  A natural partial ordering exists on the set of all weighted games and, more
broadly, on all linear games. We describe several properties of the partially
ordered sets formed by these games and utilize this perspective to enumerate
proper linear games with one generator. We introduce a geometric approach to
weighted voting by considering the convex polytope of all possible realizations
of a weighted game and connect this geometric perspective to the weighted games
poset in several ways. In particular, we prove that generic vertical lines in
$C_n$, the union of all weighted $n$-player polytopes, correspond to maximal
saturated chains in the poset of weighted games, i.e., the poset is a blueprint
for how the polytopes fit together to form $C_n$. We show how to compare the
relationships between the powers of the players using the polytope directly.
Finally, we describe the facets of each polytope, from which we develop a
method for determining the weightedness of any linear game that covers or is
covered by a weighted game.
