The Shapley value equals a player's contribution to the potential of a game. The potential is a most natural one-number summary of a game, which can be computed as the expected accumulated worth of a random partition of the players. This computation integrates the coalition formation of all players and readily extends to games with externalities. We investigate those potential functions for games with externalities that can be computed this way. It turns out that the potential that corresponds to the MPW solution introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339-356), is unique in the following sense. It is obtained as a the expected accumulated worth of a random partition, it generalizes the potential for games without externalities, and it induces a solution that satisfies the null player property even in the presence of externalities.

翻译：沙普利值等于玩家对博弈势的贡献。势是对博弈最自然的一维概括，可以通过随机划分参与者的期望累积得益来计算。这种计算整合了所有参与者的联盟形成过程，并可直接推广至具有外部性的博弈。我们研究那些可通过此方式计算的具有外部性博弈的势函数。结果表明，由Macho-Stadler等人（2007，J. Econ. Theory 135, 339-356）提出的MPW解所对应的势函数在以下意义上是唯一的：它作为随机划分的期望累积得益获得，推广了无外部性博弈的势，并且诱导出即使在存在外部性时也满足空玩家性质的解。