Balanced Collection Of Coalitions

A finite collection of coalitions $\mathcal{D}$ is balanced if its coalitions can be assigned positive weights $\delta_S>0$ so that every player is covered with total weight one: $$ \sum_{\substack{S \in \mathcal{D}\\ i \in S}} \delta_S = 1 \quad\text{for every } i \in N. $$

The condition says that the weighted incidence vectors of the coalitions in $\mathcal{D}$ add up to the incidence vector of the grand coalition.

Balanced collections are the combinatorial input to the Bondareva-Shapley theorem ([[game_theory.cooperative_game.bondareva_shapley]]).

References

• [MSZ Ch.17, Def 17.11] Maschler, Solan, Zamir, Game Theory.