigmoid_function sigmoid curve], namely,
[[FILE:PawnWinELOFormula4.jpg|none|border|text-bottom]]
where K is an unknown non-zero constant. When applying the condition that the win probability is 0.5 if there is no pawn advantage, the solution to the above seperable differential equation becomes
[[FILE:PawnWinELOFormula5.jpg|none|border|text-bottom]]
For K=4, the proposed logistic model and the data is plotted here for comparison:
[[FILE:wp2pa.PNG|none|border|text-bottom]]
=See also=
* [[Bishop versus Knight#WinningPercantages|Bishop versus Knight - Winning Percentages]]
* [[Match Statistics]]
* [[Material]]
* [[Playing Strength]]
=Publications=
* [[Shogo Takeuchi]], [[Tomoyuki Kaneko]], [[Kazunori Yamaguchi]], [[Satoru Kawai]] ('''2007'''). ''Visualization and Adjustment of Evaluation Functions Based on Evaluation Values and Win Probability''. [http://www.informatik.uni-trier.de/~ley/db/conf/aaai/aaai2007.html AAAI 2007], [https://www.aaai.org/Papers/AAAI/2007/AAAI07-136.pdf pdf]
* [[Kenneth Wingate Regan|Kenneth W. Regan]], [[Tamal T. Biswas]], [[Jason Zhou]] ('''2014'''). ''Human and Computer Preferences at Chess''. [http://www.cse.buffalo.edu/~regan/papers/pdf/RBZ14aaai.pdf pdf]
* [[Tamal T. Biswas]], [[Kenneth Wingate Regan|Kenneth W. Regan]] ('''2015'''). ''Measuring Level-K Reasoning, Satisficing, and Human Error in Game-Play Data''. [[IEEE]] [http://www.icmla-conference.org/icmla15/ ICMLA 2015], [http://www.cse.buffalo.edu/~regan/papers/pdf/BiRe15_ICMLA2015.pdf pdf preprint]
* [[Shogo Takeuchi]], [[Tomoyuki Kaneko]] ('''2015'''). ''[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7336038 Estimating Ratings of Computer Players by the Evaluation Scores and Principal Variations in Shogi]''. [http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=7335993 ACIT-CSI]
=Postingss=
* [https://www.stmintz.com/ccc/index.php?id=52542 Elo performance?] by [[Stefan Meyer-Kahlen]], [[CCC]], May 22, 1999 » [[Match Statistics]], [[Playing Strength]]
* [https://www.stmintz.com/ccc/index.php?id=261636 likelihood instead of pawnunits? + chess knowledge] by [[Ingo Lindam]], [[CCC]], October 25, 2002
* [http://www.talkchess.com/forum/viewtopic.php?t=30695 Winning percentage and centipawns] by [[Luca Hemmerich]], [[CCC]], November 18, 2009
* [http://www.talkchess.com/forum/viewtopic.php?t=43323 Pawn Advantage, Win Percentage, and Elo] by [[Adam Hair]], [[CCC]], April 15, 2012
* [http://www.talkchess.com/forum/viewtopic.php?t=44670 normal vs logistic curve for Elo model] by [[Daniel Shawul]], [[CCC]], August 02, 2012
* [http://www.talkchess.com/forum/viewtopic.php?t=50266 Houdini, much weaker engines, and Arpad Elo] by [[Kai Laskos]], [[CCC]], November 29, 2013 » [[Houdini]], [[Match Statistics]] <ref>[https://en.wikipedia.org/wiki/Arpad_Elo Arpad Elo - Wikipedia]</ref>
* [https://chesscomputer.tumblr.com/post/98632536555/using-the-stockfish-position-evaluation-score-to/embed Using the Stockfish position evaluation score to predict victory probability] by unavoidablegrain, [https://en.wikipedia.org/wiki/Tumblr Tumblr], September 28, 2014 » [[Stockfish]]
* [http://www.talkchess.com/forum/viewtopic.php?t=65764 Statistical interpretation of search and eval scores] by [[J. Wesley Cleveland]], [[CCC]], November 18, 2017 » [[Match Statistics]], [[Score]]
=External Links=
* [https://en.wikipedia.org/wiki/Elo_rating_system Elo rating system from Wikipedia]
* [http://www.ratingtheory.com/index.htm Back to Basics in Rating Theory] by [http://www.ratingtheory.com/author.htm Royal Clifford Jones, Jr.]
* [http://macechess.blogspot.de/2014/03/pawn-advantage-in-ice.html Pawn Advantage in iCE] by [[Thomas Petzke]], [http://macechess.blogspot.de/ mACE Chess], March 16, 2014 » [[iCE]]
* [https://rjlipton.wordpress.com/2015/10/06/depth-of-satisficing/ Depth of Satisficing] by [[Kenneth Wingate Regan|Ken Regan]], [https://rjlipton.wordpress.com/ Gödel's Lost Letter and P=NP], October 06, 2015 » [[Depth]], [[Match Statistics]], [[Pawn Advantage, Win Percentage, and Elo]], [[Stockfish]], [[Komodo]] <ref>[http://www.talkchess.com/forum/viewtopic.php?t=57890 Regan's latest: Depth of Satisficing] by [[Carl Lumma]], [[CCC]], October 09, 2015</ref>
* [https://rjlipton.wordpress.com/2016/12/08/magnus-and-the-turkey-grinder/ Magnus and the Turkey Grinder] by [[Kenneth Wingate Regan|Ken Regan]], [https://rjlipton.wordpress.com/ Gödel's Lost Letter and P=NP], December 08, 2016 » [[Match Statistics]] <ref>[https://rjlipton.wordpress.com/2016/11/30/when-data-serves-turkey/ When Data Serves Turkey] by [[Kenneth Wingate Regan|Ken Regan]], [https://rjlipton.wordpress.com/ Gödel's Lost Letter and P=NP], November 30, 2016</ref> <ref>[https://en.wikipedia.org/wiki/World_Chess_Championship_2016 World Chess Championship 2016 from Wikipedia]</ref> <ref>[http://www.talkchess.com/forum/viewtopic.php?t=62435 Regan's conundrum] by [[Carl Lumma]], [[CCC]], December 09, 2016</ref>
* [http://wismuth.com/elo/calculator.html Elo Win Probability Calculator] by [[Mathematician#FLabelle|François Labelle]]
=References=
<references />
'''[[Evaluation|Up one Level]]'''
