Bilateral symmetry and modified Pascal triangles in Parsimonious games

We discuss the prominent role played by bilateral symmetry and modified Pascal triangles in self twin games, a subset of constant sum homogeneous weighted majority games. We show that bilateral symmetry of the free representations unequivocally identifies and characterizes this class of games and that modified Pascal triangles describe their cardinality for combinations of m and k, respectively linked through linear transforms to the key parameters n, number of players and h, number of types in the game. Besides, we derive the whole set of self twin games in the form of a genealogical tree obtained through a simple constructive procedure in which each game of a generation, corresponding to a given value of m, is able to give birth to one child or two children (depending on the parity of m), self twin games of the next generation. The breeding rules are, given the parity of m, invariant through generations and quite simple.