Here’s an old note from my book on political representation that explores why PR systems may be good for the representation of women.
Proportional representation (PR) systems are often touted as beneficial to the inclusion of women as political representatives [almost all of this work uses a binary approach to gender]. In the literature, the argument is typically that PR systems mean larger districts, which is good for the inclusion of “minorities”. What we should be focusing on is the probability that a vote goes to a woman. There are, however, a couple of assumptions that are rarely spelled out, notably that there are few female contenders.
Situation 1 illustrates the assumption (or empirical reality) that men more likely at the top of the ballots, thus more likely to be elected. In majoritarian systems and systems with very small district size, only the top candidates are picked. So there is no inherent advantage for PR systems, but an effect that is dependent on women being selected for top positions.
Situation 2 illustrates that in a larger district, we simply reach further down the list, thus (under the same assumption that women are less likely at the top of the ballot) we are more likely to pick a woman — like any other candidate who is more likely to be further down the list.
Situation 3 shows that having more parties — a feature of many PR systems — is equivalent to having smaller districts: Assuming that all parties tend to be more likely to put men at the top of the ballot, we are no longer reaching down the lists as much (each party gets fewer votes).
So, it’s not PR as such, but district size combined with candidate selection/candidate placement that matters for the political representation of women.
Ruedin, Didier. 2013. Why Aren’t They There? The Political Representation of Women, Ethnic Groups and Issue Positions in Legislatures. Colchester: ECPR Press.