{"id":"4a75eb3c-31c5-419f-b18c-da548c9527a5","arxiv_id":"2608.13085","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On delegation-based \"liquid\" approval profiles, the paper shows strong-EJR, Liquid Representation, and Individual Representation coincide, and that PAV, MES, and sequential Phragmén all satisfy them.","lead":"The paper introduces \"liquid profiles\", approval lists built from delegation chains in liquid democracy, and shows that several strong fairness axioms become equivalent and satisfiable by simple voting rules on them. It proves that proportional committee selection in this restricted domain is computationally easy, which matters for online governance tools like LiquidFeedback.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 12's proof of PAV=seq-PAV=LDH is circular: it invokes Proposition 14, whose own proof invokes LDH=seq-PAV, leaving the central tractability claim unsupported.","rationale":"The paper's headline contribution is that strong proportionality guarantees become tractable on liquid profiles, and Theorem 12 is the load-bearing equivalence that removes PAV's NP-hardness. The reader's stated weakest assumption was the out-degree-one domain restriction, which is an external-validity concern. However, the reader also noted compressed proofs and a circular dependency in the appendix; that internal concern is the one I find most consequential. I focused on Theorem 12 rather than the domain restriction because the domain restriction is a modeling choice that the authors explicitly acknowledge in the Conclusion, whereas the circular proof, if unfixed, leaves the main theoretical result unproven. The other major results, such as Theorem 6 and Theorem 10, have more self-contained proofs, but Theorem 12 and the Proposition 14 it depends on are not independently verified. A computational enumeration over small liquid profiles would settle whether the equivalence is true; if it holds, the proof gap is likely repairable with a direct hierarchy lemma. This does not change the reader's CONDITIONAL verdict: acceptance should be conditioned on closing the circularity, but the paper need not be rejected outright.","tokens_in":26183,"tokens_out":18239,"duration_ms":184462,"concrete_test":"Enumerate all functional delegation graphs on n ≤ 6 vertices, derive their liquid approval profiles, and for every k ≤ n compute the full outcome sets of PAV, seq-PAV, and LDH by exhaustive search over all committees and all tie-breaking orders. If any profile allows a seq-PAV or LDH path that fails to be PAV-optimal, Theorem 12 is false. If the enumeration finds no counterexample, then independently prove Proposition 14 for LDH directly from Definition 11 without invoking the LDH=seq-PAV equivalence; if that proof succeeds, the circularity is closed and the CONDITIONAL verdict can be upgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that PAV becomes polynomial-time on liquid profiles—rests on Theorem 12 (LDH ≡ seq-PAV ≡ PAV). The appendix proof of Theorem 12 has two steps. First, to show LDH ≡ seq-PAV, it asserts without proof that seq-PAV 'will select a candidate c′ that is a sink or is a direct predecessor of an already selected vertex of W'; this is exactly the hierarchical property formalized later as Proposition 14. Second, to show seq-PAV ≡ PAV, it says 'Observe that, by Proposition 14, both W and W′ form a set of weakly connected components' and then performs a swap argument. But the appendix proof of Proposition 14 for LDH reads: 'For convenience, we will prove the statement for seq-PAV, which is equivalent to LDH.' That equivalence is precisely the first half of Theorem 12 being proved. Hence the argument cycles: Theorem 12(part 1) relies on an unproved hierarchy assertion; Theorem 12(part 2) relies on Proposition 14; Proposition 14 relies on Theorem 12(part 1). Since no machine-checked proof or independent verification is supplied, the main tractability consequence is not currently demonstrated. The gap may be repairable—the statements may still be true—but as written the central claim is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces 'liquid profiles,' a domain restriction for approval-based committee elections in which voters are nodes of a functional delegation graph and approve exactly the transitive closure of their delegation path. It positions this class relative to laminar and party-list profiles, characterizes several representation axioms (sJR, LR, sEJR, sEJR+, IR), and studies the behavior of CC, seq-CC, PAV, seq-PAV, MES, Sequential Phragmén, GJCR, and newly