  In this paper, we present a systematic overview of different endogenous
optimization-based characteristic functions and discuss their properties.
Furthermore, we define and analyze in detail a new, $\eta$-characteristic
function. This characteristic function has a substantial advantage over other
characteristic functions in that it can be obtained with a minimal
computational effort and has a reasonable economic interpretation. In
particular, the new characteristic function can be seen as a reduced version of
the classical Neumann-Morgenstern characteristic function, where the players
both from the coalition and from the complementary coalition use their
previously computed strategies instead of solving respective optimization
problems. Our finding are illustrated by a pollution control game with $n$
non-identical players. For the considered game, we compute all characteristic
functions and compare their properties. Quite surprisingly, it turns out that
both the characteristic functions and the resulting cooperative solutions
satisfy some symmetry relations.
