  We study minimum integer representations of weighted games, i.e.,
representations where the weights are integers and every other integer
representation is at least as large in each component. Those minimum integer
representations, if the exist at all, are linked with some solution concepts in
game theory. Closing existing gaps in the literature, we prove that each
weighted game with two types of voters admits a (unique) minimum integer
representation, and give new examples for more than two types of voters without
a minimum integer representation. We characterize the possible weights in
minimum integer representations and give examples for $t\ge 4$ types of voters
without a minimum integer representation preserving types, i.e., where we
additionally require that the weights are equal within equivalence classes of
voters.
