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：国际象棋棋局中的着法通常由专业棋手根据具体情况逐一分析，但借助大型棋局数据库的可用性，我们可以构想另一种研究方法。本文确实采用了截然不同的视角，从统计学角度分析棋局中的着法。我们首先聚焦于空间属性与棋子的位置分布，并表明一局棋中可能着法的数量与其结果呈正相关。随后，我们研究棋子的热力图，发现人类棋手之间棋子空间分布的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使用的传统方式大相径庭。这些热力图还使我们能够构建一种表征棋手使用棋子方式的距离度量。在第二部分中，我们聚焦于Stockfish评估的最佳着法与次佳着法，并研究两者评估值之差Δ。我们发现棋局中存在不同阶段。在“平稳”阶段，Δ值较小，表明双方棋手均有多条路径可选。相反，还存在以“临界点”为特征的“波动”阶段，此时Δ值显著增大。在这些临界点，棋局的最终结果可能完全取决于所选着法。我们还发现，对于大量棋局而言，Δ的分布可拟合为幂律分布P(Δ)~Δ^{-β}，其指数似乎具有普遍性（对人类棋手和引擎均成立），约为β≈1.8。因此，在棋局中遭遇临界点的概率远不可忽略。最后，我们通过提及量化理解棋局的可能研究方向作为结论，例如兵链结构、棋子间交互图，以及临界点的量化定义。