{"id":"d56e5818-22b3-444f-b688-15526f20d6d7","arxiv_id":"2606.22021","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces WCSP axiom and states a necessary and sufficient condition for the anti-plurality rule with fixed-order tie-breaking to satisfy it, expressed in terms of the numbers of agents and alternatives.","lead":"The paper introduces worst-case strategy-proofness (WCSP), a new non-manipulability axiom for voting rules that sits between full strategy-proofness and non-obvious manipulability-worst. A smart generalist might read it to understand precise conditions under which common voting rules resist manipulation in worst-case scenarios.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest_assumption correctly flags the modeling choice as the point where the necessity/sufficiency would fail to apply, but that is an external-scope issue rather than an internal flaw in the claimed characterization. With only the abstract available and no visible proof steps or counter-examples to inspect, no load-bearing technical concern can be isolated.","tokens_in":1626,"tokens_out":289,"duration_ms":19148,"concrete_test":"For the smallest pairs (|N|,|A|) satisfying and violating the claimed numerical condition, enumerate all preference profiles, compute the anti-plurality outcome under the fixed tie-breaker, and directly check the WCSP definition (no agent has a profitable deviation in the worst-case scenario over others' reports); confirm the condition correctly separates the cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a characterization result: a necessary and sufficient numerical condition (in terms of |N| and |A|) under which anti-plurality with fixed-order tie-breaking satisfies WCSP. The modeling assumptions (complete transitive preferences, finite A, fixed known tie-breaker) are standard and explicitly stated; the abstract positions WCSP between strategy-proofness and NOM-worst without internal contradiction. No derivation, counter-example construction, or edge-case handling is visible that would allow identification of a hidden assumption or incorrect step.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces worst-case strategy-proofness (WCSP) as a non-manipulability axiom that is weaker than strategy-proofness but stronger than NOM-worst. In a standard voting model with complete transitive preferences, it shows that plurality, Borda, and Dowdall rules violate WCSP, and states a necessary and sufficient numerical condition (in terms of the numbers of agents |N| and alternatives |A|) under which the anti-plurality rule with fixed-order tie-breaking satisfies WCSP.","tokens_in":1745,"tokens_out":275,"duration_ms":24499,"significance":"If the stated characterization holds and is correctly derived, the result would be a clean, parameter-free delineation of the boundary for WCSP compliance in one specific rule. This could usefully extend the literature on intermediate manipulability axioms by identifying exact size thresholds separating compliance from violation.","major_comments":[{"comment":"Abstract: the manuscript asserts the existence of a necessary and sufficient condition for anti-plurality with fixed-order tie-breaking to satisfy WCSP, but provides neither the derivation, supporting lemmas, nor any verification that the condition indeed captures WCSP. Without these elements the central claim cannot be evaluated.","section":"Abstract"}],"minor_comments":[],"recommendation":"uncertain","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and for highlighting this issue with the presentation of the central result. We address the comment below.","responses":[{"response":"The manuscript states the necessary and sufficient numerical condition on |N| and |A| under which anti-plurality with fixed-order tie-breaking satisfies WCSP. We agree, however, that the derivation, supporting lemmas, and explicit verification that the condition is indeed necessary and sufficient are not adequately developed or presented. We will revise the manuscript to supply these elements in full.","revision_made":"yes","referee_comment":"[Abstract] Abstract: the manuscript asserts the existence of a necessary and sufficient condition for anti-plurality with fixed-order tie-breaking to satisfy WCSP, but provides neither the derivation, supporting lemmas, nor any verification that the condition indeed captures WCSP. Without these elements the central claim cannot be evaluated."}],"tokens_in":1175,"tokens_out":207,"duration_ms":29761,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main contribution is the new WCSP axiom and the clean size-based condition under which anti-plurality satisfies it. The abstract positions WCSP as weaker than full strategy-proofness but stronger than NOM-worst, and it shows that plurality, Borda, and Dowdall all fail the new axiom while passing the older one. The necessary-and-sufficient condition is stated directly in terms of the numbers of agents and alternatives, which is the sort of checkable result that can be useful for applied work on voting rules.\n\nThe modeling setup is standard: complete transitive preferences over a finite set of alternatives with a known fixed-order tie-breaker. That keeps the claim grounded and avoids hidden parameters. The paper does not claim the condition extends to other rules, which is honest.