The study of biological and social complex systems has been the focus of intense interest for at least three decades 
. Elections 
, popularity 
, population growth 
, collective motion of birds 
and bacteria 
are just some examples of complex systems that physicists have tackled in these pages. An aspect rarely studied due to the lack of enough data over a long enough period is the manner in which agents learn the best strategies to deal with the complexity of the system. For example, as the number of scientific publication increases, researchers must learn how to choose which papers to read in depth 
; while in earlier times word-of-mouth or listening to a colleague’s talk were reliable strategies, nowadays the journal in which the study was published or the number of citations have become, in spite of their many caveats, indicators that seem to be gaining in popularity.
In order to understand how population-level learning occurs in the “real-word,” we study it here in a model system. Chess is a board game that has fascinated humans ever since its invention in sixth-century India 
. Chess is an extraordinary complex game with
legal positions and
distinct matches, as roughly estimated by Shannon 
. Recently, Blasius and Tönjes 
have showed that scale-free distributions naturally emerge in the branching process in the game tree of the first game moves in chess. Remarkably, this breadth of possibilities emerges from a small set of well-defined rules. This marriage of simple rules and complex outcomes has made chess an excellent test bed for studying cognitive processes such as learning 
and also for testing artificial intelligence algorithms such as evolutionary algorithms 
The very best chess players can foresee the development of a match 10–15 moves into the future, thus making appropriate decisions based on his/her expectations of what his opponent will do. Even though super computers can execute many more calculations and hold much more information in a quickly accessible mode, it was not until heuristic rules were developed to prune the set of possibilities that computers became able to consistently beat human players. Nowadays, even mobile chess programs such as Pocket Fritz™ (http://chessbase-shop.com/en/products/pocket_fritz_4
) have a Elo rating 
which is higher than the current best chess player (Magnus Carlsen with a Elo rating of 2835 – http://fide.com
The ability of many chess engines to accurately evaluate the strength of a position enables us to numerically evaluate the move-by-move white player advantage
and to determine the evolution of the advantage during the course of a chess match. In this way, we can probe the patterns of the game to a degree not before possible and can attempt to uncover population-level learning in the historical evolution of chess match dynamics. Here, we focus on the dynamical aspects of the game by studying the move-by-move dynamics of the white player’s advantage
from over seventy thousand high level chess matches.
We have accessed the portable game notation (PGN) files of 73,444 high level chess matches made free available by PGN Mentor™ (http://www.pgnmentor.com
). These data span the last two centuries of the chess history and cover the most important worldwide chess tournaments, including the World Championships, Candidate Tournaments, and the Linares Tournaments (see Table S1
). White won
of these matches, black won
ended up with in a draw. For each of these 73,444 matches, we estimated
using the Crafty™ 
chess engine which has an Elo rating of 2950 (see Methods Section A). The white player advantage
takes into account the differences in the number and the value of pieces, as well as the advantage related to the placement of pieces. It is usually measured in units of pawns, meaning that in the absence of other factors, it varies by one unit when a pawn (the pieces with lowest value) is captured. A positive value indicates that the white player has the advantage and a negative one indicates that the black player has the advantage. illustrates the move dependence of
for 50 matches selected at random from the data base. Intriguingly,
visually resembles the “erratic” movement of diffusive particles.
Diffusive dynamics of white player’s advantage.