introduced LDH, Cutoff-seq-CC, Budgeted AV, and HJCR on this domain. The headline claim is that on liquid profiles PAV, seq-PAV, and LDH coincide, making PAV polynomial-time computable, and that multiple rules satisfy the strong sEJR guarantee.","tokens_in":26509,"tokens_out":9887,"duration_ms":92317,"significance":"If the main theorems are correct, the paper makes a useful contribution: it identifies a structured domain with strong proportionality guarantees, shows a collapse of several otherwise distinct axioms, and gives concrete rule recommendations. The computational results for PAV and the axiom characterizations are interesting, and the paper includes detailed proofs in the appendix plus clear summaries of the axiomatic landscape. The central equivalence result, however, is currently not proven because of a circular argument, so the main payoff (polynomial-time PAV with sEJR) rests on unproven statements. The paper deserves a major revision rather than rejection, because the underlying claims are plausible and the gaps appear repairable.","major_comments":[{"comment":"The proof of Theorem 12 is circular. The first half asserts without proof that seq-PAV 'will select a candidate c′ that is a sink or is a direct predecessor of an already selected vertex of W' — which is exactly the hierarchical property formalized later as Proposition 14 — and the second half invokes Proposition 14 to conclude that both the seq-PAV and PAV outcomes are unions of weakly connected components. The proof of Proposition 14 for LDH, in turn, reads 'For convenience, we will prove the statement for seq-PAV, which is equivalent to LDH,' which is precisely the equivalence that Theorem 12 is proving. Consequently, none of LDH=seq-PAV, Proposition 14 for LDH/seq-PAV, and the PAV half is established by the current text.","section":"Appendix: Proof of Theorem 12"},{"comment":"Proposition 14, as stated, covers only Liquid d'Hondt, Phragmén, and MES; it does not state that PAV is hierarchical, yet the proof of Theorem 12 applies Proposition 14 to the PAV outcome W′ as well. The text therefore uses a lemma for a claim about a rule the lemma does not mention. A separate argument for the hierarchical structure of PAV outcomes on liquid profiles is required before the swap argument in the second half of the proof can go through.","section":"Appendix: Proposition 14 and Theorem 12"},{"comment":"The proof of Theorem 15 (PAV and MES satisfy sEJR) depends on the same unresolved hierarchical property for PAV: it asserts that the elections under these rules form connected components with all vertices in up(c) elected, and for PAV this is only justified through the unproven equivalences and hierarchical claims discussed above. Thus, unless the proof of Theorem 12 is repaired, the sEJR guarantee for PAV is not currently established, and the paper's claim that PAV is polynomial-time computable while satisfying strong-EJR is unsupported.","section":"Theorem 15"}],"minor_comments":[{"comment":"The proof of Observation 3 contains a duplicated and apparently misplaced third paragraph: after giving the intended two examples, it repeats an example for the first statement and contains the typo 'for a self look' instead of 'form a self-loop.' This should be cleaned up.","section":"Proof of Observation 3"},{"comment":"The proof of Theorem 11 concludes that the two solutions are 'equivalent,' but the stated characterization requires the outcome of any sJR rule to be a superset of Cutoff-seq-CC's outcome under some tie-breaking. The argument should be made precise for sink cycles, where two rules may pick different vertices of the same cycle.","section":"Proof of Theorem 11"},{"comment":"The polynomial-time membership test in Proposition 1 is presented as a sketch. Please specify the bottom-up forest reconstruction from the SCC decomposition in detail, including how the algorithm enforces the out-degree-one constraint and handles multiple bottom SCCs, so the reader can verify correctness.","section":"Proof of Proposition 1"},{"comment":"Several references are listed as forthcoming with no archive or DOI, including Brill et al. (2026), Papasotiropoulos et al. (2026), and Faliszewski et al. (2026). Please update these with stable pointers or note their availability.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is squarely within the scope of cs.GT and the results would be valuable if the proof of Theorem 12 is repaired. I see no reason to suspect the statements are false; the problem is a genuine proof-order defect in the appendix. A revision that proves a hierarchical lemma for seq-PAV and PAV independently of the LDH equivalence, and then derives Theorem 12 from it, would address the main concern. Please ensure the characterization results (Theorems 11 and 20) also receive careful treatment of tie-breaking."