If the gain produced by cooperation is negative, the super additivity condition fails to apply and thus so does the Shapley distribution. The “desirable property” number 1 of the wiki, labeled individual fairness, also does not apply. I suppose you could extend the mathematical formula to apply to negative gains, but the question would be whether that distribution satisfied some intuitively appealing set of axioms.
If the cooperative game that we compute the Shapley value from is derived from an adversarial game, superadditivity cannot fail. To get the sum of what they would’ve got separately, the players just have to play what they would’ve played separately.
If the gain produced by cooperation is negative, the super additivity condition fails to apply and thus so does the Shapley distribution. The “desirable property” number 1 of the wiki, labeled individual fairness, also does not apply. I suppose you could extend the mathematical formula to apply to negative gains, but the question would be whether that distribution satisfied some intuitively appealing set of axioms.
If the cooperative game that we compute the Shapley value from is derived from an adversarial game, superadditivity cannot fail. To get the sum of what they would’ve got separately, the players just have to play what they would’ve played separately.