{"id":"54af530b-fd33-44ef-9b76-f314937a1d48","arxiv_id":"2607.08744","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Randomized calibrated expert aggregation that Blackwell-refines a target is polynomial-time solvable; deterministic aggregation is NP-hard and has no multiplicative PTAS for proper losses.","lead":"This paper shows how to combine several calibrated Bayesian experts into one new calibrated expert that is as informative as possible given only the experts' reports and the prior. Randomized combinations can be found efficiently, while deterministic ones are computationally hard.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the linear-observability characterization as foundational; I agree it is the single most load-bearing modeling step. However, the manuscript supplies a complete proof that the condition is necessary and sufficient for uniform posterior consistency, so the assumption does not constitute an open vulnerability. All algorithmic and hardness arguments build cleanly on top of it. Consequently the ACCEPT verdict stands; no adjustment is warranted. The concrete test above would still be worth running as an independent sanity check of the hardness core.","tokens_in":46907,"tokens_out":545,"duration_ms":23301,"concrete_test":"Instantiate the hardness construction (Ex. 6.1 / Lem. 6.4) for the RestrictedSubsetSum instance c=(1,2), B=1 (A=3, m=2, L=2, n=6). Explicitly form the 5 routing vectors of f_{1},f_{2}, compute a basis for Row(A), and verify dim(S)=3 and S=S_hard by checking that every generator satisfies the block-equality + weighted-sum equation and that the three free coordinates (item values + common +block + common -block) subject to one linear relation span exactly the same space. If equality fails, the binary-vector equivalence (Lem. 6.6) collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central dichotomy (poly-time randomized Search/OPT via finite-prediction lex search + source-labelled LPs; NP-hardness / no PTAS for deterministic via binary vectors in the observable cone) rests on the exact characterization that constructible experts are precisely those assembled from atoms in K = Row(A) ∩ R_{+}^{n} (Def. 3.3, Prop. 3.1, Lem. 3.2–3.4). This is proved, not assumed: linear observability of Y(v) holds iff v ∈ S, and every such atom yields a prediction that is posterior-consistent for every latent q compatible with the inputs. The subsequent reductions (extreme-ray finite support, n+2(M+1) sparsity via pruned vertex counting in ExProg, Stern-Brocot recovery of (nW)^{10n^{3}}-bounded rationals, martingale source labels for Blackwell, and the amplified SubsetSum encoding of nontrivial binary vectors) are self-contained and appear free of hidden gaps under the stated finite-support / rational-input model. No load-bearing inconsistency or unproved leap threatens the claim.","agreement_with_reader":"partial"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces expert aggregation: given a known prior over a finite state space and a collection of calibrated (posterior-consistent) input experts, but without access to the latent Bayes probabilities, construct a new calibrated expert that is undominated under Blackwell dominance among all experts that can be assembled from the observable information, optionally refining a designated target expert, and (when a proper loss is given) nearly minimizing expected proper loss among such refinements. Constructibility is characterized exactly by membership of reporting components in the observable nonnegative cone K = Row(A) ∩ R₊ⁿ induced by the input experts’ state-wise reporting vectors. The main algorithmic dichotomy is that randomized outputs admit a polynomial-time algorithm for Search-Aggregation (via extreme-ray reduction, lexicographic optimization, small-support vertex counting, and Stern-Brocot feasibility oracles) and an additive FPTAS for OPT-Aggregation (via source-labelled martingale LPs with piecewise-linear upper approximations of regular proper losses), while deterministic outputs are NP-hard for Search-DetAgg even with two experts and a constant base-rate target, and admit no multiplicative PTAS for Brier-loss OPT-DetAgg unless P=NP (via an amplified SubsetSum construction that forces nontrivial binary vectors into the observable space).","tokens_in":47197,"tokens_out":727,"duration_ms":6966,"significance":"If the results hold, the paper cleanly separates a new, modular notion of expert aggregation (calibrated, Blackwell-undominated, and reusable without access to the original experts) from classical and robust forecast aggregation. The randomized side supplies concrete poly-time and FPTAS algorithms that work from only the prior and the input reporting rules; the deterministic side supplies matching hardness even in highly restricted two-expert instances. The linear-observability characterization (Proposition 3.1, Lemmas 3.2–3.4) and the subsequent reductions (extreme-ray finite support, n+2(M+1) sparsity, Stern-Brocot recovery, source-labelled martingales, amplified SubsetSum) are self-contained and free of circularity. These contributions are of clear interest to algorithmic game theory, information design, and forecast aggregation.","major_comments":[],"minor_comments":[{"comment":"The numerical bound (nW)^{10n^{3}} in Lemma 4.10 is extremely loose; a short remark that any poly(n) exponent suffices for the Stern-Brocot argument would improve readability without changing the claim.","section":null},{"comment":"Figure 1 and the accompanying three-state example are helpful; a second small worked example that also illustrates the source-labelled LP (LPε) would make the FPTAS more accessible.","section":null},{"comment":"Definition 5.1 (regular proper loss) is used only for the FPTAS; a one-sentence pointer that Brier and log loss satisfy it (via Corollary 5.2) already appears, but placing it earlier would help readers who skip the appendix.","section":null},{"comment":"A few typographical slips remain (e.g., “Aggrega tion”, “ExRayDec”, occasional missing spaces after periods). A light copy-edit pass would clean them.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical but the central dichotomy is cleanly proved. I see no load-bearing gap that would justify major revision or rejection; the reader’s and skeptic’s assessments align with my own reading. Fit for a theory-oriented CS/GT venue is excellent."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The new piece is the expert-aggregation problem itself: take reduced-form calibrated experts plus the prior, never see the latent Bayes probabilities, and build a new calibrated expert that Blackwell-refines a target and is undominated (or nearly loss-optimal) among constructible ones. The linear characterization is clean—observable space S = Row(A) from the input reporting vectors, cone K = S ∩ R_{+}^{n}, and constructible experts are exactly those assembled from atoms in K. That turns the whole thing into linear algebra plus martingale couplings.