Symmetry of players in a cooperative game ($v(S\cup \{i\})=v(S\cup \{j\})(S\subseteq N\setminus \{i,j\})$)

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.

A value has cooperative-game symmetry when it assigns equal payoffs to players satisfying symmetry of players in a cooperative game. Thus equality of their marginal contributions is reflected in the allocation, rather than merely being a property of the underlying game.