I think it’s not necessarily the case that free-market pairwise bargaining always leads to the Shapley value. 10Y = X -Y has an infinite number of solutions, and the only principled ways I know of for choosing solutions is either Shapley value or the fact that in this scenario, since there are no other jobs, the owner should be able to negotiate X and Y down to epsilon.
It looks like Shapley values satisfy an equilibrium property that should take into account more than just pairwise bargaining. Specifically, there is no subset of participants that can gain more than the Shapley values by excluding the others (assuming that v satisfies [superadditivity](https://en.wikipedia.org/wiki/Shapley_value#Stand-alone_test), i.e. a group is always at least as valuable as it’s subsets individually added together). We can prove this:
First, by induction see that ∑R⊂Sw(R)=v(S) for any S. And by superadditivity, w(S)≥0 for all S. Then we can do:
That means that the total value produced by the subset R is going to be less than (or equal to) the total of the Shapley values they obtain from participating in the whole group. Therefore, they can’t possibly all profit by excluding anyone since there’s not enough profit to go around. Presumably this is well known and has a name. It’s basically a direct extension of the ‘stand-alone test’ that Wikipedia lists, so maybe it’s the ‘stand-together test’?
So that makes me think Shapley values are what you might get after multi-party bargaining arrives at equilibrium. This a pretty amazing topic, and great selections of examples to explore!
Superadditivity seems rare in practice. For instance, workers should have subadditive contributions after some point. This is certainly true in the unemployment example in the post.
Perhaps there is a different scheme for dividing gains from coöperation which satisfies some of the things we want but not superadditivity, but I’m unfamiliar with one. Please let me know if you find anything in that vein, I’d love to read about some alternatives to Shapley Value.
I think it’s not necessarily the case that free-market pairwise bargaining always leads to the Shapley value. 10Y = X -Y has an infinite number of solutions, and the only principled ways I know of for choosing solutions is either Shapley value or the fact that in this scenario, since there are no other jobs, the owner should be able to negotiate X and Y down to epsilon.
It looks like Shapley values satisfy an equilibrium property that should take into account more than just pairwise bargaining. Specifically, there is no subset of participants that can gain more than the Shapley values by excluding the others (assuming that v satisfies [superadditivity](https://en.wikipedia.org/wiki/Shapley_value#Stand-alone_test), i.e. a group is always at least as valuable as it’s subsets individually added together). We can prove this:
∑i∈RvS(i)=∑i∈R∑R′⊂S1R′(s)w(R′)/|R′|≥∑i∈R∑R′⊂R1R′(i)w(R′)/|R′|First, by induction see that ∑R⊂Sw(R)=v(S) for any S. And by superadditivity, w(S)≥0 for all S. Then we can do:
So then ∑i∈RvS(i)≥∑R′⊂Rw(R′)=∑R′⊂Rw(R′)=v(R).
That means that the total value produced by the subset R is going to be less than (or equal to) the total of the Shapley values they obtain from participating in the whole group. Therefore, they can’t possibly all profit by excluding anyone since there’s not enough profit to go around. Presumably this is well known and has a name. It’s basically a direct extension of the ‘stand-alone test’ that Wikipedia lists, so maybe it’s the ‘stand-together test’?
So that makes me think Shapley values are what you might get after multi-party bargaining arrives at equilibrium. This a pretty amazing topic, and great selections of examples to explore!
Superadditivity seems rare in practice. For instance, workers should have subadditive contributions after some point. This is certainly true in the unemployment example in the post.
Perhaps there is a different scheme for dividing gains from coöperation which satisfies some of the things we want but not superadditivity, but I’m unfamiliar with one. Please let me know if you find anything in that vein, I’d love to read about some alternatives to Shapley Value.