Full Additivity with Basic Division Imply the Shapley Value

The principle of Additivity states the sum of payoffs in two separate games should equal the payoff in the combination of those games. Typically, the literature considers limited version of this principle in which both games have exactly the same set of players. In contrary, in this paper we study the broader definition of Additivity, called Full Additivity, where the games may have arbitrary sets of players. Our analysis leads to very compact axiomatizations for well-known solution concepts. In particular, we show that the Shapley value is the only fully additive value that equally divides the payoff in games in which only the grand coalition has a non-zero value. Then, we consider scenarios in which additional information is available and an equal division is no longer justified. We show that the Weighted Shapley value and the Owen value are the only fully additive values that satisfies the simple division scheme that comes from the weight vector that represents shares in the joint game and an a priori formed coalition structure, respectively.