I have been doing some reading from the adjacent field of participatory budgeting. I found this paper which defines something like Proportional Representation. Their semantics are a little different so I will translate the definition into something more similar to what we use.
Given a winner set S of K winners, a smaller winner set S containing K’ < K winners blocks the stability of S iff V(S,S’)/n >= K’/K. Where V(S,S’) is the number of voters who strictly prefer S’ to S and n is the number of voters. A winner set is stable if no replacement set blocks it.
The important thing is that S’ is a smaller set than S (ie K’<K) so the members of the group that prefers S’ each individually would have higher sum(score) with fewer winners.
This implies the Hare Quota Criterion which is basically what we use as a stand in for Proportional Representation since there is no universally accepted definition. The existing definitions of Proportional Representation are unclear and conflicting. Please don’t debate that in this thread. Instead read the wiki page and update it.
What I would like to discuss is if this concept of “stable winner set” is useful. Can we use winner set stability to get rid of the concept of Proportional Representation? There can be situations where there is no stable winner set. There can also be cases where there are many stable winner sets. All the stable winner sets make up what is called the “core”. It is somewhat like a Condorcet cycle were there are multiple “equally good winners”.
Also, can this be used to build a election method? ie the method that always finds the stable winner set if one exists and does something reasonable when there is no or multiple stable winner sets.