Here's a very late follow-up: the rationale behind linearity for Shapley values seems closely related to the rationale behind the independence axiom of VNM rationality, and under some decision theories we apparently can dispense with the latter.

This gives me the vocabulary for expressing why I find linearity constraining: if I'm about to play game $A$ or game $B$ with probabilities $p$ and $1-p$ respectively, and my payout of $A$ is lower, maybe I would prefer to get a lower payout in $B$ in exchange for a higher payout in $A$. I'm not sure how much of that is just downstream from "what if my utility isn't linear in the payout" or something like that, though.

I wasn't asking "what payment rules still satisfy the three remaining properties", I was asking "what other payment rules are there which satisfy the three remaining properties but not additivity" (with bonus questions "what other properties of Shapley values do we still get just from those three properties" and "what properties other than additivity can we add to those three properties which again pin down a unique rule").

My aim here, which I admit is nebulous, is to get a rough overview of the space of different payment rules (for example, this answer on math.stackexchange namedrops the 'pre-kernel' and 'pre-nucleolus' - I assume there's more where that came from!).

Ideally, and I know this is a cartoonish and unrealistic goal, I'd have:

• A list of "Properties Which Are Nice To Have In A Payment Rule",
• A list of "Sets of Properties Which Imply This Other Property",
• And a list of "Sets of Properties Which Specify A Unique Payment Rule".

I just found a presentation of linearity which motivates it as preserving expected payout before and after an uncertain event, which both adds usefulness-points to the property (for me) and vaguely suggests where you might not want that property, but no concrete example comes to mind.