A note on hierarchies in weighted voting games and partitions

Weighted voting games (WVGs) model situations where voters, possibly controlling different numbers of votes, vote yes or no on a proposition. A proposition passes if and only if the number of yes votes meets or exceeds a quota (q). Each winning coalition is a partition of an integer greater than or equal to (q), with parts taken from the set of all weights for that game. Results about WVGs are here interpreted as results about partitions.