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@ -31,10 +31,10 @@ ShapleyValue(Leibniz) = 1/2 * [ (Value of adding Lebniz to coalition {} / Number
If the value of creating calculus is 1, then: If the value of creating calculus is 1, then:
Value of adding Lebniz to coalition {} = 1 Value of adding Lebniz to coalition {} = 1
Number of coalitions excluding Leibniz of the same size as {} = 1 Number of coalitions excluding Leibniz of the same size as {} = 1
Value of adding Lebniz to coalition {Newton} = 0 Value of adding Lebniz to coalition {Newton} = 0
Number of coalitions excluding Leibniz of the same size as {Newton} = 1 Number of coalitions excluding Leibniz of the same size as {Newton} = 1
Value(Leibniz) = 1/2 * (1/1 + 0/1) = 1/2 Value(Leibniz) = 1/2 * (1/1 + 0/1) = 1/2
@ -63,15 +63,15 @@ If the project gets completed, what is the value of each player?
Value(AS) = 1/3 * [(Value of adding him to {} / 1) + (Value of adding him to {DB}/2) + (Value of adding him to {MS}/2) + (Value of adding him to {MS, DB}/1) ] Value(AS) = 1/3 * [(Value of adding him to {} / 1) + (Value of adding him to {DB}/2) + (Value of adding him to {MS}/2) + (Value of adding him to {MS, DB}/1) ]
Value of adding him to {} = 0 Value of adding him to {} = 0
Number of coalitions excluding him of size 0 = 1 Number of coalitions excluding him of size 0 = 1
Value of adding him to {DB} = 0 Value of adding him to {DB} = 0
Value of adding him to {MS} = 0.4 ## The mad scientist tries to talk to him, but he's not very convincing Value of adding him to {MS} = 0.4 // The mad scientist tries to talk to him, but he's not very convincing
Number of coalitions excluding him of size 1 = 2 Number of coalitions excluding him of size 1 = 2
Value of adding him to {MS, DB} = 1 Value of adding him to {MS, DB} = 1
Number of coalitions excluding him of size 2 = 1 Number of coalitions excluding him of size 2 = 1
Value(AS) = 1/3 * (0/1 + 0/2 + 0.4/2 + 1/1) = 0.4 Value(AS) = 1/3 * (0/1 + 0/2 + 0.4/2 + 1/1) = 0.4
### David Burns ### David Burns
@ -81,13 +81,13 @@ If David Burns writes the book, but there is nobody, his impact is 0. If he exis
Value(DB) = 1/3 * [(Value of adding him to {} / 1) + (Value of adding him to {AS}/2) + (Value of adding him to {MS}/2) + (Value of adding him to {MS, AS}/1) ] Value(DB) = 1/3 * [(Value of adding him to {} / 1) + (Value of adding him to {AS}/2) + (Value of adding him to {MS}/2) + (Value of adding him to {MS, AS}/1) ]
Value of adding him to {} = 0 Value of adding him to {} = 0
Number of coalitions excluding him of size 0 = 1 Number of coalitions excluding him of size 0 = 1
Value of adding him to {AS} = 0 Value of adding him to {AS} = 0
Value of adding him to {MS} = 0 Value of adding him to {MS} = 0
Number of coalitions excluding him of size 1 = 2 Number of coalitions excluding him of size 1 = 2
Value of adding him to {MS, AS} = 1-0.4 = 0.6 ## These are MARGINAL values. Value of adding him to {MS, AS} = 1-0.4 = 0.6 // These are MARGINAL values.
Number of coalitions excluding him of size 2 = 1 Number of coalitions excluding him of size 2 = 1
Value(AS) = 1/3 * (0/1 + 0/2 + 0/2 + 0.6/1) = 0.2 Value(AS) = 1/3 * (0/1 + 0/2 + 0/2 + 0.6/1) = 0.2
@ -95,23 +95,23 @@ Value(AS) = 1/3 * (0/1 + 0/2 + 0/2 + 0.6/1) = 0.2
Value(MS) = 1/3 * [(Value of adding him to {} / 1) + (Value of adding him to {DB}/2) + (Value of adding him to {AS}/2) + (Value of adding him to {AS, DB}/1) ] Value(MS) = 1/3 * [(Value of adding him to {} / 1) + (Value of adding him to {DB}/2) + (Value of adding him to {AS}/2) + (Value of adding him to {AS, DB}/1) ]
Value of adding him to {} = 0 Value of adding him to {} = 0
Number of coalitions excluding him of size 0 = 1 Number of coalitions excluding him of size 0 = 1
Value of adding him to {DB} = 0 Value of adding him to {DB} = 0
Value of adding him to {AS} = 0.4 Value of adding him to {AS} = 0.4
Number of coalitions excluding him of size 1 = 2 Number of coalitions excluding him of size 1 = 2
Value of adding him to {AS, DB} = 1 Value of adding him to {AS, DB} = 1
Number of coalitions excluding him of size 2 = 1 Number of coalitions excluding him of size 2 = 1
Value(MS) = 1/3 * (0/1 + 0/2 + 0.4/2 + 1/1) = 0.4 Value(MS) = 1/3 * (0/1 + 0/2 + 0.4/2 + 1/1) = 0.4
Total value = 0.4 + 0.2 + 0.4 = 1 (checks out) Total value = 0.4 + 0.2 + 0.4 = 1 (checks out)
### Counterfactual values: ### Counterfactual values:
Counterfactual value of AS = 1 Counterfactual value of AS = 1
Counterfactual value of MS = 1 Counterfactual value of MS = 1
Counterfactual value of DB = 0.6 Counterfactual value of DB = 0.6
Sum of the counterfactual values = 2.6 Sum of the counterfactual values = 2.6
### Why is this important ### Why is this important
@ -243,9 +243,11 @@ end <- Sys.time()
end-start end-start
sum(2/(c(1:1000))^2)``` sum(2/(c(1:1000))^2)```
```
## Sources ## Sources
https://en.wikipedia.org/wiki/Shapley_value https://en.wikipedia.org/wiki/Shapley_value
https://christophm.github.io/interpretable-ml-book/shapley.html https://christophm.github.io/interpretable-ml-book/shapley.html
https://towardsdatascience.com/understanding-how-ime-shapley-values-explains-predictions-d75c0fceca5a https://towardsdatascience.com/understanding-how-ime-shapley-values-explains-predictions-d75c0fceca5a
https://towardsdatascience.com/one-feature-attribution-method-to-supposedly-rule-them-all-shapley-values-f3e04534983d https://towardsdatascience.com/one-feature-attribution-method-to-supposedly-rule-them-all-shapley-values-f3e04534983d
https://www.eeg.tuwien.ac.at/conference/iaee2017/files/presentation/Pr_662_Jamasb_Tooraj.pdf (This last one is an interesting use fo Shapley values for international coordination)