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This paper proves that sharing information can raise the accuracy of a group's best single guess while lowering the group's chance of finding what it seeks, and that the loss is a failure of the action rule, not of the information.

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2026-08-01 16:15 UTC pith:TXC6NDYI

load-bearing objection A genuinely exact benchmark for consensus-driven discovery loss; the central paradox holds, and the only real caveat is that the market result is an anonymity benchmark, not a prediction for all self-interested equilibrium play. the 2 major comments →

arxiv 2607.18045 v1 pith:TXC6NDYI submitted 2026-07-20 cs.AI cs.GTcs.MA

The Shared Discovery Paradox: How a One-Answer Rule Turns Better Information into Worse Search

Yohei Nakajima This is my paper
classification cs.AI cs.GTcs.MA MSC 91A1090B4091B06
keywords collective discoveryconsensusprotocol lossprice of anarchycongestion gamesinformation aggregationcorrelated signalswater-filling equilibrium
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that an organization can become better informed and simultaneously less likely to discover its target, because a one-answer protocol converts a portfolio of available actions into one repeated choice. In a sixteen-box, eight-searcher benchmark, pooling noisy clues raises the best single recommendation from 20% to 38.35% accuracy, but repeating that recommendation drops group discovery from 83.22% under private clue-following to 38.35%. A coordinator using the same pooled reports and eight distinct actions reaches 85.94%, so the entire gap between consensus and the optimal portfolio is protocol loss. The same separation is then shown for self-interested searchers: the equal-split game has an exact water-filling equilibrium, a price of anarchy of 2 - 1/N, and a sole-rescue reward that makes every pure Nash equilibrium first-best.

Core claim

The central discovery is that the value of shared information and the quality of the action protocol are separable, and that a protocol which answers an estimation problem (pick the single most likely target) can destroy the solution to an allocation problem (spread attempts over the posterior). Formally, the paper defines the Shared Discovery Paradox as any protocol with higher average action quality Q but lower union success G than a diversified protocol. In the canonical instance, Q rises from 0.20 to 0.3835 while G falls from 0.8322 to 0.3835; the same pooled posterior, assigned as an eight-action portfolio, achieves 0.8594. The paper then shows that the identity LQ - G = E[(K-1)+] quant

What carries the argument

The load-bearing object is the attainable frontier V_L(F), the posterior mass of the best L-state portfolio given information F, and the protocol-loss identity L_L(Π;F) = V_L(F) - G_L(Π), which separates information value from action-rule efficiency. Alongside it, the redundancy identity LQ(Π) - G(Π) = E[(K-1)+] links average action quality to duplicated successful actions. For the market layer, the anonymous symmetric equilibrium of the equal-split game is characterized by the water-filling rule s_b = φ^{-1}(λ/π_b) for boxes with posterior mass above λ, and welfare is G = Nλ; this is what turns the congestion game into an exact potential game and yields the price-of-anarchy bound.

Load-bearing premise

The benchmark's market claim depends on selecting the anonymous symmetric equilibrium: if searchers can coordinate through role labels, conventions, or sequencing, the equal-split game also has efficient asymmetric pure equilibria, so the 0.5991 figure is an upper bound on the cost of anonymity rather than a robust prediction about self-interested search.

