Payoff Division

Even if you did find the optimal CS, it might not be worth much if its not stable. That is, if there are any agents who are motivated to depart from that CS.
We define the core of a CS to be the vector of payoffs to agents such that no subgroup is motivated to depart from the given CS.
The core does not always exists, sometimes there are several of them. The core is reminiscent of the Nash equilibrium.
Another payoff division method can be established by defining:
- i is a dummy if V_{S + i} - V_S = V_i
- i and j are interchangeable if for all S with either i or j, V_S remains the same if i is replaced with j, or viz.
We now want a set of payoffs that exhibit:
- Symmetry: If i and j are interchangeable then x_i = x_j (where x is the set of payoffs).
- Dummies: If i is a dummy then x_i = V_i.
- Additivity: For any two games V and W, x_i in V+W equals x_i in V plus x_i in W, where V+W is the game defined by (V+W)_S = V_S + W_S.
The Shapely Value satisfies all these: It sets x_i = SUM_{S in A} (|A| - |S|)!(|S| - 1)!(V_S - V_{S - i})/ |A|!
It obeys all three properties. It exists, and is unique. It is pareto efficient (all the money is distributed among the agents).
But, there is no guarantee that a subgroup is better off by splitting out. It takes a long time to compute.