(ie. all the joint strategies are trying to maximize the money earned if up against the opposing coalition in a zero-sum game and as a special case, when S=N, it says that what the entire group actually ends up doing maximizes surplus value, which is another way of stating that the {ai}i∈N are the appropriate virtual currencies to use at the {uNi}i∈N point)
If in the joint strategy for the special case of S=N the group maximizes surplus value according to some weight function, then utility profile resulting from this joint strategy should be on the Pareto frontier, so 1) should be automatically satisfied.
If it wasn’t, then you could improve a player’s utility without hurting anyone else. But that would improve the surplus value as well[1], which would mean that the S=N joint strategy didn’t maximize surplus value (contradicting 3) ).
I think this is because your utilitarian characterization is an if and only if.
Closely related to this is a result that says that any point on the Pareto Frontier of a game can be post-hoc interpreted as the result of maximizing a collective utility function.
Could be: An outcome is on the Pareto frontier if and only if it can be post-hoc interpreted as the result of maximizing a collective utility function.
I guess I’m assuming the weights are strictly positive, whereas you only assumed them to be non-negative. Does this matter/Is this the reason why we need 1)?
Thanks for the great post!
In the definition of Coalition-Perfect CoCo Equilibrium, it seems to me like part 1) is already implied by part 3).
1) means that the utility profile achieved by the joint strategy for the grand coalition is on the Pareto frontier.
If in the joint strategy for the special case of S=N the group maximizes surplus value according to some weight function, then utility profile resulting from this joint strategy should be on the Pareto frontier, so 1) should be automatically satisfied.
If it wasn’t, then you could improve a player’s utility without hurting anyone else. But that would improve the surplus value as well[1], which would mean that the S=N joint strategy didn’t maximize surplus value (contradicting 3) ).
I think this is because your utilitarian characterization is an if and only if.
Could be: An outcome is on the Pareto frontier if and only if it can be post-hoc interpreted as the result of maximizing a collective utility function.
I guess I’m assuming the weights are strictly positive, whereas you only assumed them to be non-negative. Does this matter/Is this the reason why we need 1)?