Relationship Between Method of Equal Shares and Sequential Phragm Outcomes

Determine the relationship between the outcomes produced by the Method of Equal Shares and Sequential Phragm on liquid profiles, including whether their outcomes satisfy a systematic containment, equivalence, or other structural relationship.

Background

The paper analyzes several proportional representation rules on liquid profiles, including the Method of Equal Shares (MES) and Sequential Phragm. It establishes that both rules satisfy strong extended justified representation and committee monotonicity, but it also shows that they can induce different candidate orders. The authors do not resolve how their final outcomes are related, leaving this as an explicit unresolved question for future work.

References

Our results open several directions for future research. First, from a modelling perspective, we assumed that each voter has out-degree one in the delegation graph, which could plausibly be relaxed in peer selection. One could also model diminishing trust along delegation paths, assigning higher voter satisfaction to directly approved candidates than to those reached via longer paths (as in so-called viscous democracy \citep{boldi2011visc}), and study the resulting setting using cardinal multiwinner voting rules. Second, from a structural perspective, although we establish that MES and Phragm\ may induce different candidate orders, the relationship between their outcomes remains open.

— Representation in Peer Selection: A Liquid Democracy Perspective  (2608.13085 - Grossi et al., 13 Aug 2026) in Conclusion