AI 中文总结
本文研究对称不成立时有限博弈的二阶势，构建MS势并提出MS分解，证明行动数相等的博弈中MS势增广项与Candogan等人提出的势一致。
AI 中文摘要
Monderer和Shapley（1996）证明，一个博弈是精确势博弈当且仅当参与者的交叉差异两两一致，这是关于任意两个参与者激励相互关联方式的对称条件。本文探讨当对称性不成立时，可从这些特征中构建出什么。由此得到的MS势由代表博弈共同利益元素的二阶差异构成。MS势在可分离支付项范围内是唯一的，且当高阶MS条件成立时存在。在精确势博弈类上，它能恢复出除参与者个体主效应外的势。随后，最小二乘构造将MS势扩展到所有有限博弈类。该构造引出MS分解：每个有限博弈可拆分为共同利益的MS势博弈和吸收每个参与者个体支付的残差。本文核心结果是一个恒等式：在所有参与者行动数相等的博弈中，MS势的增广项（除加性常数外）与Candogan、Menache、Ozdaglar和Parrilo（2011）提出的势一致。
英文摘要
Monderer and Shapley (1996) showed that a game is a potential game precisely when the players' second-order cross-differences agree pair by pair. This paper asks what can be built from them when the agreement fails. The resulting MS-potential, assembled from their common-interest part, is unique up to separable payoff terms and exists precisely when a higher-order MS-condition holds; on exact potential games it recovers the potential up to the players' main effects. A least-squares construction extends the MS-potential to all finite games and induces the \MS-decomposition: every game splits into a common-interest MS-potential game and a residual absorbing every player's individualistic effects. Everything read off the second differences is invariant under the transformations that leave strategic content untouched, relabelling, non-strategic translation and action duplication, where the CMOP decomposition is not; only the least-squares extension fails, since it averages and centres. The central result is an identity: when all players have equally many actions, an augmentation of the MS-potential coincides, up to the additive constant, with the potential of Candogan et al. (2011). If action counts are unequal they diverge, and no bound on that divergence is established here. Both rest on the same uniform weighting of the players' actions.