What would settle it

Run the equal-split game in the canonical 16-box environment with a minimal coordination device — for instance, a public pre-play ordering of searchers or a shared deterministic tie-breaking rule that can serve as a convention — and measure discovery. If it jumps from near 0.599 toward 0.859, the price-of-anarchy bound is not the binding constraint; alternatively, in an organization that pools information into one ranking and then lets agents act freely, track the number of distinct actions and the discovery rate: the paper predicts average-quality-up/discovery-down whenever duplication rises.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In the canonical sixteen-box benchmark, seven coordinated actions recover the 0.8322 discovery of eight independent clue-followers; the recovery budget quantifies how much duplication costs.
  • The exact mixed price of anarchy of the equal-split game is 2 - 1/N, and because the game is (1, 1-1/N)-smooth the bound extends to coarse correlated equilibria and any no-regret learning dynamics.
  • A sole-rescue reward, paying only an agent who covers the target alone, makes the game an exact potential game whose potential is social discovery; with at least as many boxes as searchers, every pure Nash equilibrium is a collision-free top-N portfolio.
  • Under a latent common-cue copying model, the centralized planner gain over private search is strictly increasing in copying probability c, and in the canonical environment the symmetric market overtakes decentralized report-following at c = 0.7884616565.
  • In the proportional large-market limit the five-protocol ordering is exact: consensus discovery vanishes while blind, market, private, and portfolio search converge to 0.500, 0.547, 0.847, and 0.874.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the 0.5991 market value relies on the anonymous symmetric equilibrium, a cheap-talk phase, role labels, or any sequencing could push actual groups toward the efficient asymmetric equilibria; the price of anonymity, not of self-interest, is the real bound.
  • The protocol-loss accounting should transfer directly to multi-agent AI systems: if agents share a pretrained model or retrieval context, adding agents can raise average prediction quality while collapsing effective channels; marginal-coverage rewards are a natural testable remedy.
  • Matching an observed duplication level (e.g., expected distinct proposals) to the benchmark's exact curve could pin down a latent copying rate and thereby a recovery budget, giving an empirical calibration device without structural econometrics.
  • The sparse-evidence limit suggests a sharp falsifiable prediction: consensus-style one-answer protocols are most dangerous in domains where evidence is sparse and the target's signal never separates from noise — early-stage science, venture sourcing, novel-hypothesis generation — and less harmful where evidence is abundant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies a sixteen-box, eight-searcher discovery benchmark with a single target, a uniform prior, and conditionally independent private clues that are correct with probability p=0.20. It claims and demonstrates a 'Shared Discovery Paradox' (Definition 1): pooling the clues raises the accuracy of the best single recommendation from 0.20 to 0.3835, yet the one-answer consensus protocol that repeats that recommendation yields group discovery 0.3835, below private clue-following (0.8322), below a blind coordinated eight-box portfolio (0.50), and far below a coordinator who uses the same pooled reports but assigns one searcher to each of the eight highest-posterior boxes (0.8594). The paper formalizes this as protocol loss L_8 = V_8 - G_consensus = 0.475952, computes the exact action-budget frontier with integer recovery budget L* = 7, and derives the identity NQ - G = E[(K-1)_+]. It then replaces the planner with an equal-split game on the pooled posterior: an exact-potential game with a unique anonymous symmetric water-filling equilibrium (canonical discovery 0.5991), a worst-case mixed price of anarchy 2 - 1/N with a smoothness proof and tight example, and a sole-rescue reward that makes every pure Nash equilibrium first-best. A latent common-cue copying model (copying probability c) yields a closed-form private-discovery function, a strict monotone-planner-gain theorem (Theorem 2), and a computational crossover c* = 0.7884616565 at which the symmetric market overtakes

Significance. If the claims hold, the paper delivers a compact and fully explicit benchmark separating information value (one-action value, action-budget frontier), action allocation (protocol loss, recovery budget), decentralized incentives (water-filling equilibrium, price of anarchy), and signal dependence (channel collapse, planner-gain monotonicity). The central paradox is established by exact enumeration over count vectors and is reproduced by the public verification script; the price-of-anarchy bound is parameter-free, properly attributed to the covering-game literature, and proved by a short smoothness argument that also covers coarse correlated equilibria; Theorem 2 is an analytic monotonicity result; the proportional limit is exact. The claim-status taxonomy in Appendix E, distinguishing proved results, exact enumerations, and computational patterns, is exemplary and materially strengthens the paper. The authors also do the right thing in disclosing caveats: the market row (0.5991) is an anonymous-symmetric-equilibrium benchmark, the copying crossover is a canonical computational result, and the decentralized-team optimum is left open. The main residual risk is that Section 4's 'below

