Wednesday, November 18, 2009

Power Laws in Chess Openings

I haven't looked closely enough at the paper to judge whether or not these are actual power laws, but the blurb on the American Physical Society website caught my eye, so I browsed through the paper at lightning speed and want to mention it here. (Note that I don't think that Physical Review Letters is necessarily the appropriate journal for such research, but anyway the situation it what it is.) The paper is called Zipf’s Law in the Popularity Distribution of Chess Openings.

Authors: Bernd Blasius and Ralf T\"{o}njes

Abstract: We perform a quantitative analysis of extensive chess databases and show that the frequencies of opening moves are distributed according to a power law with an exponent that increases linearly with the game depth, whereas the pooled distribution of all opening weights follows Zipf’s law with universal exponent. We propose a simple stochastic process that is able to capture the observed playing statistics and show that the Zipf law arises from the self-similar nature of the game tree of chess. Thus, in the case of hierarchical fragmentation the scaling is truly universal and independent of a particular generating mechanism. Our findings are of relevance in general processes with composite decisions.

Naturally, I have taken the obvious step and nominated this paper for an Ig Nobel Prize.


Jon said...

I wonder if collects move sequences for its games. If so, there would be an extensive database for contemporary board games that could be tapped and some of the hypotheses about how strategies evolve could be more directly studied (since games like Puerto Rico were set up on BSW prior to any strategy guides).

Mason said...

I definitely agree that one can go farther with this. I have actually seen a student talk in which the chess game itself was defined in terms of a bipartite network (where black and white pieces are connected if they can reach the same square on the next move), but the students did not go very far. I'm actually very keen on pursuing their idea and seeing if one can actually get somewhere with it.