Diverse Evidence, Better Forecasts: Multi-Agent Deliberation Under Information Asymmetry

# Multi-Agent Deliberation Under Information Asymmetry
Source: [https://arxiv.org/html/2607.01661](https://arxiv.org/html/2607.01661)
## Diverse Evidence, Better Forecasts: Multi\-Agent Deliberation Under Information Asymmetry

Yuante Li1, Yicheng Tao111footnotemark:1, Kate Zhang111footnotemark:1, Taozhi Wang1, Gefei Gu1, Yaxin Zhou2 1School of Computer Science, Carnegie Mellon University 2College of Engineering, Carnegie Mellon University \{yuantel, katezhan\}@cs\.cmu\.edu,yichengtao@cmu\.edu

###### Abstract

Multi\-agent systems are increasingly used for forecasting future events, as deliberation among multiple LLMs is believed to improve reasoning and calibration\. Yet existing approaches overlook a critical design choice: what information each agent receives\. When all agents are given identical evidence, deliberation collapses into herding rather than genuine belief revision, leaving multi\-agent systems little better than a single agent\. We identify this as a fundamental gap and proposedesigned information asymmetryto close it: by partitioning evidence into shared public and disjoint private subsets, each agent holds exclusive knowledge that can only reach others through deliberation\. We theoretically show that this decomposition reduces inter\-agent error correlation, and instantiate it inInfoDelphi, a framework combining relevance\-aware evidence routing, rationale\-based iterative deliberation, and confidence\-weighted aggregation\. OnPolyGym, a benchmark of 375 binary forecasting questions derived from real\-world prediction markets, InfoDelphi outperforms the strongest single\-agent and multi\-agent baselines by 12–18% in Brier score and 4–8 percentage points in accuracy\. More detailed experiments confirm that removing information asymmetry eliminates most deliberation gains, establishing diversity of input as the key enabler of effective multi\-agent reasoning\.

Diverse Evidence, Better Forecasts: Multi\-Agent Deliberation Under Information Asymmetry

Yuante Li1††thanks:Equal Contribution\.††thanks:Corresponding Author, Yicheng Tao111footnotemark:1, Kate Zhang111footnotemark:1,Taozhi Wang1, Gefei Gu1, Yaxin Zhou21School of Computer Science, Carnegie Mellon University2College of Engineering, Carnegie Mellon University\{yuantel, katezhan\}@cs\.cmu\.edu,yichengtao@cmu\.edu

## 1Introduction

Prediction markets such as Polymarket111[https://polymarket\.com/](https://polymarket.com/)aggregate the beliefs of thousands of participants into real\-time probability estimates for future events, achieving forecasting accuracy that often rivals or exceeds expert panels\(Tetlock,[2005](https://arxiv.org/html/2607.01661#bib.bib31)\)\. With every question resolved against a definitive ground truth, these markets provide a natural benchmark for machine intelligence:can an automated system reason over evidence and match human\-level forecasting?

Recent work has shown that large language models \(LLMs\) equipped with retrieved evidence can approach human\-level performance on binary forecasting tasks\(Halawiet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib3); Turtelet al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib18)\), but these advances rely primarily on single\-agent, single\-pass pipelines that treat the entire evidence corpus as a monolithic context window\. A natural extension is themulti\-agentsetting, where a panel of LLMs deliberates over the evidence before producing a collective forecast\. Such frameworks draw inspiration from well\-studied human mechanisms for aggregating distributed knowledge—expert committees, structured interviews, and the Delphi method\(Dalkey,[1969](https://arxiv.org/html/2607.01661#bib.bib27)\)—and have shown promise in general reasoning tasks\(Chenet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib26); Wanget al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib21)\)\.

![Refer to caption](https://arxiv.org/html/2607.01661v1/figures/intro.png)Figure 1:Effect of information asymmetry on a real prediction\.Left: agents with identical evidence converge to the same incorrect forecast\.Right: agents with complementary private evidence self\-correct through rationale sharing, reducing Brier score from 0\.45 to 0\.004\.Effective collective reasoning, however, requires that participants holdinformationally diverseviews\. As formalized in the Condorcet jury tradition and established empirically for human forecasters\(Lorenzet al\.,[2011](https://arxiv.org/html/2607.01661#bib.bib29)\), when agents share identical information, deliberation fails to improve collective accuracy: they reach the same conclusions independently, and iterative exchange merely reinforces the shared prior\. In LLM systems this problem runs deeper\. Recent work shows that models from different providers make correlated errors at rates above 60%\(Kimet al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib32)\), and that standard multi\-agent debate under homogeneous input behaves as amartingalewhose expected accuracy does not improve over rounds\(Choiet al\.,[2026b](https://arxiv.org/html/2607.01661#bib.bib33)\)\. Figure[1](https://arxiv.org/html/2607.01661#S1.F1)illustrates this failure mode on a real prediction market question: when all agents receive identical evidence, they converge to the same incorrect forecast \(left\), whereas agents equipped with complementary private evidence can self\-correct through deliberation \(right\)\.

To address this, we proposedesigned information asymmetry, treating information asymmetry not as a limitation to overcome but as a principle to exploit\. We introduceInfoDelphi, a multi\-agent forecasting framework that partitions evidence into a shared public pool and disjoint private subsets, giving each agent distinct expertise while preserving common ground for communication\. This mirrors effective human expert panels, where participants contribute complementary knowledge rather than reading the same briefing materials\. InfoDelphi constructs informative evidence partitions using BM25 relevance ranking\. During deliberation, agents exchange rationale excerpts rather than raw evidence, enabling efficient propagation of private signals\. Final forecasts are aggregated through confidence\-weighted averaging in logit space, emphasizing the predictions of the best\-informed agents\. Together, these design choices ensure that deliberation surfaces new information rather than reinforcing shared priors\.

Furthermore, existing forecasting benchmarks entangle retrieval quality with reasoning ability, making it impossible to assess whether gains come from better search or better deliberation\. We introducePolyGym, a controlled benchmark of 375 binary Polymarket questions with fixed, pre\-retrieved evidence shared across all methods\. OnPolyGym, InfoDelphi outperforms the strongest single\-agent and multi\-agent baselines by 12–18% in Brier score and 4–8 percentage points in accuracy\. Detailed ablations confirm that the public/private split and rationale sharing are jointly necessary, as removing either eliminates most gains, and cross\-model experiments demonstrate consistent improvements from open\-source to frontier proprietary LLMs\. These results establish designed information asymmetry as a general principle for multi\-agent reasoning under uncertainty\.