major comments (2)
  1. [Definition 1] Definition 1 defines the paradox as Q(Π_S) > Q(Π_P) and G(Π_S) < G(Π_P) without requiring the two protocols to have the same action budget L. Because Q is an average over L actions, an unequal-budget comparison can manufacture or destroy the paradox trivially: in the canonical environment, taking L_P = 1 gives G(Π_P) = Q(Π_P) = 0.20, so the stated inequalities fail even though the substantive finding is unchanged. The canonical application compares protocols with L = N = 8, and Section 2.2 already handles cross-budget questions through the frontier, so the fix is local: add a requirement that the protocol comparison is at a common action budget (or define the paradox for the best-L expansion of each protocol). As written, the formal definition is looser than the claim it supports.
  2. [Table 1 / Section 9 / Abstract] The market-layer headline — 'the equal-split game ... achieves 0.5991: strictly above consensus, but below both private search and the planner' (Abstract) — is a property of the anonymous symmetric equilibrium, not of the full equilibrium set. As the paper itself states in the Table 1 footnote and in Section 9, the equal-split game also has efficient asymmetric pure equilibria that reach the first-best portfolio whenever role labels, conventions, or sequencing are available. The 0.5991 value is therefore an anonymity benchmark — a measure of how much dispersion prices alone supply without coordination devices — rather than a robust prediction about self-interested search under every equilibrium. This does not affect the paradox or the protocol-loss diagnosis, and I agree with the authors' disclosure, but the qualifier should be surfaced in the Abstract and in the interpretive paragraphs
minor comments (5)
  1. [Theorem 1, Section 4.3] The proof line 'assigning one optimal box o_i to each searcher i (ignore zero-mass padding if M<N)' leaves the M<N case to the reader. Since the theorem is asserted for every posterior, please spell out the padding construction (add zero-mass states so the number of states reaches N) so that the stated claim is covered by the given argument.
  2. [Appendix C, Theorem 2] The sentence 'The inequality is strict for at least one k' should be made precise. The coupling shows h(k+1) ≥ h(k), with equality at k=0 because h(0)=h(1) (one copy of X0 has the same joint law as one independent clue). The strict event should be exhibited for some k ≥ 1, e.g., k = N-1, where the converted independent clue is the unique report naming the target and the common cue misses; the derivative identity then yields strict positivity on (0,1) since P(K'=N-1) > 0 for c > 0. Currently the text does not distinguish the k=0 case.
  3. [Appendix D, Proposition 4] The convergence step for the market equilibrium is compressed into one sentence. The finite (random) budget functions B_M(Λ) are decreasing in Λ; pointwise convergence in probability at each fixed Λ, combined with monotonicity and the strict monotonicity of the limit B, gives locally uniform convergence and justifies the root-squeeze Λ_M → Λ. A brief elaboration of this argument — and of the asserted covariance bound for E[r^{C_a} r^{C_b}] for two false boxes — would make the proof of a headline result self-contained.
  4. [Section 6.2] The likelihood values P(C|θ=a)=1.1427×10^{-4} and P(C|θ=b)=1.3977×10^{-4} should be identified as script-computed values, and the sentence that the Bayes-versus-count discount 'vanishes for L≥3 on the tested grid' should carry the 'computational pattern' label in the main text, as Appendix E already does.
  5. [Section 4.2, Proposition 6] The uniqueness claim for the anonymous symmetric equilibrium does not address the boundary tie π(1)/N = π(2). A parenthetical noting that the boundary is covered by the pure-corner case, or by taking the corresponding limit in the water-filling formula, would make the statement airtight. Also, unify the internal cross-reference style ('section D' / 'Appendix D'; 'section C' / 'Appendix C').

