Pawn Advantage, Win Percentage, and Elo

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An examination by Sune Fischer and Pradu Kannan in December 2007 on the approximate relations between Win Percentage, Pawn Advantage, and Elo rating advantage for computer chess resulted in following findings.

Relationship

It was found that the the approximate relationship between the winning probability W and the pawn advantage P is

The inverse relationship can be given as

From the above, the relationship between the equivalent Elo rating advantage R and the pawn advantage P can be given as

Data Acquisition

Data was taken from a collection of 405,460 computer games in 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 logistic model with its sigmoid curve, namely,

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

For K=4, the proposed logistic model and the data is plotted here for comparison:

See also

• 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. AAAI 2007, pdf
• Kenneth W. Regan, Tamal T. Biswas, Jason Zhou (2014). Human and Computer Preferences at Chess. pdf
• Tamal T. Biswas, Kenneth W. Regan (2015). Measuring Level-K Reasoning, Satisficing, and Human Error in Game-Play Data. IEEE ICMLA 2015, pdf preprint
• Shogo Takeuchi, Tomoyuki Kaneko (2015). Estimating Ratings of Computer Players by the Evaluation Scores and Principal Variations in Shogi. ACIT-CSI

Postings

• Elo performance? by Stefan Meyer-Kahlen, CCC, May 22, 1999 » Match Statistics, Playing Strength
• likelihood instead of pawnunits? + chess knowledge by Ingo Lindam, CCC, October 25, 2002
• Winning percentage and centipawns by Luca Hemmerich, CCC, November 18, 2009
• Pawn Advantage, Win Percentage, and Elo by Adam Hair, CCC, April 15, 2012
• normal vs logistic curve for Elo model by Daniel Shawul, CCC, August 02, 2012
• Houdini, much weaker engines, and Arpad Elo by Kai Laskos, CCC, November 29, 2013 » Houdini, Match Statistics
• Using the Stockfish position evaluation score to predict victory probability by unavoidablegrain, Tumblr, September 28, 2014 » Stockfish
• Statistical interpretation of search and eval scores by J. Wesley Cleveland, CCC, November 18, 2017 » Match Statistics, Score
• Why Lc0 eval (in cp) is asymmetric against AB engines? by Kai Laskos, CCC, July 25, 2018 » Asymmetric Evaluation, Leela Chess Zero

External Links

• Elo rating system from Wikipedia
• Back to Basics in Rating Theory by Royal Clifford Jones, Jr.
• Pawn Advantage in iCE by Thomas Petzke, mACE Chess, March 16, 2014 » iCE
• Depth of Satisficing by Ken Regan, Gödel's Lost Letter and P=NP, October 06, 2015  » Depth, Match Statistics, Pawn Advantage, Win Percentage, and Elo, Stockfish, Komodo
• Magnus and the Turkey Grinder by Ken Regan, Gödel's Lost Letter and P=NP, December 08, 2016 » Match Statistics
• Elo Win Probability Calculator by François Labelle

References

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