Our main contributions are as follows:

- •We proposedesigned information asymmetryfor multi\-agent reasoning, and theoretically show that public/private evidence partitioning reduces inter\-agent error correlation\.
- •We introduceInfoDelphi, which instantiates this principle through relevance\-aware evidence routing, rationale\-based deliberation, and confidence\-weighted aggregation\.
- •We constructPolyGym, a controlled forecasting benchmark that isolates information utilization from retrieval, and conduct extensive experiments showing that \(a\) deliberation without information diversity provides little benefit, \(b\) rationale sharing is critical for cross\-agent information transfer, and \(c\) gains generalize across different LLM backends\.

## 2Related Work

#### LLM\-based Forecasting\.

Early work established a substantial gap between LLM and human forecasting accuracy\.Zouet al\.\([2022](https://arxiv.org/html/2607.01661#bib.bib13)\)find that language models achieve only 65% accuracy on Autocast compared to 92% for human aggregates, whileSchoenegger and Park \([2023](https://arxiv.org/html/2607.01661#bib.bib14)\)show that GPT\-4 without augmentation does not significantly outperform a random baseline on Metaculus\. Retrieval\-augmented generation\(Lewiset al\.,[2020](https://arxiv.org/html/2607.01661#bib.bib1)\)substantially closes this gap:Yanet al\.\([2023](https://arxiv.org/html/2607.01661#bib.bib15)\)improve Autocast accuracy through zero\-shot re\-ranking and summarization of retrieved news, andHalawiet al\.\([2024](https://arxiv.org/html/2607.01661#bib.bib3)\)present a full pipeline achieving near\-human performance on Polymarket and Metaculus\. More recent work focuses on training:Turtelet al\.\([2025](https://arxiv.org/html/2607.01661#bib.bib18)\)fine\-tune on 12,100 resolved Polymarket questions, andChandaket al\.\([2025](https://arxiv.org/html/2607.01661#bib.bib17)\)train an 8B model via reinforcement learning that matches proprietary LLMs on prediction benchmarks\. On the aggregation side,Schoeneggeret al\.\([2024](https://arxiv.org/html/2607.01661#bib.bib19)\)show that ensembling diverse LLMs produces forecasts statistically indistinguishable from human forecasters\. Benchmark design has matured in parallel: ForecastBench\(Kargeret al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib5)\)and FutureX\(Zenget al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib22)\)provide continuously updated evaluations with strict temporal controls, andPalekaet al\.\([2025](https://arxiv.org/html/2607.01661#bib.bib23)\)identify systematic pitfalls including temporal leakage and piggybacking on existing forecasts\. Despite this progress, existing pipelines treat forecasting as a single\-agent process and do not study evidence distribution across a deliberating panel\.

#### Multi\-Agent Reasoning and Deliberation\.

Multi\-agent debate has emerged as a general strategy for improving LLM reasoning\(Duet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib37)\)\.Wanget al\.\([2022](https://arxiv.org/html/2607.01661#bib.bib20)\)show that sampling diverse chain\-of\-thought paths and selecting by majority vote improves accuracy, whileWanget al\.\([2024](https://arxiv.org/html/2607.01661#bib.bib21)\)propose Mixture\-of\-Agents, a layered architecture where later agents condition on all prior outputs\.Chenet al\.\([2024](https://arxiv.org/html/2607.01661#bib.bib26)\)demonstrate that a round\-table conference of diverse LLMs with confidence\-weighted voting outperforms single\-agent baselines across reasoning benchmarks\. However, recent theory suggests multi\-agent debate is not uniformly beneficial\.Choiet al\.\([2026a](https://arxiv.org/html/2607.01661#bib.bib38)\)prove that under a Dirichlet\-Categorical belief model, standard multi\-agent debate forms amartingale: expected correctness does not improve over rounds when agents receive identical inputs\.Shin \([2026](https://arxiv.org/html/2607.01661#bib.bib36)\)formalize this via the Data Processing Inequality: closed\-system deliberation forms a Markov chain, so mutual information with the ground truth can only decrease\.Kimet al\.\([2025](https://arxiv.org/html/2607.01661#bib.bib32)\)document that LLM errors are 60%\+ correlated, implying that naive ensembling hits a non\-zero error floor\. Our work addresses these limitations by introducing designed information asymmetry to break correlation and convert the martingale into productive belief revision\.

#### Information Aggregation and Collective Intelligence\.

The conditions under which groups outperform individuals are well studied\.Surowiecki \([2004](https://arxiv.org/html/2607.01661#bib.bib39)\)identify independence, diversity, and decentralized knowledge as key conditions for collective intelligence, whileHong and Page \([2004](https://arxiv.org/html/2607.01661#bib.bib40)\)show that diverse groups can outperform groups of only high\-ability individuals\. The DeGroot model\(DeGroot,[1974](https://arxiv.org/html/2607.01661#bib.bib7)\)formalizes iterative belief aggregation, and the Delphi method\(Dalkey,[1969](https://arxiv.org/html/2607.01661#bib.bib27)\)applies structured multi\-round elicitation to reduce expert disagreement\. For probabilistic forecasting, the linear opinion pool\(Stone,[1961](https://arxiv.org/html/2607.01661#bib.bib8)\)and logit\-space extremization\(Satopääet al\.,[2014](https://arxiv.org/html/2607.01661#bib.bib10)\)provide principled aggregation\. Meanwhile, social influence can suppress diversity without improving accuracy\(Lorenzet al\.,[2011](https://arxiv.org/html/2607.01661#bib.bib29)\), and informational cascades may cause agents to ignore private signals once consensus emerges\(Bikhchandaniet al\.,[1992](https://arxiv.org/html/2607.01661#bib.bib30)\)\. Our framework operationalizes the positive conditions for collective intelligence while explicitly mitigating herding and information loss through evidence partitioning and rationale\-based communication\.

## 3Methodology

### 3\.1Problem Formulation

Letqqdenote a binary forecasting question with resolution dateτ\\tauand ground truthy∈\{0,1\}y\\in\\\{0,1\\\}\. Each question is accompanied by a retrieved evidence corpusℰ=\{e1,…,en\}\\mathcal\{E\}=\\\{e\_\{1\},\\dots,e\_\{n\}\\\}, whereeie\_\{i\}consists of a title, URL, and textual snippet retrieved prior toτ\\tau\. Therefore, given the questionqqand evidence corpusℰ\\mathcal\{E\}, the goal is to estimatep​\(y=1∣q,ℰ\)p\(y=1\\mid q,\\mathcal\{E\}\), wherep∈\[0,1\]p\\in\[0,1\]denotes the predicted probability that the event resolves positively\.

