All Power Structures are Achievable in Basic Weighted Games

A major problem in decision-making is designing voting systems that are as simple as possible and able to reflect a given hierarchy of power of its members. It is known that in the class of weighted games, all hierarchies are achievable except two of them. However, many weighted games are either improper or do not admit a minimum representation in integers or do not assign a minimum weight of 1 to the weakest non-null players. These factors prevent obtaining a good representation. The purpose of the paper is to prove that for each achievable hierarchy for weighted games, there is a handy weighted game fulfilling these three desirable properties. A representation of this type is ideal for the design of a weighted game with a given hierarchy. Moreover, the subclass of weighted games with these properties is considerably smaller than the class of weighted games.