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We discuss the prominent role played by bilateral symmetry and modifiedPascal triangles in self twin games, a subset of constant sum homogeneousweighted majority games. We show that bilateral symmetry of the freerepresentations unequivocally identifies and characterizes this class of gamesand that modified Pascal triangles describe their cardinality for combinationsof m and k, respectively linked through linear transforms to the key parametersn, number of players and h, number of types in the game. Besides, we derive thewhole set of self twin games in the form of a genealogical tree obtainedthrough 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 twochildren (depending on the parity of m), self twin games of the nextgeneration. The breeding rules are, given the parity of m, invariant throughgenerations and quite simple.

Bilateral symmetry and modified Pascal triangles in Parsimonious games