### 3\.2Theoretical Motivation

We begin by formalizing why information asymmetry improves collective forecasting, drawing on the bias\-variance\-covariance decomposition of the ensemble mean squared error\.

#### Homogeneous input induces correlated errors\.

ConsiderJJagents each receiving the same evidenceℰ\\mathcal\{E\}\. Letpjp\_\{j\}denote agentjj’s forecast andyythe ground truth\. The ensemble forecastp¯=1J​∑jpj\\bar\{p\}=\\frac\{1\}\{J\}\\sum\_\{j\}p\_\{j\}has mean squared error:

MSE​\(p¯\)=Bias2\+σ¯2−Div,\\mathrm\{MSE\}\(\\bar\{p\}\)\\;=\\;\\mathrm\{Bias\}^\{2\}\+\\bar\{\\sigma\}^\{2\}\-\\mathrm\{Div\},\(1\)whereσ¯2=1J​∑jVar​\(pj\)\\bar\{\\sigma\}^\{2\}=\\frac\{1\}\{J\}\\sum\_\{j\}\\mathrm\{Var\}\(p\_\{j\}\)is the average agent variance andDiv=1J​∑jVar​\(pj−p¯\)\\mathrm\{Div\}=\\frac\{1\}\{J\}\\sum\_\{j\}\\mathrm\{Var\}\(p\_\{j\}\-\\bar\{p\}\)is the diversity term\(Woodet al\.,[2023](https://arxiv.org/html/2607.01661#bib.bib35)\)\. When all agents observe identical input, their prediction errorsϵj=pj−y\\epsilon\_\{j\}=p\_\{j\}\-yare driven by the same evidence and thus highly correlated:Cov​\(ϵi,ϵj\)≈Var​\(ϵ\)\\mathrm\{Cov\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)\\approx\\mathrm\{Var\}\(\\epsilon\)fori≠ji\\neq j\. In this regime, the diversity termDiv\\mathrm\{Div\}collapses toward zero, because agents conditioned on identical input produce nearly identical forecasts, and the ensemble provides no benefit over a single agent\. Recent empirical work confirms this: LLM errors are correlated across providers, and optimal aggregation under full correlation has a non\-zero error floor\(Kimet al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib32); Turkmenet al\.,[2026](https://arxiv.org/html/2607.01661#bib.bib34)\)\.

#### Information asymmetry reduces error correlation\.

We now show that the public/private evidence partition directly reduces this correlation\.

###### Proposition 3\.1\(Diversity\-Induced Decorrelation\)\.

Let each agent’s prediction error decompose asϵj=ρ⋅ϵpub\+\(1−ρ\)⋅ϵjpriv\\epsilon\_\{j\}=\\rho\\cdot\\epsilon^\{\\mathrm\{pub\}\}\+\(1\-\\rho\)\\cdot\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}, whereϵpub\\epsilon^\{\\mathrm\{pub\}\}is the error component driven by shared public evidence,ϵjpriv\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}is the component driven by agentjj’s private evidence, andρ∈\[0,1\]\\rho\\in\[0,1\]is the public ratio\. Letσpub2=Var​\(ϵpub\)\\sigma\_\{\\mathrm\{pub\}\}^\{2\}=\\mathrm\{Var\}\(\\epsilon^\{\\mathrm\{pub\}\}\)and assume:

1. \(i\)Cov​\(ϵipriv,ϵjpriv\)=0\\mathrm\{Cov\}\(\\epsilon\_\{i\}^\{\\mathrm\{priv\}\},\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)=0fori≠ji\\neq j\(private–private independence\);
2. \(ii\)Cov​\(ϵpub,ϵkpriv\)=0\\mathrm\{Cov\}\(\\epsilon^\{\\mathrm\{pub\}\},\\epsilon\_\{k\}^\{\\mathrm\{priv\}\}\)=0for allkk\(public–private independence\);
3. \(iii\)Var​\(ϵjpriv\)=σpriv2\\mathrm\{Var\}\(\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)=\\sigma\_\{\\mathrm\{priv\}\}^\{2\}for alljj\(symmetric private noise\)\.

Then:

Corr​\(ϵi,ϵj\)=ρ2​σpub2ρ2​σpub2\+\(1−ρ\)2​σpriv2\.\\mathrm\{Corr\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)\\;=\\;\\frac\{\\rho^\{2\}\\,\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\}\{\\rho^\{2\}\\,\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\+\(1\{\-\}\\rho\)^\{2\}\\,\\sigma\_\{\\mathrm\{priv\}\}^\{2\}\}\.\(2\)

###### Proof\.

By bilinearity of covariance and assumptions \(i\)–\(ii\), all cross\-terms vanish, givingCov​\(ϵi,ϵj\)=ρ2​σpub2\\mathrm\{Cov\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)=\\rho^\{2\}\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\. By assumptions \(ii\)–\(iii\), the cross\-term in the variance expansion also vanishes, soVar​\(ϵj\)=ρ2​σpub2\+\(1−ρ\)2​σpriv2\\mathrm\{Var\}\(\\epsilon\_\{j\}\)=\\rho^\{2\}\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\+\(1\{\-\}\\rho\)^\{2\}\\sigma\_\{\\mathrm\{priv\}\}^\{2\}, which equalsVar​\(ϵi\)\\mathrm\{Var\}\(\\epsilon\_\{i\}\)by symmetry\. Dividing covariance byVar​\(ϵi\)​Var​\(ϵj\)=Var​\(ϵj\)\\sqrt\{\\mathrm\{Var\}\(\\epsilon\_\{i\}\)\\mathrm\{Var\}\(\\epsilon\_\{j\}\)\}=\\mathrm\{Var\}\(\\epsilon\_\{j\}\)gives \([2](https://arxiv.org/html/2607.01661#S3.E2)\)\. ∎

Proposition[3\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1)shows that inter\-agent error correlation decreases monotonically asρ\\rhodecreases from 1 to 0\. As a concrete instance, at the defaultρ=0\.5\\rho=0\.5with equal\-variance components \(σpub=σpriv\\sigma\_\{\\mathrm\{pub\}\}=\\sigma\_\{\\mathrm\{priv\}\}\), correlation equals0\.25/\(0\.25\+0\.25\)=0\.50\.25/\(0\.25\+0\.25\)=0\.5, half the all\-public baseline \(ρ=1\\rho\{=\}1, correlation=1=1\)\. Atρ=0\\rho=0\(all\-private\), agents are fully decorrelated; atρ=1\\rho=1\(all\-public, standard debate\), correlation reaches its maximum\.

