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A simple cooperative game is a monotone transferable utility game in which each coalition either loses, with value zero, or wins, with value one. In the usual nontrivial convention the empty coalition loses and the grand coalition wins. The Shapley value is then the probability that a player changes a losing coalition into a winning one when players enter in uniformly random order.

The characteristic function of a coalitional game is superadditive when disjoint coalitions can attain at least the sum of their separate values by joining. This follows when their separate strategies can be played together without interference. Superadditivity is an additional property of an abstract transferable utility game, not a consequence of merely assigning values to subsets: $v(\{1\})=v(\{2\})=v(\{1,2\})=1$ is a normalized counterexample.

Players $i,j$ are symmetric when interchanging them does not affect any coalition value. The displayed condition is equivalent to invariance under their transposition. If every pair is symmetric, arbitrary permutations preserve the game, and coalitions of equal cardinality have equal values. This is a property of the transferable utility game, rather than a requirement that a chosen payoff allocation treat players equally.