Show that cohesivity is necessary, but not sufficient, for the existence of a core.
A partition {S1, S2,..., SK} is a family of coalitions in which each player I belongs to one and only one of the coalitions. A balanced family of coalitions B is a set of coalitions {S1, S2,..., Sk} together with a corresponding set of positive numbers λ1, λ2,..., λK called weights such that for every player i ˆˆ N, the sum of the weights of the coalitions to which player i belongs sum to one. That is, for every player i ˆˆ N,

Compared to a simple partition, the coalitions in a balanced family do not require the exclusive allegiance of their members.

Transcribed Image Text:

Σλ.-1, for every leN