Transposed Second-Order Symmetry and Second-Order Marginality: An Exact Axiomatic Classification of Shapley-Type Solutions

Source abstract

We study the relation between mixed second differences of a transferable-utility cooperative game and payoff changes caused by player deletion. Standard second-order symmetry compares two entries in a fixed recipient row of the deletion-effect matrix. We introduce its transposed counterpart, transposed second-order symmetry, which compares two recipients in a fixed deleted-player column. The Shapley value satisfies this property, and the associated payoff comparison admits a positive-kernel representation in terms of mixed second differences. We prove that efficiency, second-order marginality, and transposed second-order symmetry do not by themselves characterize the Shapley value. Instead, they characterize exactly a centered-bias family obtained by adding a player-specific constant and subtracting its average on the active player set; the parameter is unique up to a common additive constant. Adding the zero-game property removes this affine freedom and yields the Shapley value, while normalization only on two-player zero games is already sufficient. We also establish the logical independence of the four axioms in the normalized characterization and clarify the relation between transposed second-order symmetry, standard second-order symmetry, and balanced contributions.