I like this angle! It seems useful to compare the Shapley value in this domain to the Banzhaf value. (Brief, dense description: If Shapley value attributes value to pivotal actors during the sequential process of coalition formation (averaged across all permutations of coalition formation orderings), Banzhaf value attributes value to critical actors without which any given coalition would fail. See Shapley-Shubik power index and Banzhaf power index for similar concepts in a slightly different context.)
Focusing on just the properties where they differ:
Efficiency: I’ve sometimes seen this called “full allocation” which is suggestive. It’s basically just whether the full value of the coalition is apportioned to actors of the coalition or if some of it is leftover.
2-Efficiency: “The 2-Efficiency property states that the allocation rule that satisfies it is
immune against artificial merging or splitting of players.”
Total power: “The Total power property establishes that the total payoff obtained for the
players is the sum of all marginal contributions of every player normalized by
2n−1.”
I’d have to think about this more carefully, but it’s not immediately obvious to me which set of properties is better for the purpose at hand.
Is it possible to use Banzhaf values for generic attribution questions outside of voting? If so, can you link to some posts/papers that describe how to use it in such cases. The first set of things that came up are all voting-related.
Unless I’m very confused, yes. Unfortunately, it does seem that almost all of the discussion of it is pretty theoretical and about various axiomatic characterizations. Here’s an interesting application paper I found though: The Shapley and Banzhaf values in microarray games. They have a short description of their use of the Banzhaf value (equation 2)---not sure how helpful it is.
I like this angle! It seems useful to compare the Shapley value in this domain to the Banzhaf value. (Brief, dense description: If Shapley value attributes value to pivotal actors during the sequential process of coalition formation (averaged across all permutations of coalition formation orderings), Banzhaf value attributes value to critical actors without which any given coalition would fail. See Shapley-Shubik power index and Banzhaf power index for similar concepts in a slightly different context.)
This paper has a nice table of properties:
(“Additivity’ is the same as “linearity” here.)
Focusing on just the properties where they differ:
Efficiency: I’ve sometimes seen this called “full allocation” which is suggestive. It’s basically just whether the full value of the coalition is apportioned to actors of the coalition or if some of it is leftover.
2-Efficiency: “The 2-Efficiency property states that the allocation rule that satisfies it is immune against artificial merging or splitting of players.”
Total power: “The Total power property establishes that the total payoff obtained for the players is the sum of all marginal contributions of every player normalized by 2n−1.”
I’d have to think about this more carefully, but it’s not immediately obvious to me which set of properties is better for the purpose at hand.
Is it possible to use Banzhaf values for generic attribution questions outside of voting? If so, can you link to some posts/papers that describe how to use it in such cases. The first set of things that came up are all voting-related.
Unless I’m very confused, yes. Unfortunately, it does seem that almost all of the discussion of it is pretty theoretical and about various axiomatic characterizations. Here’s an interesting application paper I found though: The Shapley and Banzhaf values in microarray games. They have a short description of their use of the Banzhaf value (equation 2)---not sure how helpful it is.