With fair majority voting, Balinski offers a way to achieve proportional representation (PR) in the context of single winner elections, but it has two disadvantages:
It is has the complexity of PR, plus the complexity of single member districts, plus the complexity of reconciling the two.
The price of preserving districts is that neither parties nor voters can choose the best candidates within parties. For example, a district in which all parties field their worst candidates in all voters’ opinions has the same number of winners as one in which all parties field their best candidates.
I think a better way to reconcile PR and single winner would be to let each voter decide which seat he wants to (approval) vote on and add his contribution to representation to, and let each seat have vote weight equal to the number of voters that chose it. Electoral voting would be just like in open-list PR, except there would be as many lists as seats and voters could choose as many candidates as they wish on their chosen list.