### Power of chess

Sergei Maslov writes a viewpoint Power laws in chess about the paper Zipf’s Law in the Popularity Distribution of Chess Openings (Bernd Blasius and Ralf Tönjes, *Phys. Rev. Lett.* 103, 218701 (2009)). Blasius and Tönjes have observed that chess opening move popularity follows a power law with exponent -2. For longer sequences of openings the popularity still follows a power law, but with a more negative exponent.

An interesting point made by Maslov is that the branching ratios of the game tree seems to approach the distribution 2/(pi sqrt(1-r^2)), which has a peak close to 1 - most positions have very few likely subsequent positions. This may be due to the database used mainly contains games by skilful players, who do their best to move the game into situations where the opponent has few choices. So if he is right, here is a signature of intelligence in the game tree statistics.

Apropos power laws, I just read Power-law distributions in empirical data by Aaron Clauset, Cosma Rohilla Shalizi & M. E. J. Newman (*SIAM Review* 51, 661-703 (2009)). A nice extension of their previous very useful paper about MLE estimation of power laws, and a stern reminder to use careful statistical methods before claiming power law properties. I suspect Blasius and Tönjes would do well checking their data with maximum likeliehood rather than logarithmic bins before claiming as they do, "Stretching over 6 orders of magnitude, the here-reported distributions are among the most precise examples for power laws known today in social data sets."

Posted by Anders3 at November 19, 2009 05:47 PM