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

🕒 Published (v1): 2026-07-20 15:14 UTC · Source: Arxiv · link

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Nakajima (BabyAGI); exactly solvable multi-agent shared-discovery model exposes coverage vs belief tradeoff

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TL;DR¶

Pooling dispersed information into a single shared recommendation can improve individual action quality while catastrophically reducing group discovery. In a 16-box, 8-searcher benchmark, consensus search drops discovery from 0.8322 (private clues) to 0.3835; a coordinated portfolio using the same pooled information recovers 0.8594. The failure is a protocol failure—compressing a portfolio of actions into one repeated choice—not an information failure.

Problem¶

Organizations and multi-agent systems that aggregate information into a single "best" recommendation cause all agents to converge on one action, destroying the coverage diversity that makes parallel search effective. Existing literature treats information aggregation and action allocation as separate problems; this paper shows their interaction produces a measurable paradox—better beliefs, worse outcomes—in sparse-evidence settings like frontier science, early-stage investing, and novel-hypothesis generation.

Method¶

An exactly solvable Bayesian benchmark with \(M = 16\) boxes, \(N = 8\) searchers, and noisy private clues (\(p = 0.20\) hit probability, conditionally independent). Four protocols are compared under exact enumeration:

Private clue-following: each searcher opens their own clue's box.
Shared consensus: all 8 agents follow the posterior mode (plurality winner).
Shared portfolio: a coordinator assigns the \(L\) actions to the top-\(L\) posterior boxes.
Equal-split market: self-interested searchers choose boxes and split the unit prize equally if multiple find the jackpot; analyzed as a potential/congestion game.

Protocol loss is defined as \(\mathcal{L}_L(\Pi; \mathcal{F}) = V_L(\mathcal{F}) - G_L(\Pi)\), separating information quality from action-allocation efficiency. A sole-rescue rule (prize paid only to a searcher who covers the target alone) is analyzed as an incentive correction. A latent common-cue copying model introduces correlation probability \(c\) among signals to study herd behavior.

Key Contributions¶

Exact quantification of the paradox: \(G^{\text{consensus}} = 0.3835 < G^{\text{private}} = 0.8322 < G^{\text{planner}} = 0.8594\); even blind diversified search (\(G^{\text{blind}} = 0.50\)) beats informed consensus.
Protocol loss accounting: the full 47.60 pp gap between consensus and the optimal portfolio is protocol loss, not information loss; the recovery budget is \(L^* = 7\) (seven coordinated actions match eight decentralized clue-followers).
Partial conformity is monotonically harmful: each additional agent deferring to the consensus strictly decreases discovery; dosing conformity cannot recover diversity.
Equal-split game is an exact potential game with a unique anonymous symmetric equilibrium following a water-filling rule; discovery equals \(N\lambda\) where \(\lambda\) is the common equilibrium payoff.
Exact mixed price of anarchy: \(\text{PoA} \leq 2 - 1/N\) for all mixed Nash equilibria (tight).
Sole-rescue rule achieves first-best: every pure Nash equilibrium assigns one searcher to each of the \(N\) highest-posterior boxes, fully implementing the planner portfolio.
Proportional sparse-evidence limit: as \(M \to \infty\) with \(N_M/M \to \alpha\), the ordering \(G^{\text{consensus}} \to 0 < G^{\text{blind}} \to 0.50 < G^{\text{market}} \to 0.547 < G^{\text{private}} \to 0.847 < G^{\text{portfolio}} \to 0.874\) survives exactly.
Copying threshold: in the correlated-signals model, the market overtakes decentralized private search at copying probability \(c = 0.788462\).

Results¶

Canonical protocol ordering: \(0.20 = Q(\Pi^P) < G^{\text{consensus}} = 0.3835 < G^{\text{private}} = 0.8322 < G^{\text{planner}} = 0.8594\)
Equal-split market equilibrium: \(G^{\text{market}} = 0.5991\), with only 2.67 distinct searches in expectation
Consensus improves individual action quality (0.3835 > 0.20) while destroying group coverage
Recovery budget: 7 coordinated actions suffice to match 8 decentralized searchers
Proportional large-market limits: consensus \(\to 0\); blind \(\to 0.500\); market \(\to 0.547\); private \(\to 0.847\); portfolio \(\to 0.874\)
Market discovery decline with scale: \(M \in \{16, 64, 256, 1024\}\) yields \(\{0.5991, 0.5740, 0.5582, 0.5494\}\)
Sole-rescue rule: pure-strategy implementation of the first-best planner portfolio at every Nash equilibrium

Limitations¶

Benchmark is deliberately small and stylized (16 boxes, uniform prior, symmetric independent signals); extensions to non-uniform or correlated priors require further analysis.
The sole-rescue incentive correction achieves implementation in pure Nash equilibria but does not address equilibrium selection or mixed-strategy behavior.
The intermediate copying regime \(\lambda_M \asymp \log M\) (multiple-comparisons threshold) is left open.
The water-filling equilibrium is the anonymous symmetric equilibrium; efficient asymmetric pure equilibria also exist but require coordination devices the model explicitly excludes.
Results assume one action per agent and a binary (jackpot/no jackpot) discovery signal; richer action spaces or partial-discovery payoffs are not analyzed.

Relevance to Agentic AI / LLM Agents¶

This paper provides a rigorous theoretical grounding for a failure mode that is directly instantiated in multi-agent LLM systems: when multiple agents share a single ranked output (e.g., a retrieved document list, a tool recommendation, or a consensus answer from an orchestrator), they may all act on the same top-1 choice, destroying the coverage advantage of parallelism. The "protocol loss" framework offers a concrete vocabulary—portfolio value vs. one-answer compression—for diagnosing why agentic systems that pool information before acting underperform systems that preserve diversity. The sole-rescue incentive correction (rewarding unique discovery) and the water-filling equilibrium suggest principled reward-shaping strategies for multi-agent RL settings where agents must explore diverse hypotheses. The copying model directly maps to scenarios where multiple agents draw from the same retrieved context or shared memory, causing correlated failures that mimic herd behavior.