The Executive Council of a major international organization has five members. Countries 1 and 2 are big countries; and countries 3, 4, and 5 are small countries. The Executive Council can approve an action on any issue relating to international security provided both big countries and at least one small country vote for the proposed action. The power structure on this Council may be described mathematically by a coalitional game in which any coalition that can approve an action (by the votes of its members alone) has worth 1, and any coalition that cannot approve an action has worth 0.
a. What is the core of this game?
b. What is the Shapley value of this game?
c. Suppose that the three small countries form closer relations among themselves in the Executive Council, so there is an effective nested coalition structure {{1},{2},{3,4,5}1. What is the Owen value of this game with this coalition structure? Does the union help the small countries, according to the Owen value?
d. Compute the Owen value of this game with the nested coalition structure {{1,2},{3,4,5}}. If the small countries form a union, do the big countries gain by also forming a union, according to the Owen value?