Characterizations of proportional division value in TU-games via fixed-population consistency

We study the proportional division value in TU-games, which distributes the grand coalition worth in proportion to the players’ stand-alone worths. Focusing on fixed-population consistency, we characterize the proportional division value through three types of axioms: axioms concerning transformations of the grand coalition worth, composition axioms, and nullified-game consistency. The first type captures how payoffs respond when the grand coalition worth changes. The composition axioms require that the outcome be the same whether a game is solved directly or through a two-step procedure based on a reference allocation and an associated adjustment of the game. Nullified-game consistency requires that, when some players’ payoffs are fixed, the solution for the remaining players in the adjusted game coincides with their original payoffs. Together with efficiency and/or fairness-related conditions where appropriate, these axioms characterize the proportional division value.