Circularity Check

0 steps flagged

No significant circularity: the central derivation is self-contained, and the only self-referential element is a reproducible verification repository, not a load-bearing citation loop.

full rationale

The derivation chain is self-contained. Definition 1 names a pattern; its canonical instance is established by exact enumeration (Appendix A, Eq. 28) rather than by assuming the ordering. Proposition 3 is a set-inclusion argument: the plurality winner carries at least one report, so consensus success implies at least one private clue success; the strict values 0.3835 vs 0.8322 are computed, not imposed. The market value 0.5991 solves the water-filling equations (Section 4.2, Eq. 14) from the pooled posterior; it is not fitted to the quantity it claims to explain. The 2−1/N price-of-anarchy bound is explicitly attributed to Gairing (2009) and also proved directly in Theorem 1; it is external support, not a self-citation. The sole-rescue implementation follows from the potential being covered mass (Proposition 8), and the copying results are either proved (Theorem 2) or explicitly labeled as computational patterns on a grid (Section 6.4). The only self-referential element is the author's verification repository, which is independent reproducible code and therefore counts as real evidence under the review rules. The Table 1 market row is flagged by the paper itself as an equilibrium-selection benchmark, not a claim about all Nash equilibria, so the existence of efficient asymmetric equilibria does not make the 0.5991 row a disguised fit or prediction. No step reduces a claimed output to an input by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 8 axioms · 1 invented entities

The central claims are derived from an explicitly stated Bayesian search model. The only non-standard inputs are the benchmark constants and the equilibrium-selection assumption for the market; these are disclosed. No parameters are fitted to external data.

free parameters (4)
  • canonical benchmark constants M=16, N=8, p=0.20 = M=16, N=8, p=0.20
    Chosen by hand to make the paradox visible while keeping exact enumeration feasible; not fitted to data, and the main propositions are stated for general M,N,p.
  • uniform randomization at cutoff ties = uniform tie-breaking
    Modeling choice in the exact frontier formula (Appendix A); affects numerical values at small budgets.
  • copying probability c = c in [0,1]; crossover c*=0.788462
    Model parameter for the common-cue process; the crossover value is solved, not fitted.
  • proportional scaling alpha and r = alpha=1/2, r=3.75
    Large-market limit constants chosen to match the canonical benchmark; the limit formulas are stated for general alpha,r.
axioms (8)
  • domain assumption Uniform prior and conditionally independent clues with P(X_i=theta|theta)=p, P(X_i=b|theta)=q for b≠theta.
    Defines the information structure of the benchmark (Section 2).
  • domain assumption Each searcher opens exactly one box; group succeeds if any opened box is the target.
    The discovery objective and action constraint (Section 2).
  • domain assumption Equal split of the unit prize among successful searchers.
    Defines the congestion game in Section 4; alternative credit rules would change payoffs.
  • ad hoc to paper Anonymous symmetric equilibrium as the market selection.
    The 0.5991 market value is computed at this equilibrium; the authors acknowledge efficient asymmetric pure equilibria exist (Table 1 note, Section 9).
  • domain assumption M>=N and strictly positive posterior mass for sole-rescue implementation.
    Needed for Proposition 8 to guarantee every pure Nash equilibrium is collision-free and top-N (Section 5).
  • domain assumption One latent common cue X0 with copying probability c.
    The dependence model in Section 6; a single common source is a modeling choice.
  • domain assumption Sparse-evidence scaling N_M/M->alpha with fixed likelihood ratio r>1.
    The proportional large-market limit (Section 3.2) requires this scaling; under abundant evidence the paradox vanishes (Remark 2).
  • standard math Standard potential game and smoothness results (Rosenthal, Roughgarden).
    Used for existence of pure equilibria and robust PoA bounds (Section 4).
invented entities (1)
  • Latent common cue X0 independent evidence
    purpose: Generates correlated reports so that copying collapses effective discovery channels.
    The paper provides a falsifiable handle: duplication levels in observed proposals can be matched to the exact E[D](c) curve to infer c, which then implies recovery budget and planner gain (Section 8). It is a model construct, not a physical entity.

