Consistency for the additive efficient normalization of semivalues

► The B-reduced game is defined and extension of Sobolev’s reduced game. ► B-reduced and path-independent linear reduced game only coincide on Sobolev’s case. ► The additive efficient normalization of semivalues is characterized by B-consistency. ► The ESE-value is characterized by path-independently linear consistency. ► The relationship between ESE-values and the least square values is derived. This paper contributes to consistency for the additive efficient normalization of semivalues. Motivated from the additive efficient normalization of a semivalue being a B-revision of the Shapley value, we introduce the B-reduced game which is an extension of Sobolev’s reduced game. Then the additive efficient normalization of a semivalue is axiomatized as the unique value satisfying covariance, symmetry, and B-consistency. Furthermore, by means of the path-independently linear consistency together with the standardness for two-person games, the additive efficient normalization of semivalues is also characterized. Accessorily, the additive efficient normalization of semivalues is directly verified as the linear consistent least square values (see Ruiz et al., 1998).