You can make it work without an explicit veto. Bob convinces Alice that Carol will be a valuable contributor to the team. In fact, Carol does nothing, but Bob follows a strategy of “Do nothing unless Carol is present”. This achieves the same synergies:
A+B: $0 (Venture needs action from both A and B, B chooses to take no action)
A+C: $0 (Venture needs action from both A and B)
B+C: $0 (Venture needs action from both A and B)
A+B+C: $300
In this way Bob has managed to redirect some of Alice’s payouts by introducing a player who does nothing except remove a bottleneck he added into his own playstyle in order to exploit Alice.
Shapley values are constructed such that introducing a null player doesn’t change the result. You are doing something different by considering the wrong counterfactual (one where C exists but isn’t part of the coalition, vs one when it doesn’t exist)
Sounds like you agree with both me and Ninety-Three about the descriptive claim that the Shapley Value has, in fact, been changed, and have not yet expressed any position regarding the normative claim that this is a problem?
Your example is wrong becuase you are not leaving the A+B case unchanged.
I agree that “being more people” is a problem in coalitional dynamics with vetos, but I don’t think this is a problem with the Shapley value solution. I agree that when trying to apply the Shapley value solution, you should make sure to set C’s value as zero (even though it might hurt egos), etc.
Your example is wrong becuase you are not leaving the A+B case unchanged.
On what basis do you claim that the A+B case should be unchanged? The entire point of the example is that Carol now actually has the power to stop A+B and thus they actually can’t do anything without her on board.
If you are intending to make some argument along the lines of “a veto is only a formal power, so we should just ignore it” then the example can trivially be modified so that B’s resources are locked in a physical vault with a physical lock that literally can’t be opened without C. The fact that B can intentionally surrender some of his capabilities to C is a fact of physical reality and exists whether you like it or not.
You can make it work without an explicit veto. Bob convinces Alice that Carol will be a valuable contributor to the team. In fact, Carol does nothing, but Bob follows a strategy of “Do nothing unless Carol is present”. This achieves the same synergies:
A+B: $0 (Venture needs action from both A and B, B chooses to take no action)
A+C: $0 (Venture needs action from both A and B)
B+C: $0 (Venture needs action from both A and B)
A+B+C: $300
In this way Bob has managed to redirect some of Alice’s payouts by introducing a player who does nothing except remove a bottleneck he added into his own playstyle in order to exploit Alice.
Shapley values are constructed such that introducing a null player doesn’t change the result. You are doing something different by considering the wrong counterfactual (one where C exists but isn’t part of the coalition, vs one when it doesn’t exist)
Sounds like you agree with both me and Ninety-Three about the descriptive claim that the Shapley Value has, in fact, been changed, and have not yet expressed any position regarding the normative claim that this is a problem?
No, the shapley value hasn’t been changed. The correct way to do this would have been, that, if:
A: $0
B: $0
A+B: $300
then
A: $0
B: $0
A+B: $0 (Carol vetoes)A+B: $300A+C: $0
B+C: $0
A+B+C: $300
Your example is wrong becuase you are not leaving the A+B case unchanged.
I agree that “being more people” is a problem in coalitional dynamics with vetos, but I don’t think this is a problem with the Shapley value solution. I agree that when trying to apply the Shapley value solution, you should make sure to set C’s value as zero (even though it might hurt egos), etc.
On what basis do you claim that the A+B case should be unchanged? The entire point of the example is that Carol now actually has the power to stop A+B and thus they actually can’t do anything without her on board.
If you are intending to make some argument along the lines of “a veto is only a formal power, so we should just ignore it” then the example can trivially be modified so that B’s resources are locked in a physical vault with a physical lock that literally can’t be opened without C. The fact that B can intentionally surrender some of his capabilities to C is a fact of physical reality and exists whether you like it or not.