Yes, it’s a coordination problem. I understand the claim”Agents individually optimizing for cost-effectiveness in terms of Shapley value globally optimize for total cost-effectiveness.” in the way that they don’t coordinate but optimize individually.
What about this example? Intervention A Value of {}: 0 Value of {1}: 0 Value of {2}: 0 Value of {1,2}: 100 -> Shapley value of 1 is: 50, shapley value of 2 is: 50. Intervention B Value of {0}: 0 Value of {1}: 60 Value of {2}: 0 Value of {1,2}: 60 -> Shapley value of 1 is: 60, shapley value of 2 is: 0. Player 1 would go for intervention B, player 2 would go for intervention A. Result: value of A = 0; value of B = 60 → total utility 60. It would be better if both players decide for A.
My assumption was that player 1 and player 2 each have one year and can dedicate that year either to Intervention A or Intervention B. In the joint game, Player 1 would choose Intervention A if they were alone and Intervention B if Player 2 was also involved. If we always construct a joint game in this way—so that, depending on the coalition formed, the interventions are chosen and divided in a way that achieves the best overall outcome—then this joint game, by definition, leads to the best overall outcome.
Additionally, I am unclear on what “optimizing for cost-effectiveness in terms of Shapley value” is supposed to mean. In order to optimize something, there must be multiple options—so, multiple games, right? Even if we include the joint game, it would still be best for Player 2, in terms of the Shapley value, to play the game “Intervention A” with a Shapley value of 50.
But I think it’s not very useful to continue discussing this unless the claim “Agents individually optimizing for cost-effectiveness in terms of Shapley value globally optimize for total cost-effectiveness.” is precisely defined.
Yes, it’s a coordination problem. I understand the claim”Agents individually optimizing for cost-effectiveness in terms of Shapley value globally optimize for total cost-effectiveness.” in the way that they don’t coordinate but optimize individually.
What about this example?
Intervention A
Value of {}: 0
Value of {1}: 0
Value of {2}: 0
Value of {1,2}: 100
-> Shapley value of 1 is: 50, shapley value of 2 is: 50.
Intervention B
Value of {0}: 0
Value of {1}: 60
Value of {2}: 0
Value of {1,2}: 60
-> Shapley value of 1 is: 60, shapley value of 2 is: 0.
Player 1 would go for intervention B, player 2 would go for intervention A. Result: value of A = 0; value of B = 60 → total utility 60. It would be better if both players decide for A.
These are two different games. The joint game would be
Value of {}: 0 Value of {1}: 60 Value of {2}: 0 Value of {1,2}: 100`
and in that game player one is indeed better off in shapley value terms if he joins together with 2.
I’ll let you reflect on how/whether adding an additional option can’t decrease someone’s shapley value, but I’ll get back to my job :)
My assumption was that player 1 and player 2 each have one year and can dedicate that year either to Intervention A or Intervention B. In the joint game, Player 1 would choose Intervention A if they were alone and Intervention B if Player 2 was also involved. If we always construct a joint game in this way—so that, depending on the coalition formed, the interventions are chosen and divided in a way that achieves the best overall outcome—then this joint game, by definition, leads to the best overall outcome.
Additionally, I am unclear on what “optimizing for cost-effectiveness in terms of Shapley value” is supposed to mean. In order to optimize something, there must be multiple options—so, multiple games, right? Even if we include the joint game, it would still be best for Player 2, in terms of the Shapley value, to play the game “Intervention A” with a Shapley value of 50.
But I think it’s not very useful to continue discussing this unless the claim “Agents individually optimizing for cost-effectiveness in terms of Shapley value globally optimize for total cost-effectiveness.” is precisely defined.