I don’t exactly claim to have identified a problem with the counterfactual function, in itself. The counterfactual is perfectly well defined, and I like it, and it has done nothing wrong. I understand this. It is clear to me that it can’t be added just like that. The function, per se, is fine.
What I’m claiming is that, because it can’t be aggregated, it is not the right function to think about in terms of assigning impact to people in the context of groups. I am arguing about the area of applicability of the function, not about the function. I am claiming that, if you are optimizing for counterfactual impact in terms of groups, pitfalls may arise.
It’s like, when you first see for the same time: −1 = sqrt(-1)*sqrt(-1) = sqrt((-1)*(-1)) = sqrt(1) = 1, therefore −1 = 1, and you can’t see the mistake. It’s not that the sqrt function is wrong, it’s that you’re using it outside it’s limited fiefdom, so something breaks. I hope the example proved amusing.
I’m not only making statements about the counterfactual function, I’m also making statements about the concept which people have in your head which is called “impact”, and how that concept doesn’t map to counterfactual impact some of the time, and about how, if you had to map that concept to a mathematical function, the Shapley value is a better candidate.
I don’t exactly claim to have identified a problem with the counterfactual function, in itself. The counterfactual is perfectly well defined, and I like it, and it has done nothing wrong. I understand this. It is clear to me that it can’t be added just like that. The function, per se, is fine.
What I’m claiming is that, because it can’t be aggregated, it is not the right function to think about in terms of assigning impact to people in the context of groups. I am arguing about the area of applicability of the function, not about the function. I am claiming that, if you are optimizing for counterfactual impact in terms of groups, pitfalls may arise.
It’s like, when you first see for the same time: −1 = sqrt(-1)*sqrt(-1) = sqrt((-1)*(-1)) = sqrt(1) = 1, therefore −1 = 1, and you can’t see the mistake. It’s not that the sqrt function is wrong, it’s that you’re using it outside it’s limited fiefdom, so something breaks. I hope the example proved amusing.
I’m not only making statements about the counterfactual function, I’m also making statements about the concept which people have in your head which is called “impact”, and how that concept doesn’t map to counterfactual impact some of the time, and about how, if you had to map that concept to a mathematical function, the Shapley value is a better candidate.