Abstract We provide a concise characterization of the class of positively weighted Shapley values by three properties, two standard properties, efficiency and marginality, and a relaxation of the balanced contributions property called the weak balanced contributions property. Balanced contributions: the amount one player gains or loses when another player leaves the game equals the amount the latter player gains or loses when the former player leaves the game. Weakly balanced contributions: the direction (sign) of the change of one player’s payoff when another player leaves the game equals the direction (sign) of the change of the latter player’s payoff when the former player leaves the game. Given this characterization, the symmetric Shapley value can be “extracted”from the class of positively weighted Shapley values by either replacing the weak balanced contributions property with the standard symmetry property or by strengthening the former into the balanced contributions property.

中文翻译：

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弱平衡贡献和加权沙普利值

摘要 我们通过三个属性、两个标准属性、效率和边际性以及平衡贡献属性的松弛（称为弱平衡贡献属性）提供了一类正加权沙普利值的简明特征。平衡贡献：当另一个玩家离开游戏时，一个玩家获得或失去的数量等于前一个玩家离开游戏时后一个玩家获得或损失的数量。弱平衡贡献：另一个玩家离开游戏时一个玩家收益变化的方向（符号）等于前一个玩家离开游戏时后一个玩家收益变化的方向（符号）。鉴于这种特征，