![Refer to caption](https://arxiv.org/html/2607.01661v1/figures/framework.png)Figure 2:Overview of theInfoDelphiframework\. \(a\)Evidence Routing: documents are ranked by BM25 relevance and distributed round\-robin, constructing a shared public pool and disjoint private subsets for each agent\. \(b\)Multi\-Agent Deliberation: agents exchange structured forecasts and rationales over two rounds via a history summarizer, enabling private information to propagate\. \(c\)Confidence\-Weighted Aggregation: final forecasts are combined by prediction\-extremity weights \(wj=\|logit​\(pj\)\|w\_\{j\}=\|\\mathrm\{logit\}\(p\_\{j\}\)\|\) into a single probability estimate\.###### Corollary 3\.1\.1\(Optimal Public Ratio Trade\-off\)\.

Decreasingρ\\rhoreduces error correlation \(improving ensemble gain\) but also reduces the shared context necessary for agents to interpret each other’s rationales\. An optimalρ∗∈\(0,1\)\\rho^\{\*\}\\in\(0,1\)therefore exists that balances diversity against communicability\.

#### Rationale sharing enables cross\-agent information transfer\.

Even with disjoint evidence, agents cannot benefit from each other’s private signals unless they communicate theirreasoning, not just their conclusions\. By the Data Processing Inequality\(Shin,[2026](https://arxiv.org/html/2607.01661#bib.bib36)\), a deliberation system where agents only exchange numeric estimates forms a Markov chainℰj→pj\(1\)→pj\(2\)→⋯\\mathcal\{E\}\_\{j\}\\to p\_\{j\}^\{\(1\)\}\\to p\_\{j\}^\{\(2\)\}\\to\\cdots, soI​\(y;pj\(r\+1\)\)≤I​\(y;pj\(r\)\)I\(y;p\_\{j\}^\{\(r\+1\)\}\)\\leq I\(y;p\_\{j\}^\{\(r\)\}\): information with the ground truth can only decrease\. Sharing rationales breaks this Markov structure: agentiinow observes a summary of agentjj’s reasoning overℰjpriv\\mathcal\{E\}\_\{j\}^\{\\mathrm\{priv\}\}, effectively receiving a lossy projection ofjj’s private evidence\. This converts the system from closed to open, allowingI​\(y;pj\(r\+1\)\)\>I​\(y;pj\(r\)\)I\(y;p\_\{j\}^\{\(r\+1\)\}\)\>I\(y;p\_\{j\}^\{\(r\)\}\), which is a necessary condition for iterative deliberation to improve over a single round\.

### 3\.3The InfoDelphi Framework

#### Overview\.

Figure[2](https://arxiv.org/html/2607.01661#S3.F2)illustrates the end\-to\-end InfoDelphi pipeline\. Our theoretical analysis \(Section[3\.2](https://arxiv.org/html/2607.01661#S3.SS2)\) identifies three conditions for effective collective forecasting: agents must holddiverse signals\(to reduce error correlation\), sharecommon ground\(to enable meaningful communication\), and exchangereasoning\(to transfer private information\)\. InfoDelphi instantiates these conditions through three corresponding stages: \(1\)Evidence routingpartitionsℰ\\mathcal\{E\}into a shared public pool and disjoint private subsets via BM25 relevance ranking, simultaneously satisfying diversity and common ground; \(2\)Iterative deliberationenables agents to exchange rationale excerpts across rounds, converting private signals into shared knowledge; \(3\)Confidence\-weighted aggregationcombines final\-round forecasts in logit space, amplifying agents whose private evidence supports stronger predictions\.

#### Evidence Routing\.

We consider a panel ofJJagents\{A1,…,AJ\}\\\{A\_\{1\},\\dots,A\_\{J\}\\\}and a public ratioρ∈\[0,1\]\\rho\\in\[0,1\]\. The corpus is partitioned into a shared public subsetℰpub\\mathcal\{E\}^\{\\mathrm\{pub\}\}andJJdisjoint private subsets:

ℰ=ℰpub∪⨆j=1Jℰjpriv,\|ℰpub\|=⌊ρ⋅\|ℰ\|⌋\.\\mathcal\{E\}\\;=\\;\\mathcal\{E\}^\{\\mathrm\{pub\}\}\\;\\cup\\;\\bigsqcup\_\{j=1\}^\{J\}\\mathcal\{E\}\_\{j\}^\{\\mathrm\{priv\}\},\\ \|\\mathcal\{E\}^\{\\mathrm\{pub\}\}\|\\;=\\;\\lfloor\\rho\\cdot\|\\mathcal\{E\}\|\\rfloor\.\(3\)AgentAjA\_\{j\}observes the private viewℰj=ℰpub∪ℰjpriv\\mathcal\{E\}\_\{j\}=\\mathcal\{E\}^\{\\mathrm\{pub\}\}\\cup\\mathcal\{E\}\_\{j\}^\{\\mathrm\{priv\}\}\.

To maximize information diversity across private subsets, we allocate documents viaBM25 relevance routing: each documentei∈ℰ∖ℰpube\_\{i\}\\in\\mathcal\{E\}\\setminus\\mathcal\{E\}^\{\\mathrm\{pub\}\}receives a relevance scoresi=BM25​\(ei,q\)s\_\{i\}=\\mathrm\{BM25\}\(e\_\{i\},q\)\. The top\-\|ℰpub\|\|\\mathcal\{E\}^\{\\mathrm\{pub\}\}\|scoring documents formℰpub\\mathcal\{E\}^\{\\mathrm\{pub\}\}: since these are the most directly relevant to the question, they serve as the shared “language” through which agents can interpret each other’s reasoning during deliberation\. The remaining documents are sorted bysis\_\{i\}and assigned to agents in round\-robin order, so that adjacent relevance levels are spread across distinct private views\. This design ensures that each agent holds a unique slice of peripheral evidence while sharing access to core information, enabling diversity without sacrificing common ground\.

