Curated. This was a quite nice introduction. I normally see Shapley values brought up in a context that’s already moderately complicated, and having a nice simple explainer is helpful!
I’d like it if the post went into a bit more detail about when/how Shapley values tend to get used in real world contexts.
This is (mostly) a crosspost of my (pending review? and so i can’t link to it?) comment from the EA forums replying to a commenter also asking for actual uses of Shapley values
The first real world example that comes to mind… isn’t about agents bargaining. Namely, statistical models. The idea is that you have some subparts that each contribute to the prediction, and want to know which are the most important, and so you can calculate shapley values (“how well does this model do if it only uses age and sex to predict life expectancy, but not race”, etc. for the other coalitions).
Here’s a microecon stack exchange question that asks a similar thing as you. The only non stats answer states that a bank used Shapley values to determine capital allocation in investments. It sounds like they didn’t have a problem using a ‘time machine’ because they had the performance of the investments and so could simply evaluate what returns they would’ve gotten had they invested differently. But I haven’t read it thoroughly, so for all I know they stopped using it soon after, or had some other way to evaluate counterfactuals, etc.
Also the Lightcone (so, including you?) fundraising post mentioned trying to charge for half the surplus produced when setting Lighthaven prices (i.e. the 2 player special case of the Shapley value).
Of course, the 2 player case is much easier than even the 3 player case because you only need to know the other person’s willingness to pay (that is, their value oven BATNA) and can then estimate your own costs (in total, one advantage-over-batna that doesn’t just involve you needs to be determined) while for 3 players you need 3*2 = 6 comparisons and for n players you need ∑nk=2k∗(nk) total comparisons (each player giving the benefit they get for if that coalition occurred) of which ∑nk=2(nk−1) are your comparisons (which, to be clear, aren’t trivial, but at least you know your own preferences and situation and don’t have to ask others about them). The first sum is ∑nk=2n!(k−1)!(n−k)!=∑nk=2n(n−1k−1)=n∑nk=2(n−1k−1)=n(2n−1−1) which is faster than exponential growth, while the second sum is ∑nk=2(nk−1)=2n−2=2(2n−1−1) which means that discounting the comparisons that are about the value you get doesn’t make the asymptotics better. This suggests that even just the communication costs get pretty high pretty fast unless you have a compact way to encode how much value you get out of the interactions (like in the bank example, I think you only need to be told the individual performance history, and then can just compute the value in each investment counterfactual). So if there’s nonlinear relationships between people (read: real life most of the time) my intuition is that you are screwed?
Curated. This was a quite nice introduction. I normally see Shapley values brought up in a context that’s already moderately complicated, and having a nice simple explainer is helpful!
I’d like it if the post went into a bit more detail about when/how Shapley values tend to get used in real world contexts.
This is (mostly) a crosspost of my (pending review? and so i can’t link to it?) comment from the EA forums replying to a commenter also asking for actual uses of Shapley values
The first real world example that comes to mind… isn’t about agents bargaining. Namely, statistical models. The idea is that you have some subparts that each contribute to the prediction, and want to know which are the most important, and so you can calculate shapley values (“how well does this model do if it only uses age and sex to predict life expectancy, but not race”, etc. for the other coalitions).
Here’s a microecon stack exchange question that asks a similar thing as you. The only non stats answer states that a bank used Shapley values to determine capital allocation in investments. It sounds like they didn’t have a problem using a ‘time machine’ because they had the performance of the investments and so could simply evaluate what returns they would’ve gotten had they invested differently. But I haven’t read it thoroughly, so for all I know they stopped using it soon after, or had some other way to evaluate counterfactuals, etc.
Also the Lightcone (so, including you?) fundraising post mentioned trying to charge for half the surplus produced when setting Lighthaven prices (i.e. the 2 player special case of the Shapley value).
Of course, the 2 player case is much easier than even the 3 player case because you only need to know the other person’s willingness to pay (that is, their value oven BATNA) and can then estimate your own costs (in total, one advantage-over-batna that doesn’t just involve you needs to be determined) while for 3 players you need 3*2 = 6 comparisons and for n players you need ∑nk=2k∗(nk) total comparisons (each player giving the benefit they get for if that coalition occurred) of which ∑nk=2(nk−1) are your comparisons (which, to be clear, aren’t trivial, but at least you know your own preferences and situation and don’t have to ask others about them). The first sum is ∑nk=2n!(k−1)!(n−k)!=∑nk=2n(n−1k−1)=n∑nk=2(n−1k−1)=n(2n−1−1) which is faster than exponential growth, while the second sum is ∑nk=2(nk−1)=2n−2=2(2n−1−1) which means that discounting the comparisons that are about the value you get doesn’t make the asymptotics better. This suggests that even just the communication costs get pretty high pretty fast unless you have a compact way to encode how much value you get out of the interactions (like in the bank example, I think you only need to be told the individual performance history, and then can just compute the value in each investment counterfactual). So if there’s nonlinear relationships between people (read: real life most of the time) my intuition is that you are screwed?