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!
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!