#### Iterative Deliberation with Rationale Sharing\.

At each roundr∈\{1,…,R\}r\\in\\\{1,\\dots,R\\\}, agentAjA\_\{j\}produces a structured forecast:

fj\(r\)=\(pj\(r\),ℓj\(r\),rj\(r\)\),f\_\{j\}^\{\(r\)\}\\;=\\;\\bigl\(p\_\{j\}^\{\(r\)\},\\,\\ell\_\{j\}^\{\(r\)\},\\,r\_\{j\}^\{\(r\)\}\\bigr\),\(4\)wherepj\(r\)∈\[0,1\]p\_\{j\}^\{\(r\)\}\\in\[0,1\]is a probability estimate,ℓj\(r\)∈\{0,1\}\\ell\_\{j\}^\{\(r\)\}\\in\\\{0,1\\\}is a categorical prediction, andrj\(r\)r\_\{j\}^\{\(r\)\}is a free\-text rationale\. The agent conditions on its private view and a history summary of prior\-round forecasts:

fj\(r\)∼LLM​\(q,ℰj,h\(r−1\)\),f\_\{j\}^\{\(r\)\}\\;\\sim\\;\\mathrm\{LLM\}\\\!\\left\(q,\\;\\mathcal\{E\}\_\{j\},\\;h^\{\(r\-1\)\}\\right\),\(5\)whereh\(0\)=∅h^\{\(0\)\}=\\varnothingand, forr≥2r\\geq 2:

h\(r−1\)=\{\(pk\(r−1\),ℓk\(r−1\),rk\(r−1\)\)\}k≠j\.h^\{\(r\-1\)\}\\;=\\;\\bigl\\\{\(p\_\{k\}^\{\(r\-1\)\},\\,\\ell\_\{k\}^\{\(r\-1\)\},\\,r\_\{k\}^\{\(r\-1\)\}\)\\bigr\\\}\_\{k\\neq j\}\.\(6\)Crucially,h\(r−1\)h^\{\(r\-1\)\}includes rationale excerpts from other agents, not only their numeric estimates\. As established in Section[3\.2](https://arxiv.org/html/2607.01661#S3.SS2), this is what allows private evidence to propagate across agent boundaries\.

We study two deliberation protocols:independent forecasting, where agents reason solely from their private view; anditerative deliberation\(the full InfoDelphi protocol\), where agents additionally observe all other agents’ Round 1 forecasts and rationales before producing their final prediction\.

#### Confidence\-Weighted Aggregation\.

GivenJJagents’ final\-round forecasts\{fj\(R\)\}j=1J\\\{f\_\{j\}^\{\(R\)\}\\\}\_\{j=1\}^\{J\}, we aggregate them viaConfidence\-Weighted \(CW\) pooling:

p^CW\\displaystyle\\hat\{p\}\_\{\\mathrm\{CW\}\}=σ​\(∑j=1Jwj⋅logit​\(pj\(R\)\)∑j=1Jwj\),\\displaystyle\\;=\\;\\sigma\\\!\\left\(\\frac\{\\displaystyle\\sum\_\{j=1\}^\{J\}w\_\{j\}\\cdot\\mathrm\{logit\}\(p\_\{j\}^\{\(R\)\}\)\}\{\\displaystyle\\sum\_\{j=1\}^\{J\}w\_\{j\}\}\\right\),wj\\displaystyle w\_\{j\}=\|logit​\(pj\(R\)\)\|\\displaystyle=\\bigl\|\\mathrm\{logit\}\(p\_\{j\}^\{\(R\)\}\)\\bigr\|\(7\)whereσ​\(⋅\)\\sigma\(\\cdot\)is the sigmoid function andwj=\|logit​\(pj\(R\)\)\|w\_\{j\}=\|\\mathrm\{logit\}\(p\_\{j\}^\{\(R\)\}\)\|weights each agent by the extremity of its prediction\. Aggregation is performed in logit space to respect bounded probabilities\. This is equivalent to weighted geometric averaging of odds ratios and coincides with the logarithmic opinion pool under uniform weights\(Genest and Zidek,[1986](https://arxiv.org/html/2607.01661#bib.bib12)\)\. Agents with stronger private evidence produce more extreme forecasts, so up\-weighting them amplifies the most informed signals\.

## 4Experimental Setup

#### Dataset\.

Current LLM forecasting systems\(Halawiet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib3); Turtelet al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib18)\)primarily emphasizeretrieval quality, while an equally important capability isinformation utilization: how agents interpret, distribute, and collaboratively reason over retrieved evidence\. To study this dimension in isolation, we constructPolyGym, a controlled forecasting benchmark built from real\-world Polymarket questions\. Specifically,PolyGymcontains 375 binary forecasting markets with temporally filtered evidence corpora, where each question is paired with up to 30 pre\-retrieved documents collected before market resolution\. Unlike live forecasting benchmarks, all methods operate over the same fixed evidence pool, eliminating variability from changing web environments and unstable search results\. This design enables reproducible evaluation of multi\-agent deliberation strategies under controlled information conditions\. Importantly,PolyGymexplicitly supports the study ofdesigned information asymmetry, as it isolates whether performance gains arise from improved reasoning and coordination rather than from superior retrieval alone\. More details in Appendix[B](https://arxiv.org/html/2607.01661#A2)\.

#### Baselines\.

We compare against two categories of baselines \(details in Appendix[E](https://arxiv.org/html/2607.01661#A5)\)\. ❶Single\-agent methods: Zero\-shot\(Kojimaet al\.,[2022](https://arxiv.org/html/2607.01661#bib.bib47)\), Sequential Bayesian\(Shiet al\.,[2023](https://arxiv.org/html/2607.01661#bib.bib45)\), Chain\-of\-Thought\(Weiet al\.,[2022](https://arxiv.org/html/2607.01661#bib.bib46)\), Self\-Consistency\(Wanget al\.,[2022](https://arxiv.org/html/2607.01661#bib.bib20)\), Halawi et al\.\(Halawiet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib3)\), and Superforecaster\(Kargeret al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib5)\)\. ❷Multi\-agent methods with homogeneous input: Standard Debate\(Duet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib37)\), MoA\(Wanget al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib21)\), Crowd Ensemble\(Schoeneggeret al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib19)\), and AIA Forecaster\(Aluret al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib44)\)\.

#### Metrics\.

We report two complementary metrics\. TheBrier scoreBS=1N​∑k=1N\(p^k−yk\)2\\mathrm\{BS\}=\\frac\{1\}\{N\}\\sum\_\{k=1\}^\{N\}\(\\hat\{p\}\_\{k\}\-y\_\{k\}\)^\{2\}measures probabilistic calibration\.Accuracyreports the fraction of questions for which the predicted label𝟏​\[p^≥0\.5\]\\mathbf\{1\}\[\\hat\{p\}\\geq 0\.5\]matches the ground\-truthyy\.

#### Implementation\.

All experiments usegpt\-5\.4\-minivia the OpenRouter API222[openrouter\.ai](https://openrouter.ai/)as the primary model\. We useJ=3J=3agents,R=2R=2deliberation rounds, andρ=0\.5\\rho=0\.5as default settings, with temperature 0\.7, random seed 42, and parallelization across 15 workers\.

## 5Results

### 5\.1Main Results

Table[1](https://arxiv.org/html/2607.01661#S5.T1)compares InfoDelphi under independent and iterative deliberation protocols with baselines\.

