Balanced Coalitional Game

A coalitional game is balanced if every balanced collection of coalitions satisfies the weighted worth inequality $$ \sum_{S \in \mathcal{D}} \delta_S v(S) \le v(N), $$ whenever the positive weights $\delta_S$ cover each player with total weight one.

Intuitively, if a fractional schedule of coalitions uses each player once in total, then the weighted value of that schedule cannot exceed the worth of the grand coalition.

References

• [MSZ Ch.17, Thm 17.14] Maschler, Solan, Zamir, Game Theory. Balancedness condition in the Bondareva-Shapley theorem.