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Disclosure up front: this is the first systematic proportionality theory for liquid democracy, and the axiomatic core is worth taking seriously. The new domain restriction (liquid profiles) is sensible and well-motivated, the collapse of LR, sEJR, sEJR+, and IR in Theorem 6 is a genuine result with detailed proofs, and the CC/Cutoff-seq-CC and HJCR characterizations give useful handles on the axioms. The paper does a real service in showing that strong axioms unsatisfiable or NP-hard in general multiwinner elections become tractable on a natural domain.\n\nThe soft spot is the proof of Theorem 12, and it is not minor. Theorem 12 is load-bearing: it claims LDH = seq-PAV = PAV, and the paper uses it to conclude PAV is polynomial-time on liquid profiles. The appendix proof has a circular dependency. The first half asserts without proof that seq-PAV picks a sink or direct predecessor of an already selected vertex; this is exactly the hierarchical property stated later as Proposition 14. The second half invokes Proposition 14 to claim PAV and seq-PAV committees are unions of weakly connected components. But the appendix proof of Proposition 14 says 'we will prove the statement for seq-PAV, which is equivalent to LDH'—the very equivalence being proved in Theorem 12. So the argument loops. The same gap leaks into Theorem 15, whose proof assumes the hierarchical structure for PAV and MES.\n\nThis is likely repairable: the hierarchical property for LDH can probably be proved directly from its quotient rule, and then Theorem 12 and Proposition 14 can be reordered. But as written, the central tractability claim is not established. Everything else I checked—Theorem 6, the CC results, the HJCR characterization—has real proofs and seems sound. The out-degree-one assumption is acknowledged in the conclusion as a limitation; that is honest rather than a flaw.\n\nWho is this for? Researchers in computational social choice working on liquid democracy or domain restrictions. It deserves a serious referee and probably acceptance after a proper revision, but the reviewer should demand a repaired proof of Theorem 12/Proposition 14 and a clean proof ordering. With that fixed, this would be a solid paper.","headline":"A genuinely useful paper whose main tractability result currently rests on a circular proof; the axiomatic core is solid and the fix looks straightforward.","tokens_in":27035,"tokens_out":2267,"would_cite":false,"duration_ms":22723,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91B12","91B14","68Q25"],"pacs":[],"model":"deepseek-v4-flash","headline":"From single-chain trust in delegation graphs, this paper shows that the strongest proportional-representation axioms become satisfiable and polynomial-time computable on liquid profiles.","keywords":["liquid democracy","peer selection","liquid profiles","approval-based committee elections","proportional representation","justified representation","multiwinner voting"],"falsifier":"Enumerate all liquid profiles on small voter sets—all functional delegation graphs on, say, up to eight vertices, with committee sizes up to three—and compute, under one fixed tie-breaking order, the committees returned by Liquid d'Hondt, sequential PAV, and PAV; a single profile where the three outcomes differ would refute the central equivalence of Theorem 12 and with it the polynomial-time claim for PAV, and because the claim is a finite statement about exactly this class of graphs, exhaustive search would settle it.","tokens_in":26004,"feed_emoji":"🗳️","tokens_out":19391,"duration_ms":149093,"temperature":0.7,"pith_summary":"This paper introduces a domain restriction for peer selection, called liquid profiles: a profile is liquid when it is exactly the transitive closure of a delegation graph in which each voter names one trusted agent, so each voter approves precisely the chain of people that agent trusts. The principal claim is that within this domain the strongest proportionality guarantees from committee voting—strong extended justified representation (sEJR), strong EJR+, individual representation, and the new graph-based liquid representation (LR)—all become equivalent and are always satisfiable (Theorem 6). The computational result behind this is Theorem 12: on liquid profiles, the