Some new results on generalized additive games

A Generalized Additive Game (GAG) (Cesari et al. in Int J Game Theory 46(4):919–939, 2017) is a Transferable Utility (TU) game (N, v), where each player in N is provided with an individual value, and the worth v(S) of a coalition $$S \subseteq N$$ S ⊆ N is obtained as the sum of the individual values of players in another subset $$\mathcal {M}(S)\subseteq N$$ M ( S ) ⊆ N . Based on conditions on the map $$\mathcal {M}$$ M (which associates to each coalition S a set of beneficial players $$\mathcal {M}(S)$$ M ( S ) not necessarily included in S), in this paper we characterize classes of GAGs that satisfy properties like monotonicity, superadditivity, (total) balancedness, PMAS-admissibility and supermodularity, for all nonnegative vectors of individual values. We also illustrate the application of such conditions on $$\mathcal {M}$$ M over particular GAGs studied in the literature (e.g., glove games (Shapley and Shubik in Int Econ Rev 10:337–362, 1969), generalized airport games (Norde et al. in Eur J Oper Res 136(3):635–654, 2002), fixed tree games (Bjørndal et al. in Math Methods Oper Res 59(2):249–270, 2004), link-connection games (Moretti in Multi-agent systems and agreement technologies, vol 10767. Springer, Cham, 2008; Nagamochi et al. in Math Oper Res 22(1):146–164, 1997), simple minimum cost spanning tree games (Norde et al. in Eur J Oper Res 154(1):84–97, 2004; Tijs et al. in Eur J Oper Res 175(1):121–134, 2006) and graph coloring games (Deng et al. in Math Program 87(3):441–452, 2000; Hamers et al. in Math Program 145(1–2):509–529, 2014)).