Representation in Peer Selection: A Liquid Democracy Perspective

Abstract: We study the peer selection problem in which a group of peers must select a fixed-size committee from among themselves. To this end, we introduce a class of approval-based preference profiles, inspired by delegation graphs in liquid democracy and the transitivity of trust therein, which we call ``liquid profiles''. Under this new domain restriction, each voter designates a single trusted agent and, in turn, approves all those trusted by that individual. After establishing relationships to standard restricted preference profiles and examining computational properties of liquid profiles, we focus on understanding how novel and established proportionality axioms behave in this domain and how they relate to one another. Then, we study their satisfaction under various voting rules, including new ones that are directly inspired by the structure of liquid profiles. Our findings reveal that particularly strong proportionality guarantees are attainable in this new domain, showing promise for applications to real-world peer selection problems via transitive delegations.

• The paper introduces liquid profiles, models approval preferences as transitive delegation graphs, and proves that recognizing them is possible in polynomial time.
• The paper shows that Liquid Representation, strong EJR, strong EJR+, and Individual Representation coincide on liquid profiles, making existence and verification tractable.
• The paper proves that rules including PAV, MES, and sequential Phragmén achieve strong EJR, while MES also becomes committee-monotonic and polynomially computable.

Introduction and Motivation

The paper studies peer selection—choosing a committee of size $k$ from a group of voters who are themselves the candidates—through the lens of liquid democracy (LD). In LD, each participant either votes directly or delegates to a proxy, with delegations being transitive; deployed systems such as LiquidFeedback effectively select representatives among peers via these delegation structures. The authors formalize this connection by introducing liquid profiles: approval-based preference profiles in which a voter approves exactly those peers reachable from them in a functional delegation graph of out-degree one, i.e., approval sets are transitive closures of delegation paths. This is the first work interfacing liquid democracy research with the theory of proportional representation in approval-based committee elections (2608.13085).

The paper pursues three questions: how liquid profiles relate to established domain restrictions in multiwinner voting; which proportionality axioms are attainable on this domain and how they interrelate; and which voting rules deliver such guarantees efficiently.

Liquid Profiles as a Domain Restriction

A profile is liquid if it is witnessed by a delegation graph $G=(V,\mathcal{R})$ whose transitive closure yields the approval relation. The authors establish several structural results:

• Recognition is tractable: deciding whether an arbitrary approval profile is liquid can be done in polynomial time, via strongly connected component decomposition and a bottom-up forest reconstruction.
• Party-list profiles are exactly liquid profiles of depth at most 1, where depth measures the longest path to a cycle.
• Laminarity and liquidity are incomparable as domain restrictions: neither contains the other. The liquid-and-laminar profiles are precisely those whose components are kite graphs (a cycle with attached paths) plus adjacent vertices—a very restricted subclass.

These placements situate liquid profiles within the domain-restriction literature in computational social choice while showing they are genuinely new: strictly more expressive than party-list profiles, yet structurally disciplined by the out-degree-one constraint.

An Axiomatic Collapse

The central axiomatic contribution is a collapse theorem. The paper first shows that the two known strengthenings of Justified Representation—strong JR (sJR) and semi-strong JR (ssJR)—coincide on liquid profiles. In the general multiwinner setting sJR is unsatisfiable, so this result already signals that the domain is exceptionally well-behaved.

The main result establishes that four axioms are all equivalent on liquid profiles:

1. Liquid Representation (LR), a new axiom defined directly on the delegation graph: any candidate supported by at least $\ell n/k$ voters entitles its supporters to $\ell$ elected vertices from the candidate together with everything it approves ($c \cup \mathsf{up}(c)$), or all of them if fewer than $\ell$ exist.
2. Strong EJR (sEJR), requiring every member of every $\ell$-cohesive group to have satisfaction at least $\ell$—unsatisfiable in general committee elections.
3. Strong EJR+ (sEJR+), a new efficiently checkable strengthening.
4. Individual Representation (IR) of Brill et al., which guarantees each voter satisfaction equal to the largest $\ell$ for which they belong to an $\ell$-cohesive group.

This equivalence has immediate computational consequences: IR committees always exist on liquid profiles (unlike the general setting), deciding existence—which is NP-hard for general profiles—is polynomial-time, and EJR verification, co-NP-hard in general, becomes tractable.

