# 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 [1] 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: