Compressing many private clues into one shared recommendation raises the top pick's accuracy but collapses group discovery as agents repeat that single choice; allowing coordinated portfolios or changing rewards (pay only sole discoverers) restores exploration and can achieve first-best discovery.

Main Finding

Pooling agents' private signals into a single recommended choice can increase the quality of that recommendation while drastically reducing group discovery. In a small, exactly solvable benchmark (16 boxes, one target, 8 searchers, private clue accuracy p = 0.20) the paper shows that consensus (everyone follows the pooled posterior mode) raises average action quality above an individual clue but compresses the group's action portfolio so much that discovery probability falls below decentralized private search. The loss is not an information failure but a protocol (allocation) failure: compressing many available actions into one repeated choice creates redundant effort and destroys coverage.

Key finite numbers (canonical instance) - Private clue-following (each opens own clue): G_private = 0.83222784 - Shared consensus (everyone follows pooled posterior mode): G_consensus = 0.38346871 - Planner (assigns 8 distinct highest-posterior boxes): G_planner = 0.85942125 - Blind coordinated portfolio (no information, 8 distinct boxes): G_blind = 0.50 - Market (equal-split game, symmetric equilibrium on pooled posterior): G_market = 0.599099 - Protocol loss under pooling (same info, 8-action budget): 0.85942125 − 0.38346871 = 0.475952 - Recovery budget: assigning L* = 7 coordinated searches to pooled top boxes recovers private-search performance.

Key Points

• Definition: Shared Discovery Paradox — a protocol where shared information increases average action quality but reduces union success (coverage).
  • Relation: L · Q(Π) − G(Π) = E[(K − 1)+], so concentrating on one box raises Q by increasing redundant hits K.
• Mechanism: Pooling transforms a diversified set of candidate actions into a single repeated action; the resulting collisions (duplicates) are the source of failure.
• Benchmarks and ordering (canonical): 0.20 (single private clue quality) < G_consensus = 0.3835 < G_private = 0.8322 < G_planner = 0.8594.
• Protocol loss is large: in the canonical pooled-information regime the entire 47.6-percentage-point gap between consensus and planner is attributable to protocol loss.
• Diversification value: even uninformed but coordinated diversification (blind portfolio) beats informed consensus (0.50 > 0.3835).
• Recovery budget: with pooled information, a planner needs only 7 coordinated actions (not 8) to match decentralized private-search coverage.
• Self-interested searchers (equal-split game):
  • Game is an exact potential (Φ(a) = Σ_b π_b H_{n_b(a)}, H_k harmonic numbers). At any profile total expected payoff = discovery probability.
  • Anonymous symmetric equilibrium has a water-filling characterization: agents trade off posterior mass vs expected dilution from rivals.
  • Canonical symmetric-market discovery: G_market = 0.599099; expected distinct searches ≈ 2.67 (so symmetry still yields heavy collisions).
  • Exact mixed Price of Anarchy: factor = 2 − 1/N (for N agents). For N = 8, this gives 1.875. Uniform symmetric mixing yields the familiar e/(e−1) limit in the uniform-posterior case.
• Incentive fix: a sole-rescue reward (pay only the unique finder) aligns agent incentives with social discovery. Under sole-rescue, every pure Nash equilibrium attains the planner portfolio (full implementation of the planner outcome in pure strategies, given mild conditions).
• Correlated/latent copying: if agents draw from a common latent cue with copying probability c, effective discovery channels collapse, making decentralized report-following worse and increasing the planner's relative gain. In the canonical environment the equal-split market overtakes decentralized private report-following at c ≈ 0.788462.
• Asymptotics (proportional sparse-evidence limit): scale M, N with N/M → α ∈ (0,1) while holding likelihood ratio r fixed. Then:
  • G_consensus → 0 (plurality fails by multiple comparisons),
  • G_blind → α,
  • G_private → 1 − e^{−α r},
  • G_planner → 1 − (1 − α) e^{−α(r − 1)},
  • G_market → α Λ (Λ solved by a water-filling / Poisson-count equation). For canonical α = 1/2, r = 3.75, limits are 0 < 0.50 < 0.547011 < 0.846645 < 0.873580 (consensus → 0).
• Paradox is a sparse-evidence phenomenon: when evidence is abundant (target's signal separates cleanly), consensus works well; when evidence is sparse and many comparable alternatives exist, consensus compresses coverage and fails.

Data & Methods

• Exact finite-instance analysis: canonical benchmark with M = 16, N = 8, independent private clues p = 0.20; exact enumeration of posterior distributions (plurality ties resolved uniformly) gives the finite benchmark numbers reported above.
• Analytical decomposition: define attainable frontier V_L(F) = E[max_{|S|≤L} P(θ ∈ S | F)] and protocol loss L_L(Π; F) = V_L(F) − G_L(Π). This cleanly separates information value from allocation efficiency.
• Game-theoretic analysis of equal-split (congestion/covering) game:
  • Derive expected payoff of choosing box b when others mix s: u_b(s) = π_b φ(s_b) with φ(s) = [1 − (1 − s)^N]/(N s).
  • Prove existence of an exact potential and characterize symmetric equilibria via water-filling conditions.
  • Price-of-anarchy bounds derived with short search-specific proofs.
• Asymptotic approximations: Poisson limits for report counts and analysis of the market equilibrium in the proportional-sparse-evidence limit.
• Correlated/copying model: latent common-cue copying probability c incorporated into posterior; show planner gain strictly increases with c; numerically solve thresholds where market overtakes private search.
• Reproducibility: code, data, and an interactive guide are provided at github.com/yoheinakajima/shared-discovery-paradox; numerical claims are reproducible to stated precision.

Implications for AI Economics

• Centralized recommendations can harm collective exploration: aggregating many agents' signals into a single top recommendation (e.g., "best candidate" ranking delivered to many workers / models / evaluators) risks concentrating effort and reducing the chance of finding rare or diverse targets. Design of AI-assisted organizational decision workflows must treat aggregation outputs as inputs to allocation, not as final decisions.
• Distinguish information aggregation from action allocation: better estimated rankings do not automatically yield better group outcomes. Systems that surface a single "best" option (rank-1) for many downstream actors should be supplemented with allocation protocols that preserve coverage (portfolios, randomized assignment, forced distinctness).
• Incentive design matters: when agents self-select among pooled recommendations (e.g., crowdworkers, independent developers, submission reviewers), equal-split rewards produce a partial correction (market reduces but does not eliminate collisions). Specific payment rules (sole-rescue / unique-finder bonuses) can fully align private incentives with social discovery and implement the planner outcome in pure equilibria.
• Markets vs planner vs randomized protocols:
  • Markets (price splitting) can partially restore dispersion but leave inefficiencies (price of anarchy ≤ 2 − 1/N).
  • Central assignment (planner) achieves near-best outcomes but requires authoritative allocation.
  • Simple randomization or enforced distinctness (e.g., coordinate top-L different options) can be cheap and effective—seven coordinated actions suffice in the canonical example to match decentralized private search.
• Robustness to correlated information: when multiple agents draw from overlapping information sources (training data re-use, shared crawled signals, common APIs), effective discovery channels collapse; pooled reports become more correlated and the cost of compression rises. Monitoring and diversifying information sources (or adjusting allocation rules to account for copying) is critical.
• Applicability domains: early-stage discovery tasks with scarce signals—scientific hypothesis generation, frontier research funding, early-stage venture screening, bug-hunting in software, novel product ideation—are particularly vulnerable. In contexts with abundant evidence (many strong signals), consensus policies are less risky.
• Practical prescriptions for AI-enabled organizations:
  • Preserve or enforce portfolio diversity: allocate agents to different high-posterior candidates rather than repeating the single top candidate.
  • Use tie-breaking and distinctness constraints in recommender outputs (e.g., recommend top-L with assignment).
  • Employ incentive schemes that reward unique contributions (sole-rescue or diminishing sharing of rewards).
  • Reduce copying by encouraging independent evidence generation and tracking provenance of signals.
  • Measure effective channel count (how many independent signals are available) and choose allocation protocol accordingly.
  • When authority to assign is absent, use market mechanisms or randomized assignments to reduce collisions; be mindful of the residual price-of-anarchy inefficiency.
• Research implications for AI economics: the paper provides an exact, reusable benchmark separating information quality from allocation and incentives. This framework can be used to (i) quantify losses from various organizational protocols, (ii) design incentive-compatible mechanisms for exploration, and (iii) study the interaction of correlated data sources with allocation protocols in larger-scale, realistic AI systems.