Zipf’s Law in the Popularity Distribution of Chess Openings

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.

Figure 1

(a) Schematic representation of the weighted game tree of chess based on the ScidBase [6] for the first three half moves. Each node indicates a state of the game. Possible game continuations are shown as solid lines together with the branching ratios $r_d$. Dotted lines symbolize other game continuations, which are not shown. (b) Alternative representation emphasizing the successive segmentation of the set of games, here indicated for games following a 1.d4 opening until the fourth half move $d=4$. Each node $\sigma$ is represented by a box of a size proportional to its frequency $n_{\sigma }$. In the subsequent half move these games split into subsets (indicated vertically below) according to the possible game continuations. Highlighted in (a) and (b) is a popular opening sequence 1.d4 Nf6 2.c4 e6 (Indian defense).

Figure 2

(a) Histogram of weight frequencies $S(n)$ of openings up to $d=40$ in the Scid database and with logarithmic binning. A straight line fit (not shown) yields an exponent of $\alpha =2.05$ with a goodness of fit $R^2>0.9992$. For comparison, the Zipf distribution Eq. (8) with $\mu =1$ is indicated as a solid line. Inset: number $C(n)=\sum _{m=n+1}^{N}S(m)$ of openings with a popularity $m>n$. $C(n)$ follows a power law with exponent $\alpha =1.04$ ($R^2=0.994$). (b) Number $S_d(n)$ of openings of depth $d$ with a given popularity $n$ for $d=16$ and histograms with logarithmic binning for $d=4$, $d=16$, and $d=22$. Solid lines are regression lines to the logarithmically binned data ($R^2>0.99$ for $d<35$). Inset: slope ${\alpha }_d$ of the regression line as a function of $d$ and the analytical estimation Eq. (6) using $N=1.4\times {10}^6$ and $\beta =0$ (solid line).

Figure 3

(a) Probability density $q(r)$ of branching ratios $r$ sampled from all games in the Scid database with a bin size of $\Delta r=0.01$ and arcsine distribution Eq. (2) (black solid line). Every edge of the weighted game tree, from nodes of size $n_{d-1}$ to $n_d$, contributes to the bin corresponding to $r=n_d/n_{d-1}$ with weight $r$. We disregarded clusters with $n_d<100$ so that, in principle, a cluster could contribute to any of the bins. We found $q(r)$ to be depth independent. (b) Distribution of opening popularities $S_d(n)$ for $d=22$ obtained from the Scid database (black) and from a direct simulation of the multiplicative process Eq. (2), with branching ratios $q(r)$ taken from a uniform or arcsine distribution. Further indicated is the theoretical result Eq. (5) (dashed line). Similar results are obtained for other values of $d$.

Figure 4

Inequality [11, 21] of the distribution $S_d(n)$. (a) Proportion $W$ of games that is concentrating in the fraction $Q$ of the most popular openings, for several levels of the game depth $d$. (b) $Q$ as a function of $d$ for three different values of $W$ (solid lines) and Gini coefficient $G=1-2{\int }_0^{1}Q(W)dW$ as a function of game depth (dotted line).