Thanks for this post. I’m also pretty enthusiastic about Shapley values, and it is overdue for a clear presentation like this.
The main worry I have is related to the first one GeorgeBridgwater notes: the values seem very sensitive to who one includes as a co-operative counterparty (and how finely we individuate them). As your example with vaccine reminders shows, different (but fairly plausible) accounts of this can change the ‘raw’ CE estimate by a factor of five.
We may preserve ordering among contributors if we twiddle this dial, but the more typical ‘EA problem’ is considering different interventions (and thus disjoint sets of counter-parties). Although typical ‘EA style’ CE estimates likely have expected errors in their exponent rather than their leading digit, a factor of 5 (or maybe more) which can hinge on relatively arbitrary decisions on how finely to individuate who we are working with looks pretty challenging to me.
The Banzhaf value should avoid this problem since it has the property of 2-Efficiency: “The 2-Efficiency property states that the allocation rule that satisfies it is immune against artificial merging or splitting of players.”
I’d like to hear more about this if you have the time. It seems to me that it’s hard to find a non-arbitrary way splitting of players.
Say a professor and a student work together on a paper. Each of them spends 30 hours on it and the paper would counterfactually not have been written if either of them had not contributed this time. The Shapley values should not be equivalent, because the ‘relative size’ of the players’ contributions shouldn’t be measured by time input.
Similarly, in the India vaccination example, players’ contribution size is determined by their money spent. But this is sensitive to efficiency: one should not be able to get a higher Shapley value just from spending money inefficiently, right? Or should it, because this worry is addressed by Shapley cost-effectiveness?
(This issue seems structurally similar to how we should allocate credence between competing hypotheses in the absence of evidence. Just because the two logical possibilities are A and ~A, does not mean a 50⁄50 credence is non-abitrary. Cf. Principle of Indifference)
The {ij}-merged game [...] considers that a pair of players merge or unite in a new player p which lies outside the original set of players. [...] In the {i ▹ j}-amalgamation game, player j delegates his/her role to player i, who belongs to the original set of players.
These lead to the corresponding properties:
A value, f , satisfies 2-merging efficiency if, for every (N,v)∈g and i,j∈N, fp(Np,vp)=fi(N,v)+fj(N,v).
A value, f, satisfies 2-amalgamation efficiency if, for every (N,v)∈g and i,j∈N, fi(Ni⊳j,vi⊳j)=fi(N,v)+fj(N,v).
So basically they’re just saying that players can’t artificially boost or reduce their assigned values by merging or amalgamating—the resulting reward is always just the sum of the individual rewards.
I don’t think it directly applies in the case of your professor and student case. The closest analogue would be if the professor and student were walking as part of a larger group. Then 2-efficiency would say that the student and professor collectively get X credit whether they submit their work under two names or one.
Sorry for the delayed reply. Does that help at all?
Thanks! Late replies are better than no replies ;)
I don’t think this type of efficiency deals with the practical problem of impact credit allocation though! Because there the problem appears to be that it’s difficult to find a common denominator for people’s contributions. You can’t just use man hours, and I don’t think the market value of man hours would do that much better (although it gets in the right direction).
Thanks for this post. I’m also pretty enthusiastic about Shapley values, and it is overdue for a clear presentation like this.
The main worry I have is related to the first one GeorgeBridgwater notes: the values seem very sensitive to who one includes as a co-operative counterparty (and how finely we individuate them). As your example with vaccine reminders shows, different (but fairly plausible) accounts of this can change the ‘raw’ CE estimate by a factor of five.
We may preserve ordering among contributors if we twiddle this dial, but the more typical ‘EA problem’ is considering different interventions (and thus disjoint sets of counter-parties). Although typical ‘EA style’ CE estimates likely have expected errors in their exponent rather than their leading digit, a factor of 5 (or maybe more) which can hinge on relatively arbitrary decisions on how finely to individuate who we are working with looks pretty challenging to me.
The Banzhaf value should avoid this problem since it has the property of 2-Efficiency: “The 2-Efficiency property states that the allocation rule that satisfies it is immune against artificial merging or splitting of players.”
I’d like to hear more about this if you have the time. It seems to me that it’s hard to find a non-arbitrary way splitting of players.
Say a professor and a student work together on a paper. Each of them spends 30 hours on it and the paper would counterfactually not have been written if either of them had not contributed this time. The Shapley values should not be equivalent, because the ‘relative size’ of the players’ contributions shouldn’t be measured by time input.
Similarly, in the India vaccination example, players’ contribution size is determined by their money spent. But this is sensitive to efficiency: one should not be able to get a higher Shapley value just from spending money inefficiently, right? Or should it, because this worry is addressed by Shapley cost-effectiveness?
(This issue seems structurally similar to how we should allocate credence between competing hypotheses in the absence of evidence. Just because the two logical possibilities are A and ~A, does not mean a 50⁄50 credence is non-abitrary. Cf. Principle of Indifference)
This is the best explanation I could find: Notes on a comment on 2-efficiency and the Banzhaf value.
It describes two different kinds of 2-efficiency:
These lead to the corresponding properties:
So basically they’re just saying that players can’t artificially boost or reduce their assigned values by merging or amalgamating—the resulting reward is always just the sum of the individual rewards.
I don’t think it directly applies in the case of your professor and student case. The closest analogue would be if the professor and student were walking as part of a larger group. Then 2-efficiency would say that the student and professor collectively get X credit whether they submit their work under two names or one.
Sorry for the delayed reply. Does that help at all?
Thanks! Late replies are better than no replies ;)
I don’t think this type of efficiency deals with the practical problem of impact credit allocation though! Because there the problem appears to be that it’s difficult to find a common denominator for people’s contributions. You can’t just use man hours, and I don’t think the market value of man hours would do that much better (although it gets in the right direction).