I think the unemployed worker example is the following: If I’m an unemployed worker, I tell the owner I’ll work for $X-1 dollars rather than the current worker who is working for $X dollars. The owner accepts, of course. So to keep that from happening, the worker offers me $Y to not work and let him keep the job. I go around to all the other workers and makes the same argument, getting Y from each. At the end of the day—viola I’m making 10Y dollars to not work and all the workers are making X—Y dollars and I’ll refuse any Y up until 10Y = X—Y and we make the same amount (otherwise I’ll undercut them).
That’s not enough to replicate the result since it doesn’t prescribe the value the owner gets.
Presumably in real life we can somehow include the negative term for the downsides of employment and the unemployed will accept a lower pay? Also, it’s interesting to think of unemployment benefits as “paying people to not compete with me in the job market” but I guess that kind of makes sense.
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!
I think the unemployed worker example is the following: If I’m an unemployed worker, I tell the owner I’ll work for $X-1 dollars rather than the current worker who is working for $X dollars. The owner accepts, of course. So to keep that from happening, the worker offers me $Y to not work and let him keep the job. I go around to all the other workers and makes the same argument, getting Y from each. At the end of the day—viola I’m making 10Y dollars to not work and all the workers are making X—Y dollars and I’ll refuse any Y up until 10Y = X—Y and we make the same amount (otherwise I’ll undercut them).
That’s not enough to replicate the result since it doesn’t prescribe the value the owner gets.
Presumably in real life we can somehow include the negative term for the downsides of employment and the unemployed will accept a lower pay? Also, it’s interesting to think of unemployment benefits as “paying people to not compete with me in the job market” but I guess that kind of makes sense.
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!