  We introduce a new class of totally balanced cooperative TU games, namely p
-additive games. It is inspired by the class of inventory games that arises
from inventory situations with temporary discounts (Toledo, 2002) and contains
the class of inventory cost games (Meca et al. 2003). It is shown that every
p-additive game and its corresponding subgames have a nonempty core. We also
focus on studying the character concave or convex and monotone of p-additive
games. In addition, the modified SOC-rule is proposed as a solution for
p-additive games. This solution is suitable for p-additive games since it is a
core-allocation which can be reached through a population monotonic allocation
scheme. Moreover, two characterizations of the modified SOC-rule are provided.
