I want to know about the meaning of cost allocation in a coalition. I know we have some solutions for this(Shapley or Nucleolus value).
Consider following interpretation from cost allocation:
Payoff that each player gets in that coalition(Money).
for example N friends work on a project and share the money base on their contribution. Everything is okay :)
Now consider following example:
We have a football team and calculate value for each player(for example shapley value).
- What is the meaning of these values? (You didn't share anything between players. I think it's just a number)
- What is the difference between shapley and nucleolus in this case?
I couldn't understand cost allocation in this case and it's more confusing when I realize that shapley value isn't in core but nucleolus is(in another example)!!!
I think you confound costs and benefits. In a cost allocation problem the Shapley value or nucleolus determines an amount of money (costs) or transferable utility a player has to pay to support a project, and not an amount that she/he can get. Players receive something like an amount of money or transferable utility in case of a benefit game. However, in a cost game (negative benefit) they have to pay/contribute something to start the project. Note in addition that each cost game is strategically equivalent to a savings game (benefit game), and vice versa.
This means for your football team example, that you can consider on one side the total benefit of the football team that it has generated over a season to allocate by the Shapley value or nucleolus to each player (here I mean an investor of the team and not an individual football player) her/his return of investment. On the other side, you can also look on the total overhead cost that the team generates, and you might ask how should we distribute the overhead to each cost center. In case of a relegated team you can replace overhead cost by loss.
Furthermore, the Shapley value and nucleolus are fairness values based on a different set of principles of distributive justice (axioms). Thus, if partners accept the rules of distributive justice that characterizes the Shapley value (nucleolus), then a fair cost allocation should be given by the Shapley value (nucleolus). Hence, these values can be interpreted as different outcomes from negotiations based on different rules of arbitration.
Finally, that the Shapley value might be not in the core simply means that it is more difficult to establish a fair division based on its rules of arbitration, and one probably fails to establish mutual cooperation into the grand coalition. Here, the nucleolus has the advantage that it can be better defended by counter-objections.