new Liquid d'Hondt rule, sequential PAV, and proportional approval voting (PAV) return identical committees, so the NP-hardness of PAV in general elections disappears and polynomial-time rules such as PAV, MES, and Sequential Phragmén all meet the sEJR guarantee. A reader should care because the motivating scenario is liquid democracy, which already produces exactly such transitive-trust ballots, and the paper shows these are precisely the cases where the hardest proportionality targets become cheap.","feed_headline":"Transitive trust makes the hardest voting rule tractable","feed_subtitle":"Delegation chains turn the hardest proportional-voting rule into a fast, simple one.","key_machinery":"The central object is the liquid profile: an approval profile over a set of peers for which there exists a delegation graph with exactly one outgoing edge per vertex, such that each voter approves exactly the vertices reachable from them under the graph's transitive closure. The property that carries the argument is hierarchy: a rule is hierarchical if, whenever it elects a non-sink vertex, it also elects every vertex that vertex reaches (Proposition 14), and this structure is what lets the paper transfer guarantees from graph structure to axioms and rules. The named new rule, Liquid d'Hondt, works directly on the graph, electing at each step the candidate maximizing the quotient $\\frac{|down(c)|}{|up(c) \\cap W_t| + 1}$, where $down(c)$ is the set of voters who reach $c$ and $up(c)$ is the set of candidates $c$ reaches; Theorem 12's identification of this rule with PAV and seq-PAV is the step that turns a hard global optimization into a greedy, polynomial-time computation.","core_discovery":"On liquid profiles, the paper argues, the obstacles that make proportional representation hard in general committee elections disappear. Its central theorem is that strong EJR, strong EJR+, individual representation, and liquid representation are all equivalent in this domain (Theorem 6), so committees in which every member of every cohesive group is fully represented always exist—whereas in general multiwinner elections these axioms are typically unsatisfiable. The load-bearing identity is Theorem 12, which equates Liquid d'Hondt—a rule applying d'Hondt apportionment quotients directly to the delegation graph—with sequential PAV and with PAV itself, making PAV polynomial-time computable on liquid profiles. Since PAV and MES satisfy sEJR (Theorem 15) and MES additionally satisfies committee monotonicity (Theorem 19), the paper concludes that liquid profiles support concrete rule recommendations: Cutoff-seq-CC when diversity in the spirit of strong JR is the goal, Hierarchical Justified Cohesive Rule when the stronger proportionality guarantee is wanted, with any standard proportional rule used to complete the committee.","pith_inferences":["If the Theorem 12 collapse—Liquid d'Hondt, seq-PAV, and PAV agreeing on every liquid profile—extends to delegation graphs where a voter splits trust among several proxies or where trust decays with distance (the viscous-democracy variants the authors list as future work), greedy graph-based rules would deliver PAV-grade proportionality well beyond the single-chain case; this is an extension the pa","The kite-graph characterization of liquid profiles that are also laminar (Proposition 4) hints at a wider domain restriction: delegation networks that are disjoint unions of kites might inherit the strong-EJR guarantees even when the full liquid assumption fails, a hypothesis one could test by running HJCR and MES on such graphs.","For deployed liquid-democracy systems, where ballots pile up at the top of delegation trees, the hierarchical rules amount to a formal justification of the common practice of letting the most-delegated peer of a component represent the whole branch; the theorems say that practice is not a heuristic but the exact shape of the proportional-optimum committee.","The independence of core stability from strong JR (Theorem 9) means an outcome can be core-stable while leaving a component's most-trusted sink unelected, which sits uneasily with the delegation semantics; in my reading, the open question the authors flag—whether core-stable committees exist on liquid profiles at all—becomes most interesting when restricted to hierarchical committees."],"forward_implications":["PAV, the proportionality standard that is NP-hard to compute in general committee elections, becomes polynomial-time computable on liquid profiles because its outcome coincides with its sequential variant and with