#### ➠Overall performance\.

InfoDelphi achieves the best Brier score \(0\.178\) and accuracy \(77\.9%\) across all methods\. The improvement stems from two complementary mechanisms:information partitioningcreates diverse signals that reduce correlated errors, andrationale\-based deliberationallows agents to correct each other using their private evidence\. Notably, even without deliberation \(R=1R\{=\}1\), InfoDelphi already achieves the second\-highest accuracy \(74\.1%\), confirming that the diversity introduced by evidence routing provides valuebeforeany inter\-agent communication occurs\. Deliberation then amplifies this by enabling agents to propagate their private insights to others\.

Table 1:Results onPolyGym\(N=375N=375\)\. Methods are grouped by category\.Boldis the best;underlineis the second best\.
#### ➠Information diversity is the key enabler\.

Across all baselines, neither scaling the number of agents, improving prompting strategies, nor adding sophisticated aggregation mechanisms yields meaningful gains without information diversity\. Multi\-agent methods with homogeneous input \(MoA, Crowd Ensemble, AIA Forecaster\) fail to outperform a single Chain\-of\-Thought agent despite deploying 3–10 agents with various aggregation strategies\. Similarly, single\-agent methods with elaborate prompting \(Halawi’s 7\-step scratchpad, Superforecaster’s structured reasoning\) performworsethan simple Chain\-of\-Thought, suggesting that more elaborate reasoning on noisy evidence amplifies rather than mitigates errors\. The one partial exception is Standard Debate, which benefits from iterative belief revision, yet it remains well below InfoDelphi because agents sharing identical evidence can only reinforce existing interpretations rather than introduce genuinely new perspectives\. These results are consistent with the prediction of Proposition[3\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1): without information asymmetry, inter\-agent error correlation remains high and aggregation provides diminishing returns\.

### 5\.2Ablation Studies

We ablate each design dimension against the default InfoDelphi configuration \(J=3J\{=\}3agents,R=2R\{=\}2rounds,ρ=0\.5\\rho\{=\}0\.5, BM25 routing, confidence\-weighted aggregation, with rationale sharing\)\. Each ablation varies one component while holding others fixed \(detailed settings are provided in Appendix[E](https://arxiv.org/html/2607.01661#A5)\)\.

Table 2:Ablation results onPolyGym\. Each row varies one component from the default InfoDelphi configuration\.Boldis the best\.#### ➠Creating asymmetry is necessary but not sufficient\.

Section[3\.2](https://arxiv.org/html/2607.01661#S3.SS2)argues that effective collective reasoning requires three conditions: diverse signals, common ground, and reasoning exchange\. The ablation results confirm this hierarchy\. Simply partitioning evidence without deliberation already outperforms most homogeneous baselines; adding a second round yields a further−\-0\.028 Brier reduction, demonstrating that information asymmetry creates thepotentialfor improvement while deliberationrealizesit\. However, a third round reverses this gain: with a finite evidence pool, two rounds suffice to propagate most private signals, and further deliberation leads to convergence toward a group consensus rather than continued refinement\.

#### ➠How agents communicate determines whether deliberation helps\.

The largest degradation occurs when rationale sharing is removed, reducing deliberation to a closed system\. This validates the DPI argument: exchanging only numeric estimates preserves the Markov structure and prevents private information from flowing across agent boundaries\. With rationale sharing, agents receive a lossy projection of each other’s private evidence; without it, information about ground truth can only decrease over rounds\.

#### ➠The balance between diversity and common ground\.

The public ratioρ\\rhoreveals an asymmetric failure mode: too little shared context \(ρ=0\.3\\rho\{=\}0\.3\) is as harmful as too much \(ρ=1\.0\\rho\{=\}1\.0\), but for different reasons\. The former breaks communication while the latter eliminates diversity\. Even at the optimal split \(ρ=0\.5\\rho\{=\}0\.5\), themethodof allocation still matters: relevance\-based routing outperforms random by 0\.010 Brier\. This suggests that effective information asymmetry is not just abouthow muchto split, but about ensuring the public pool serves as meaningful common ground while private subsets contain genuinely complementary signals\.

#### ➠Confidence\-weighted aggregation amplifies informed agents\.

Replacing confidence\-weighted aggregation with uniform weighting degrades Brier by 0\.018, showing that not all agents contribute equally after deliberation\. Agents with stronger private evidence produce more extreme predictions, and up\-weighting them concentrates influence in the most informed voices \(validated in Section[5\.4](https://arxiv.org/html/2607.01661#S5.SS4)\)\.

#### ➠Summary\.

The ablations reveal that InfoDelphi’s gains are driven by theprincipleof designed information asymmetry rather than any single algorithmic component\. Rationale sharing and the public/private split are the load\-bearing design choices, while the routing algorithm and aggregation scheme provide further gains but are not irreplaceable, implying that the framework generalizes beyond its current form: alternative evidence partitioning strategies \(e\.g\., dense retrieval\) or aggregation methods can be substituted while preserving the benefit of information\-diverse deliberation\.

### 5\.3Cross\-Model Generalization

Table 3:Cross\-model generalization of InfoDelphi \(mean±\\pmstd over 3 runs\)\.To verify that InfoDelphi’s gains are not specific to a single model, we evaluate the full framework on additional backends: DeepSeek\-V3\.2\(Liuet al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib51)\)and Llama\-4\-Scout\-17B\(Adcocket al\.,[2026](https://arxiv.org/html/2607.01661#bib.bib52)\), as shown in Table[3](https://arxiv.org/html/2607.01661#S5.T3)\. All three models benefit from iterative deliberation, with the gain scaling with model capability\. This pattern suggests that stronger models extract more value from rationale sharing, as interpreting others’ reasoning requires sufficient comprehension ability\. Notably, even the smallest model benefits from information partitioning alone \(its independent mode already outperforms most homogeneous multi\-agent baselines\), confirming that the framework is effective across the full spectrum of model capabilities\.

### 5\.4Detailed Analysis

#### ➠Why does confidence weighting work in the asymmetric setting?

Our confidence\-weighted aggregation rests on the assumption that agents with stronger private evidence will produce more extreme predictions\. Figure[3](https://arxiv.org/html/2607.01661#S5.F3)validates this: accuracy increases monotonically from 62\.7% in the least extreme quartile to 94\.7% in the most extreme, while Brier score falls from 0\.235 to 0\.052\. However, a deeper question is why CW worksbetter under information asymmetrythan under homogeneous input\. We find that individual agents are equally extreme in both settings \(mean\|p−0\.5\|≈0\.36\|p\-0\.5\|\\approx 0\.36\), but their predictions point indifferent directionsunder InfoDelphi: inter\-agent variance is 1\.7×\\timeshigher than in the homogeneous setting\. In other words, information asymmetry does not make agents more confident—it makes them confident aboutdifferent things\. CW aggregation therefore synthesizes diverse high\-confidence signals grounded in complementary evidence, rather than merely amplifying a shared bias\.

