Well, the Shapely value still meets your criteria, since they're only removal of constraints, not addition of new ones. If you don't care about linearity/additivity, what DO you care about that the Shapely calculations don't include?
Separately, can you explain why you don't care about linearity? Where do the unaccounted-for differences come from when two sub-games fail to add up to the total game?
I’m not the asker, but I think I get where they’re coming from. For a long time, linear and logistic regression were the king & queen of modeling. Then the introduction of non-linear functions like random forest and gradient boosters made us far more able to fit difficult data. So the original question has me wondering if there’s a similar possible gain in going from linearity to non-linearity in interpretability algorithms.
Yep, that's pretty much it, but with the added bonus of a concrete motivating example. Thanks!
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 or game with probabilities and respectively, and my payout of is lower, maybe I would prefer to get a lower payout in in exchange for a higher payout in . 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:
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.
I originally asked this on math.stackexchange; after reading Diffractor's Unifying Bargaining sequence (Part 1 here) I'm wondering if there are more insights floating about, so I'm repeating it here.
Shapley values seem to be the standard answer to "how should a coalition split the rewards of their cooperation", but I'm curious about alternatives.
The standard characterization of Shapley values says that Shapley values are the unique coalition payments which satisfy a bunch of properties. Three of them (efficiency, symmetry, and null player) seem pretty necessary for any "reasonable" or "practical" coalition payment rule, but the last one (linearity) does not.
If I didn't care for linearity (or its close synonyms, additivity and aggregation):
Alternatively, are any of the other properties also reasonable to drop (for instance, symmetry)? What do you end up with?