Liquid d'Hondt (Theorem 12).","Committees satisfying strong EJR—unsatisfiable in the general setting—always exist on liquid profiles and are returned by PAV, MES, and Sequential Phragmén, all in polynomial time (Theorems 6, 15 and 17).","Because LR, sEJR, sEJR+, and IR coincide on liquid profiles (Theorem 6), the decision problem of whether an IR committee exists—NP-hard in general—becomes polynomial-time there, and so does verification of EJR (Proposition 7).","MES, which fails committee monotonicity in general, is committee monotonic on liquid profiles, where it reduces to a budgeted approval-voting procedure (Theorems 18 and 19).","The two new characterizing rules, Cutoff-seq-CC and HJCR, identify the minimal committees that any sJR- or sEJR-satisfying rule must elect, so a designer can secure the guarantee first and complete the committee with any standard rule (Theorems 11 and 20)."],"supporting_citations":[{"why":"supplies the JR, EJR, and core-stability definitions the paper strengthens, and the general-setting hardness and unsatisfiability results that liquid profiles overturn.","marker":"(Aziz et al., 2017)"},{"why":"introduces individual representation, its independence from perfect representation, and the NP-hardness of IR committees that Theorem 6 and Theorem 15 make polynomial-time on liquid profiles.","marker":"(Brill et al., 2025)"},{"why":"introduces EJR+, whose strong variant participates in the four-way axiomatic collapse of Theorem 6.","marker":"(Brill and Peters, 2023)"},{"why":"established that PAV, MES, and Phragmén coincide on party-list profiles, the baseline that Proposition 13 shows breaks once liquid profiles widen the domain.","marker":"(Brill et al., 2018)"},{"why":"defines perfect representation, the axiom whose independence from IR is strengthened to liquid profiles in Theorem 8.","marker":"(Sánchez-Fernández et al., 2017)"},{"why":"source of the Chamberlin–Courant rule that Theorem 10 proves satisfies strong JR and coincides with seq-CC on liquid profiles.","marker":"(Chamberlin and Courant, 1983)"},{"why":"first suggested reading approval ballots as transitive trust derived from delegation structure, the interpretation liquid profiles formalize.","marker":"(Boldi et al., 2011)"},{"why":"standard reference for the general multiwinner setting, including the counterexample showing strong JR is unsatisfiable there, which motivates the liquid-profile contrast.","marker":"(Lackner and Skowron, 2022)"}],"fun_headline_variants":["Transitive trust makes PAV easy to compute","Liquid profiles make PAV and d'Hondt equivalent","Strong proportional representation is guaranteed on liquid profiles","On liquid profiles, all strong axioms are equivalent","Delegation chains unlock strong proportional representation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every voter designates exactly one trusted agent and that the voter's ballot lists precisely the people reachable through that single chain of trust; if voters approve anyone outside the chain, split their trust among several proxies, or stop trusting after one step, the paper's guarantees are not claimed to hold—and the authors acknowledge this modeling choice as a limitation in their Conclusion.","fun_headline_variants_meta":{"raw":{"variants":["Transitive trust makes PAV easy to compute","Liquid profiles make PAV and d'Hondt equivalent","Strong proportional representation is guaranteed on liquid profiles","On liquid profiles, all strong axioms are equivalent","Delegation chains unlock strong proportional representation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001505,"raw_usage":{"total_tokens":6014,"prompt_tokens":902,"completion_tokens":5112,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":518,"completion_tokens_details":{"reasoning_tokens":5040}},"tokens_in":518,"tokens_out":5112,"duration_ms":36796,"temperature":1.0,"reasoning_tokens":5040,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:06:30.675059+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate all liquid profiles on small voter sets—all functional delegation graphs on, say, up to eight vertices, with committee sizes up to three—and compute, under one fixed tie-breaking order, the committees returned by Liquid d'Hondt, sequential PAV, and PAV; a single profile where the three outcomes differ would refute the central equivalence of Theorem 12 and with it the polynomial-time claim for PAV, and because the claim is a finite statement about exactly this class of graphs, exhaustive search would settle it.","supporting_citations":[],"review_version":1}