![Refer to caption](https://arxiv.org/html/2607.01661v1/x2.png)Figure 3:Accuracy and Brier score by prediction extremity quartile \(\|p−0\.5\|\|p\-0\.5\|\)\. More extreme predictions correlate strongly with higher accuracy, validating the confidence\-weighted aggregation design\.
#### ➠Information asymmetry preserves diversity through deliberation\.

A key concern with iterative deliberation is herding: agents may converge toward a consensus regardless of their private evidence, collapsing the diversity that makes multi\-agent systems valuable\. Figure[4](https://arxiv.org/html/2607.01661#S5.F4)quantifies this by measuring inter\-agent prediction variance before and after deliberation under three evidence allocation strategies\. Under All Info \(ρ=1\.0\\rho\{=\}1\.0\) and Random Split \(ρ=0\.5\\rho\{=\}0\.5\), deliberation reduces variance by 90–92%, indicating near\-complete convergence\. Under relevance\-based routing \(ρ=0\.5\\rho\{=\}0\.5\), variance reduction is only 73%, preserving substantially more inter\-agent diversity\. This confirms that information asymmetry acts as a structural safeguard against herding: agents holding private evidence that they judge to be strongly predictive maintain distinct perspectives even after observing others’ reasoning, because their exclusive information provides an independent basis for conviction that social pressure alone cannot override\.

![Refer to caption](https://arxiv.org/html/2607.01661v1/x3.png)Figure 4:Mean inter\-agent prediction variance before \(Round 1\) and after \(Round 2\) deliberation\. BM25 routing preserves substantially more diversity \(73% reduction\) compared to Random Split and All Info\.
#### ➠Statistical significance\.

We verify that InfoDelphi’s improvements are statistically significant via paired Wilcoxon signed\-rank tests on per\-question Brier scores \(Table[4](https://arxiv.org/html/2607.01661#S5.T4)\)\. InfoDelphi significantly outperforms all baselines \(p<0\.05p<0\.05\), with the strongest significance against methods that scale agents without information diversity \(Crowd Ensemble, AIA Forecaster:p<0\.001p<0\.001\)\.

Table 4:Statistical significance of InfoDelphi vs\. baselines \(paired Wilcoxon signed\-rank test\)\.

## 6Conclusion

We presentedInfoDelphi, a multi\-agent forecasting framework built on a simple but overlooked insight: collective reasoning only outperforms individual reasoning when participants hold genuinely different information\. By treating information asymmetry as a first\-class design principle, InfoDelphi converts redundant deliberation into productive belief revision, achieving consistent gains across model scales and families onPolyGym\. Probabilistic forecasting represents a particularly demanding test of this principle, requiring synthesis of noisy, contradictory evidence into calibrated estimates under irreducible uncertainty\. That designed information asymmetry yields robust improvements in this setting suggests broad applicability to multi\-agent systems where agents reason over shared evidence, from collaborative question answering to scientific hypothesis evaluation to complex decision support\.

## Limitations

#### Theoretical assumptions\.

Proposition[3\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1)rests on three idealizing assumptions: a linear error decomposition, public–private independence, and symmetric private noise\. In practice, documents retrieved for the same question share topical correlations, partially violating the independence assumptions and making the decorrelation bound approximate\. The linear decomposition further assumes that public and private error components combine with fixed weightsρ\\rhoand1−ρ1\{\-\}\\rho, a functional form that LLMs, as non\-linear functions of their inputs, need not satisfy\.

#### Scope\.

API cost scales linearly with agents × rounds \(about 12$ for 375 questions\), which limits our ability to conduct large\-scale experiments with more agents or deliberation rounds that would be needed to precisely characterize convergence behavior\. Additionally, as our primary contribution is the multi\-agent paradigm of designed information asymmetry, both evidence routing and aggregation employ standard off\-the\-shelf methods\. More sophisticated alternatives may yield further gains\.

## Ethics considerations

This work presents a forecasting framework and does not directly enable disinformation or harm beyond risks inherent to LLM\-based systems generally\. However, automated probability estimates generated at scale could in principle be misused to manipulate market sentiment or create false impressions of consensus\. We recommend that InfoDelphi be used as a decision\-support tool with human oversight in high\-stakes deployment contexts\.

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## Appendix ADiscussion

#### Key insight\.

Our central finding is thatdesigned information asymmetryis necessary for effective multi\-agent deliberation\. When all agents receive identical evidence, iterative discussion provides little measurable benefit, as agents merely reinforce correlated beliefs rather than contribute complementary knowledge\. This aligns with the martingale characterization of homogeneous\-input debate and suggests that the widely adopted practice of giving all agents the same context fundamentally limits the value of multi\-agent interaction\.

#### Future work\.

Natural extensions include \(1\) adaptive routing that dynamically adjusts evidence allocation based on emerging disagreement, \(2\) more expressive routing algorithms beyond BM25, such as dense retrieval or learned semantic partitioning, \(3\) learned aggregation weights replacing self\-reported confidence, \(4\) evaluation on open\-ended and multi\-choice forecasting tasks, and \(5\) heterogeneous model panels that leverage model\-level diversity in addition to evidence\-level asymmetry\.

#### Potential Risks\.

Designed information asymmetry encourages agents to maintain distinct beliefs, but it may also amplify biased or misleading evidence partitions\. If private evidence subsets contain low\-quality, adversarial, or outdated documents, agents may become confidently polarized around incorrect conclusions rather than self\-correcting through deliberation\. In high\-stakes domains such as finance or geopolitics, this could reinforce speculative narratives or misleading impressions of certainty\. In addition, rationale sharing introduces a potential attack surface, where persuasive but flawed rationales propagate errors across agents\. Future deployments should therefore incorporate evidence verification, source reliability estimation, and safeguards against adversarial persuasion\.

## Appendix BPolyGymBenchmark Details

#### Motivation\.

Agentic forecasting systems for prediction markets require two distinct capabilities: \(1\)retrieval, identifying relevant information from the web, and \(2\)utilization, reasoning over that information to produce calibrated probability estimates\. Existing benchmarks predominantly evaluate these capabilities jointly \(Table[5](https://arxiv.org/html/2607.01661#A2.T5)\), making it difficult to determine whether performance gains arise from better retrieval or better collective reasoning\. Also, live benchmarks\(Kargeret al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib5); Zenget al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib22)\)introduce variability from changing web environments and unstable evidence availability\. Recent work shows that even retrieval\-augmented forecasting systems continue to exhibit substantial performance degradation over time, suggesting that access to information alone does not solve the utilization challenge\(Daiet al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib49)\)\.