\n\nWhat they deliver is a sharp randomized/deterministic split. Randomized Search-Aggregation is poly-time: extreme-ray reduction to a finite prediction set, lex objective whose optimum is undominated, small-support vertex bound n+2(M+1) via pruned binding constraints in the exponential ExProg, then Stern-Brocot search with FeasProg oracles that recover the bounded-rational predictions. OPT gets an additive FPTAS by piecewise-linear upper approximation of the Bayes risk plus source-labelled LPs that enforce Blackwell exactly. Deterministic is hard even for two experts and a constant base-rate target: they encode Restricted SubsetSum into the observable binary vectors of an amplified hard instance, so Search-DetAgg is NP-hard and Brier OPT-DetAgg has no multiplicative PTAS. Proofs look complete and self-contained under the finite-support rational model; the stress-test found no load-bearing gap, and I agree.\n\nSoft spots are minor and mostly modeling. Everything is finite-support and rational-input; continuous or irrational cases are left open. Regularity of the proper loss is needed for the FPTAS, but that is standard and covers Brier/log. The multi-target extension is sketched and works for fixed |T|. No experiments, no code, but this is pure theory and does not claim otherwise. Citations sit correctly inside robust aggregation and information design.\n\nThis is for people who care about algorithmic information design, modular expert systems, or Blackwell orderings under partial observability. It deserves a serious referee. I would bring it to reading group and expect to cite the dichotomy.","headline":"Clean algorithmic dichotomy for calibrated expert aggregation: poly-time randomized search/OPT via observable cones, NP-hard deterministic even with two experts.","tokens_in":47757,"tokens_out":526,"would_cite":true,"duration_ms":7440,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"When output experts may randomize, both finding an undominated calibrated refinement of a target and nearly minimizing proper loss among such refinements can be done efficiently; deterministic outputs make even the search problem NP-hard.","keywords":["expert aggregation","Blackwell dominance","calibration","forecast aggregation","proper loss","computational complexity","observable linear information"],"falsifier":"Exhibit two distinct Bayes-probability vectors that induce identical input-expert reports yet assign different conditional outcome means to some nonnegative vector outside the claimed observable cone; such a counter-example would falsify the completeness of the linear characterization.","tokens_in":47796,"feed_emoji":"🔀","tokens_out":611,"duration_ms":10972,"temperature":0.7,"pith_summary":"The paper reframes forecast aggregation as expert aggregation: given calibrated input experts and a known prior over states, but without the latent Bayes probabilities, build a new calibrated expert that Blackwell-refines a target and cannot be improved further with the available information. Constructibility is completely determined by an observable linear system whose nonnegative cone tells exactly which output prediction components can be certified. With randomization allowed, both the search for an undominated refinement and additive approximation of proper-loss optimization admit polynomial-time algorithms. Forcing the output expert to be deterministic collapses the problem to an integrality question that is already NP-hard with two experts and a constant base-rate target, and admits no multiplicative PTAS for Brier loss unless P equals NP. The result therefore draws a sharp computational line between randomized and deterministic calibrated aggregation.","feed_headline":"Randomized experts aggregate in poly-time; deterministic ones don't","feed_subtitle":"A linear cone decides which calibrated refinements are possible, drawing a sharp complexity line.","key_machinery":"The observable nonnegative cone K = Row(A) ∩ R₊ⁿ, where A is the signal-state matrix whose rows are the input experts’ state-wise reporting-probability vectors; membership in K exactly certifies which output atoms can be given identifiable posterior means, and every algorithmic and hardness result is built on linear optimization or binary-vector search inside this cone.","core_discovery":"A calibrated output expert is constructible from the observed inputs if and only if its reporting components lie in the observable nonnegative cone generated by the row space of the input experts’ state-wise reporting matrix; under this characterization, randomized aggregation admits efficient algorithms for both undominated refinement search and additive proper-loss optimization, while deterministic aggregation is computationally intractable even in the simplest two-expert setting.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Randomized expert aggregation is poly-time; deterministic is NP-hard","Observable cone decides constructible calibrated refinements","Randomized refinements admit efficient algorithms; deterministic do not","Linear row-space cone unlocks poly-time randomized expert aggregation","Calibrated output experts exist via cone; determinism is intractable"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The claim that every constructible expert is completely characterized by components lying inside the observable nonnegative cone rests on the premise that linear consequences of the input experts’ calibration equations recover every identifiable outcome mass.","fun_headline_variants_meta":{"raw":{"variants":["Randomized expert aggregation is poly-time; deterministic is NP-hard","Observable cone decides constructible calibrated refinements","Randomized refinements admit efficient algorithms; deterministic do not","Linear row-space cone unlocks poly-time randomized expert aggregation","Calibrated output experts exist via cone; determinism is intractable"]},"model":"grok-4.5","effort":"low","cost_usd":0.006874,"raw_usage":{"total_tokens":1727,"prompt_tokens":824,"num_sources_used":0,"completion_tokens":86,"cost_in_usd_ticks":68740000,"prompt_tokens_details":{"text_tokens":824,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":817,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":824,"tokens_out":86,"duration_ms":7795,"temperature":1.0,"reasoning_tokens":817,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T05:00:39.814792+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit two distinct Bayes-probability vectors that induce identical input-expert reports yet assign different conditional outcome means to some nonnegative vector outside the claimed observable cone; such a counter-example would falsify the completeness of the linear characterization.","supporting_citations":[],"review_version":2}