The Matryoshka of Homogeneous Weighted Majority Games

We define a sequence of three progressively smaller subsets of the class of constant sum homogeneous weighted majority: parsimonious games, bilaterally symmetric games and uniform games. Parsimonious games are characterized by a parsimony property concerning the number of minimal winning coalitions. The key concept to identify the other two classes is the truncated type representation of a parsimonious game. Precisely, bilaterally symmetric games are parsimonious games whose representation is a symmetric one, while uniform games present not only symmetry but also uniformity at a level k of such type a representation. We found that k−uniform games are strictly linked to k−Fibonacci sequences. This justifies the choice to call such games k Fibonacci and Fibonacci games the union over all integers k of such sets.