I have a game that has a special form of payoff function of a coalitional game.
There is a set $N$ (of $n$ players) and a coalition function $v$ that maps subsets of players to the real numbers: $ v\colon \mathbf{P}(N) \to \mathbb {R} $, with $ v(\emptyset )=0$.
And in my setting, $v(S) = g(S) - c, S\subset N$. where $c$ is a constant and $g(\emptyset) = c$. And even nicely $g(S\cup \{i\}) = q_ig(S), \forall i \notin S$. And $q_i$ is a variable only depends on player $i$.
I'm wondering if there is a name for this type of game?
And one area I'm mostly interested in, in the game, is about splitting players.
Suppose i split/replace a player $i$ with the variable $q_i$ into two players $m,n$, with their variable $q_m, q_n$ where $q_m, q_n \neq 1$ and $q_m * q_n = q_i$. So now we have a game $N+1$ player. Using something like shapely value to divide the payoff is not going to be consistent, namely the value of $i$ in the $N$ game is different from the value for sum of $m$ and $n$ in the $N+1$ game.
Wondering what kind of payoff I can use for each player so it's consistent with or without such splitting ? (players get the same with or without splitting).