\n\nThe obvious limitation is scope. Only anti-plurality receives the characterization, and the abstract gives no indication that the result generalizes or changes how we think about larger classes of mechanisms. Without the full proof I cannot verify the derivation steps, but nothing in the stated claim looks circular or dependent on fitted quantities.\n\nThis is a narrow but precise incremental paper in social choice. Readers who care about non-manipulability distinctions in voting models will get value from the axiom definition and the size condition. It is the kind of work that belongs in a specialized journal and deserves referee time rather than a desk reject.","headline":"The paper defines WCSP as an axiom strictly between strategy-proofness and NOM-worst, then gives a numerical necessary-and-sufficient condition for anti-plurality with fixed tie-breaking.","tokens_in":2206,"tokens_out":362,"would_cite":false,"duration_ms":13689,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The anti-plurality rule with fixed-order tie-breaking satisfies worst-case strategy-proofness exactly when the numbers of agents and alternatives meet a specific numerical condition.","keywords":["worst-case strategy-proofness","anti-plurality rule","voting rules","strategy-proofness","non-obvious manipulability","tie-breaking","social choice"],"falsifier":"For any specific pair of agent count n and alternative count m that the derived condition claims satisfies WCSP, exhibit one preference profile and one misreport under which the deviator obtains a strictly better outcome in every possible tie resolution.","tokens_in":2518,"feed_emoji":"🗳","tokens_out":603,"duration_ms":25589,"temperature":0.7,"pith_summary":"The paper defines worst-case strategy-proofness, an axiom requiring that no voter can gain in the worst-case outcome by misreporting preferences. This axiom lies strictly between full strategy-proofness and the weaker non-obvious manipulability-worst. Several common rules, including plurality, Borda, and Dowdall, satisfy the weaker notion yet fail WCSP. For the anti-plurality rule with fixed-order tie-breaking the paper supplies a necessary and sufficient condition stated purely in terms of the counts of agents and alternatives.","feed_headline":"Anti-plurality rule satisfies WCSP only under specific agent-alternative counts","feed_subtitle":"A necessary and sufficient numerical condition determines when the rule resists profitable worst-case deviations.","key_machinery":"Worst-case strategy-proofness (WCSP), which demands that truthful reporting remains optimal for every agent even when all ties are resolved against the deviator.","core_discovery":"The central claim is that the anti-plurality rule with fixed-order tie-breaking meets worst-case strategy-proofness if and only if the numbers of agents and alternatives obey a stated numerical relation; the relation is derived directly from the requirement that no profitable worst-case deviation exists under the rule.","pith_inferences":["The numerical condition may indicate for which small electorates anti-plurality can be used without fear of worst-case manipulation.","Similar conditions could be derived for other scoring rules or for randomized tie-breaking.","The gap between WCSP and NOM-worst suggests that many rules remain vulnerable only when agents consider the most adverse tie outcomes."],"forward_implications":["Plurality, Borda, and Dowdall rules all violate WCSP.","Anti-plurality satisfies WCSP precisely under the identified numerical condition on agents and alternatives.","WCSP is strictly stronger than NOM-worst yet weaker than full strategy-proofness."],"fun_headline_variants":["Anti-plurality WCSP tied to agent and alternative counts","Condition on numbers controls anti-plurality WCSP","Anti-plurality rule gets WCSP at specific sizes only","WCSP for anti-plurality requires matching agent-alternative count"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The model assumes agents hold complete transitive preferences over a finite set of alternatives and that the tie-breaking rule is fixed and known in advance.","fun_headline_variants_meta":{"raw":{"variants":["Anti-plurality WCSP tied to agent and alternative counts","Condition on numbers controls anti-plurality WCSP","Anti-plurality rule gets WCSP at specific sizes only","WCSP for anti-plurality requires matching agent-alternative count"]},"model":"grok-4.3","cost_usd":0.007562,"raw_usage":{"total_tokens":3403,"prompt_tokens":541,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":75624500,"prompt_tokens_details":{"text_tokens":541,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2794,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":541,"tokens_out":68,"duration_ms":25184,"temperature":1.0,"reasoning_tokens":2794,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T10:59:10.980686+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"For any specific pair of agent count n and alternative count m that the derived condition claims satisfies WCSP, exhibit one preference profile and one misreport under which the deviator obtains a strictly better outcome in every possible tie resolution.","supporting_citations":[],"review_version":1}