  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 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.
