We use logistic regression to estimate the value of the pieces in standard chess and several chess variants, namely Chess 960, Atomic chess, Antichess, and Horde chess. We perform our regressions on several years of data from Lichess, the free and open-source internet chess server. We use the published player ratings to control for the confounding effect of differential player skill. We adjust for the attenuation bias in regressions due to the noise in observed ratings. We find that major piece values, relative to the value of a pawn, are fairly consistent with historical valuation systems. However we find slightly higher value to bishops than knights. We find that piece values are smaller, in absolute value, in Atomic and Antichess than standard chess. We also present approximate values of the pieces to equalize odds when players of varying skill face off. We briefly consider self-play experiments using the Stockfish engine, which give a contrasting view of piece value.

翻译：我们使用逻辑回归来估计标准国际象棋以及几种变体（包括 Chess 960、原子象棋、反象棋和部落象棋）中的棋子价值。我们对来自免费开源国际象棋服务器 Lichess 的多年数据进行回归分析。我们利用已公布棋手等级分来控制不同棋手技能水平的混杂效应，并针对观测等级分中的噪声导致的回归衰减偏差进行调整。我们发现，相对于兵的价值，主要棋子的价值与历史估值体系相当一致，但主教的价值略高于骑士。在原子象棋和反象棋中，棋子价值的绝对值小于标准国际象棋。我们还给出了当不同技能水平的棋手对抗时均衡局面的近似棋子价值。我们简要探讨了使用 Stockfish 引擎进行的自我对弈实验，这些实验提供了关于棋子价值的对比视角。