Moves in chess games are usually analyzed on a case-by-case basis by professional players, but thanks to the availability of large game databases, we can envision another approach of the game. Here, we indeed adopt a very different point of view, and analyze moves in chess games from a statistical point of view. We first focus on spatial properties and the location of pieces and show that the number of possible moves during a game is positively correlated with its outcome. We then study heatmaps of pieces and show that the spatial distribution of pieces varies less between human players than with engines (such as Stockfish): engines seem to use pieces in a very different way as human did for centuries. These heatmaps also allow us to construct a distance between players that characterizes how they use their pieces. In a second part, we focus on the best move and the second best move found by Stockfish and study the difference $\Delta$ of their evaluation. We found different regimes during a chess game. In a `quiet' regime, $\Delta$ is small, indicating that many paths are possible for both players. In contrast, there are also `volatile' regimes characterized by a `tipping point', for which $\Delta$ becomes large. At these tipping points, the outcome could then switch completely depending on the move chosen. We also found that for a large number of games, the distribution of $\Delta$ can be fitted by a power law $P(\Delta)\sim \Delta^{-\beta}$ with an exponent that seems to be universal (for human players and engines) and around $\beta\approx 1.8$. The probability to encounter a tipping point in a game is therefore far from being negligible. Finally, we conclude by mentioning possible directions of research for a quantitative understanding of chess games such as the structure of the pawn chain, the interaction graph between pieces, or a quantitative definition of critical points.

翻译：在国际象棋中，棋手的走法通常由专业棋手逐一分析。然而，借助大型棋局数据库，我们可以构想另一种研究棋局的方法。本文确实采用了一种截然不同的视角，从统计学角度分析国际象棋的走法。我们首先关注空间特性及棋子的位置，发现对局中可能的走法数量与棋局结果呈正相关。随后，我们研究棋子的热力图，并表明人类棋手之间的空间分布差异小于人类与引擎（如Stockfish）之间的差异：引擎使用棋子的方式似乎与人类数百年来习惯的方式截然不同。这些热力图还使我们能够构建一种表征棋手如何使用其棋子的距离度量。在第二部分中，我们聚焦于Stockfish评估的最佳走法与次佳走法，并研究其估值差$\Delta$。我们发现国际象棋对局中呈现不同的动态阶段。在“平稳”阶段中，$\Delta$较小，表明双方棋手均存在多条可行路径。相反，还存在以“临界点”为特征的“波动”阶段，此时$\Delta$值显著增大。在这些临界点上，棋局结果可能完全取决于所选走法而反转。此外，我们发现在大量对局中，$\Delta$的分布可用幂律$P(\Delta)\sim \Delta^{-\beta}$拟合，其指数似乎具有普适性（对人类棋手和引擎均成立），约$\beta\approx 1.8$。因此，在棋局中遭遇临界点的概率远非可忽略。最后，我们提及了定量理解国际象棋棋局的潜在研究方向，例如兵链结构、棋子间相互作用图，或临界点的量化定义。