[fastball]

Why can't you just split elected representatives into two groups, regardless of stated political affiliation? One group then cuts, the other chooses.

[specialist]

Defections. Electeds will happily maximize their own personal power at someone else's expense. There are many examples of getting opposition support for inequity in exchange for safe seats.

People outside of (partisan) politics don't often get to see that the party orgs and the politicians are usually in conflict. More so on the left than on the right.

[splitrocket]

Interestingly enough, this works even for defectors. They still have to participate in a turn-based game against a competitor.

There is a whole field of study around efficient "cake cutting" algorithms, many of which contemplate and account for defectors.

https://en.wikipedia.org/wiki/Fair_cake-cutting