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：本研究采用逻辑回归方法，对标准国际象棋及若干变体（包括Chess 960、原子象棋、弃兵棋和部落棋）中的棋子价值进行估算。回归分析基于免费开源网络象棋服务器Lichess多年积累的对局数据，并利用公开的棋手评级数据控制不同棋手技术水平带来的混杂效应。针对观测评级噪声导致的回归衰减偏误，我们进行了相应校正。研究发现：主要棋子相对于兵的价值与历史估值体系基本一致，但象的价值略高于马；在原子象棋和弃兵棋中，棋子的绝对价值均低于标准国际象棋。本文同时提出了用于平衡不同水平棋手对局胜率的棋子近似价值参数。此外，通过Stockfish引擎进行的自我对弈实验为棋子价值提供了对比视角。