They definitely aren’t Cartel independent! Let’s take your example, and imagine that our “cartel” is Alice and Bob, forming a combined coalition player (“AlicoBob, sitting in a tree, k-i-s-s-i-n-g”)
AlicoBob by themself can make $100. Bert by himself can make $0. AlicoBob + Bert can make $100.
The synergy is $0, so the Shapley value is that AlicoBob gets everything and Bert goes home sad.
However, Alice could’ve also formed a cartel with Bert, and then Bob would go home sad. So there’s like an equilibrium thing going on here where both Bob and Bert want to be the sole partners of Alice and leave the other one out, and so what I expect would happen in real life is that if for some reason one of the two got there first that they would naturally form a coalition and demand more concessions from the excluded party, while Alice would then also demand more concessions from Bob because she can threaten to go and collude with Bert instead.
This is basically the “stand alone core” property that is sometimes logically impossible to satisfy, so I guess it’s not too sad that the Shapley value doesn’t live up to it.
They definitely aren’t Cartel independent! Let’s take your example, and imagine that our “cartel” is Alice and Bob, forming a combined coalition player (“AlicoBob, sitting in a tree, k-i-s-s-i-n-g”)
AlicoBob by themself can make $100. Bert by himself can make $0. AlicoBob + Bert can make $100. The synergy is $0, so the Shapley value is that AlicoBob gets everything and Bert goes home sad.
However, Alice could’ve also formed a cartel with Bert, and then Bob would go home sad. So there’s like an equilibrium thing going on here where both Bob and Bert want to be the sole partners of Alice and leave the other one out, and so what I expect would happen in real life is that if for some reason one of the two got there first that they would naturally form a coalition and demand more concessions from the excluded party, while Alice would then also demand more concessions from Bob because she can threaten to go and collude with Bert instead.
This is basically the “stand alone core” property that is sometimes logically impossible to satisfy, so I guess it’s not too sad that the Shapley value doesn’t live up to it.