Notably, the collapse does not extend to all proportionality notions. Core Stability remains independent of sJR (hence sEJR) even on liquid profiles, and the separating example is instructive: on a three-vertex path with $G=(V,\mathcal{R})$0, electing the middle vertex is Core Stable but fails sJR, since only electing the sink represents everyone. Similarly, Perfect Representation stays independent of IR on this domain, strengthening a known separation. The authors argue that CS- or PR-satisfying outcomes can be intuitively inferior (e.g., leaving a sink unelected), supporting their focus on the sEJR-family axioms.

Voting Rules: Guarantees and Equivalences

On the algorithmic side, the paper delivers a series of positive results summarized below.

Rule	Guarantee	Polynomial time	Committee monotonic
CC ≡ seq-CC	sJR	✓	—
Cutoff-seq-CC (new)	sJR-characterizing	✓	—
PAV ≡ seq-PAV ≡ Liquid d'Hondt (new)	sEJR	✓	✓
MES ≡ Budgeted AV (new)	sEJR	✓	✓
Sequential Phragmén	sEJR	✓	✓
HJCR (new)	sEJR-characterizing	✓	—

Several findings deserve emphasis. First, the Chamberlin–Courant rule satisfies not merely JR but sJR on liquid profiles, and coincides with its sequential variant—removing CC's well-known computational hardness. The new Cutoff-seq-CC rule essentially characterizes sJR outcomes: any sJR-satisfying rule must return a superset of its output.

Second, the paper introduces Liquid d'Hondt (LDH), a d'Hondt-style greedy rule operating directly on delegation graphs via per-seat representation quotients over shrinking subtrees. Remarkably, LDH is equivalent to seq-PAV, which is itself equivalent to the (NP-hard in general) PAV rule on this domain. Thus PAV becomes polynomial-time computable under liquid profiles, extending the known party-list equivalence.

Third, the party-list equivalence between PAV, MES, and Phragmén breaks on liquid profiles: MES reduces instead to a "Budgeted Approval Voting" procedure, and Phragmén follows distinct dynamics. Yet both still satisfy sEJR. A striking consequence concerns committee monotonicity: MES, which fails this property in standard committee elections, satisfies it on liquid profiles—resolving, within this domain, the long-standing tension between EJR-type guarantees and monotonicity. Sequential Phragmén, which violates EJR in general, also attains sEJR here.

Fourth, the Greedy Justified Cohesive Rule fails sEJR because it lacks a hierarchical structure (electing a vertex without electing everything above it). The authors repair this with the Hierarchical Justified Cohesive Rule (HJCR), which processes each component from the sink-cycle outward and characterizes sEJR: every sEJR-satisfying rule returns a superset of HJCR's outcome. Together with Cutoff-seq-CC, this provides principled minimal cores for committees: use Cutoff-seq-CC when diversity-style (but stronger-than-JR) guarantees suffice, and HJCR when EJR-strength proportionality is desired, completing the committee with any standard rule thereafter.

Limitations and Open Questions

The paper's model imposes one delegation per voter (out-degree one), excluding richer delegation patterns; relaxing this is left open. Trust is treated as binary along delegation chains, so diminishing trust à la viscous democracy [boldi2011viscous] and cardinal satisfaction models are unexplored. Although MES and Phragmén induce different candidate orders, the precise relationship between their outcomes remains unresolved. Most notably, whether core-stable outcomes always exist on liquid profiles—an open problem even in general committee elections—is not settled here. Finally, the laminar structure identified for liquid-and-laminar profiles suggests, but does not establish, a broader domain restriction yielding similar positive results beyond peer selection.

Conclusion

This paper introduces liquid profiles as a novel, tractably recognizable domain restriction for peer selection motivated by transitive delegations in liquid democracy. On this domain, strong representation axioms that are unsatisfiable or hard in general committee elections become simultaneously satisfiable, polynomial-time verifiable, and—in the case of LR, sEJR, sEJR+, and IR—equivalent. Standard rules acquire stronger guarantees (CC satisfies sJR; PAV, MES, and Phragmén satisfy sEJR, with MES gaining committee monotonicity), and two new characterizing rules delineate the minimal requirements of these axioms. The work supplies the first systematic theory of proportional representation for liquid democracy and demonstrates that delegation structure can serve as a powerful domain restriction in multiwinner voting.