pith-pipeline@v1.3.0-alltime-deepseek · 20886 in / 18339 out tokens · 173015 ms · 2026-08-01T16:15:12.498607+00:00 · methodology

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read the original abstract

Organizations often pool dispersed information into one ranking and then allow many agents to act on that shared view. In a discovery problem, this can improve beliefs while reducing coverage. We develop an exactly solvable benchmark with sixteen boxes, one target, eight searchers, and noisy private clues. Pooling raises the accuracy of the best single recommendation from 0.20 to 0.3835, but repeating that recommendation lowers group discovery from 0.8322 under decentralized clue-following to 0.3835. A coordinated eight-action portfolio using the same pooled reports reaches 0.8594, and seven coordinated actions recover the decentralized benchmark. The paradox is a protocol failure, not an information failure: a one-answer rule compresses a portfolio of available actions into one repeated choice. We then replace the planner with self-interested searchers who split a prize. The equal-split game is a potential game. Its anonymous symmetric equilibrium obeys a water-filling rule. In the canonical instance it achieves 0.5991: strictly above consensus, but below both private search and the planner. The exact mixed price of anarchy is 2 - 1/N. A sole-rescue reward, which pays only an agent who covers the target alone, makes every pure Nash equilibrium first-best. Finally, a latent common-cue model shows how correlated reports collapse effective discovery channels. The centralized planner gain rises strictly with copying, and in the canonical environment the symmetric market overtakes decentralized report-following at copying probability c = 0.788462. In a proportional large-market limit the five-protocol ordering survives exactly: consensus discovery vanishes while blind, market, private, and portfolio search converge to 0.500, 0.547, 0.847, and 0.874. The contribution is a compact benchmark that separates information, allocation, incentives, and dependence into exact, reusable quantities.

Figures

Figures reproduced from arXiv: 2607.18045 by Yohei Nakajima.

Figure 1
Figure 1. Figure 1: The exact pooled-information frontier for the canonical benchmark. The one-action consensus value is only the left endpoint. Seven coordinated actions recover the performance of eight decentralized clue-followers. 2.2 The full action-budget frontier Consensus evaluates pooled information at a one-action budget. Discovery is instead a budget-indexed frontier. Exact enumeration gives Actions 𝐿 Pooled discove… view at source ↗
Figure 2
Figure 2. Figure 2: Exact ex-ante discovery under four protocols in the independent sixteen-box benchmark. The labels inside the bars report the expected number of distinct boxes searched. The dashed line marks the blind coordinated portfolio at 8/16 = 0.50: informed consensus falls below a protocol that ignores the clues entirely. Equal sharing supplies partial dispersion, but not enough to match natural coverage from indepe… view at source ↗
Figure 3
Figure 3. Figure 3: Exact canonical planner gain 𝑉8 (𝑐) − 𝐺private (𝑐). As common-source copying creates more duplicated capacity, identifying and reallocating it becomes more valuable. planner gain Δplanner(𝑐) is strictly increasing for 𝑐 ∈ (0, 1). At full copying, Δplanner(1) = (𝑁 − 1)𝑞. (24) The proof is in section C. Its key identity is Δplanner(𝑐) = E  1{private search misses} 𝑁 − 𝐷 𝑀 − 𝐷  . (25) All named boxes preced… view at source ↗
Figure 4
Figure 4. Figure 4: Exact canonical comparison under common-source copying. The symmetric equal-split market initially trails decentralized report-following, then overtakes it when channel collapse becomes sufficiently severe. The crossover is a canonical computational result, not a general theorem. 7 Why this is a benchmark A useful benchmark is not a universal law. It is a small model that isolates a mechanism, supplies sta… view at source ↗

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