![Refer to caption](https://arxiv.org/html/2607.01661v1/x4.png)Figure 5:PolyGymData Construction PipelineTable 5:Comparison of forecasting evaluation setups\.PolyGymuniquely provides fixed evidence and full reproducibility, enabling controlled study of information utilization and multi\-agent deliberation\.Therefore, our work specifically studies a utilization question: given the same global evidence pool, can a multi\-agent system reason more effectively by distributing information asymmetrically and exchanging rationales? To isolate this utilization dimension, we constructPolyGymas a controlled benchmark where all methods operate over the same pre\-retrieved evidence corpus, while allowing different agents to observe different evidence subsets\. This design enables direct evaluation of whether information asymmetry itself improves deliberative forecasting, independent of retrieval quality\. Conceptually,PolyGymis related to controlled retrieval settings in QA research\(Chenet al\.,[2021](https://arxiv.org/html/2607.01661#bib.bib41); Yanget al\.,[2018](https://arxiv.org/html/2607.01661#bib.bib42)\)that separate retrieval quality from downstream reasoning\.

#### Construction\.

PolyGymis derived from a large\-scale Polymarket dataset\(Semedov,[2025](https://arxiv.org/html/2607.01661#bib.bib43)\)consisting of approximately 40,000 events and 100,000 prediction markets\. Events are assigned to canonical domains \(Crypto, Sports, Science & Technology, Politics, Business\) using metadata fields\. We apply popularity\-weighted sampling to ensure topical diversity, yielding 500 binary markets\. All questions and evidence documents inPolyGymare in English\.

#### Evidence\.

For each market, we retrieve up to 30 documents via the Tavily search API333[https://www\.tavily\.com/](https://www.tavily.com/)using queries derived from the question text\. A strict temporal cutoff ensures all evidence precedes the market resolution date\. Retrieved documents are deduplicated and truncated to 500 characters\. By pre\-constructing a shared evidence corpus, we focus evaluation on how agents interpret, partition, and aggregate information rather than on search strategy differences\.

#### Temporal filtering\.

To reduce contamination from LLM prior knowledge, we retain only markets with resolution date≥\\geq2025\-09\-01, yielding the final set of 375 questions used in all experiments\. More details are provided in Appendix[E\.3](https://arxiv.org/html/2607.01661#A5.SS3)\.

#### Licenses and intended use\.

The source Polymarket dataset\(Semedov,[2025](https://arxiv.org/html/2607.01661#bib.bib43)\)is licensed under the Open Data Commons Database Contents License \(DbCL\) v1\.0\. Evidence documents retrieved via the Tavily Search API are used solely for non\-commercial research purposes under its API terms of service and are not independently redistributed\.PolyGymis intended for research use only\. The accompanying code is released under the MIT License\.

## Appendix CProof of Proposition[3\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1)

We provide the complete derivation\. Letϵj=ρ⋅ϵpub\+\(1−ρ\)⋅ϵjpriv\\epsilon\_\{j\}=\\rho\\cdot\\epsilon^\{\\mathrm\{pub\}\}\+\(1\-\\rho\)\\cdot\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}for agentj∈\{1,…,J\}j\\in\\\{1,\\dots,J\\\}\.

#### Covariance between agents\.

Fori≠ji\\neq j, expanding by bilinearity of covariance:

Cov​\(ϵi,ϵj\)\\displaystyle\\mathrm\{Cov\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)=ρ2​Var​\(ϵpub\)\\displaystyle=\\rho^\{2\}\\mathrm\{Var\}\(\\epsilon^\{\\mathrm\{pub\}\}\)\+ρ​\(1−ρ\)​Cov​\(ϵpub,ϵjpriv\)\\displaystyle\\quad\+\\rho\(1\{\-\}\\rho\)\\mathrm\{Cov\}\(\\epsilon^\{\\mathrm\{pub\}\},\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)\+ρ​\(1−ρ\)​Cov​\(ϵipriv,ϵpub\)\\displaystyle\\quad\+\\rho\(1\{\-\}\\rho\)\\mathrm\{Cov\}\(\\epsilon\_\{i\}^\{\\mathrm\{priv\}\},\\epsilon^\{\\mathrm\{pub\}\}\)\+\(1−ρ\)2​Cov​\(ϵipriv,ϵjpriv\)\.\\displaystyle\\quad\+\(1\{\-\}\\rho\)^\{2\}\\mathrm\{Cov\}\(\\epsilon\_\{i\}^\{\\mathrm\{priv\}\},\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)\.\(8\)
By the independence assumptions \(private components are independent of each other and of the public component\):Cov​\(ϵipriv,ϵjpriv\)=0\\mathrm\{Cov\}\(\\epsilon\_\{i\}^\{\\mathrm\{priv\}\},\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)=0andCov​\(ϵpub,ϵjpriv\)=0\\mathrm\{Cov\}\(\\epsilon^\{\\mathrm\{pub\}\},\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)=0\. Therefore:

Cov​\(ϵi,ϵj\)=ρ2​Var​\(ϵpub\)\.\\mathrm\{Cov\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)=\\rho^\{2\}\\mathrm\{Var\}\(\\epsilon^\{\\mathrm\{pub\}\}\)\.\(9\)

#### Individual variance\.

Sinceϵpub\\epsilon^\{\\mathrm\{pub\}\}andϵjpriv\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}are independent, the cross term vanishes:

Var​\(ϵj\)=ρ2​Var​\(ϵpub\)\+\(1−ρ\)2​Var​\(ϵjpriv\)\.\\mathrm\{Var\}\(\\epsilon\_\{j\}\)=\\rho^\{2\}\\mathrm\{Var\}\(\\epsilon^\{\\mathrm\{pub\}\}\)\+\(1\{\-\}\\rho\)^\{2\}\\mathrm\{Var\}\(\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)\.\(10\)

#### Correlation\.

Letσpub2=Var​\(ϵpub\)\\sigma\_\{\\mathrm\{pub\}\}^\{2\}=\\mathrm\{Var\}\(\\epsilon^\{\\mathrm\{pub\}\}\)and assume agents are symmetric \(Var​\(ϵjpriv\)=σpriv2\\mathrm\{Var\}\(\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}\)=\\sigma\_\{\\mathrm\{priv\}\}^\{2\}for alljj\), which impliesVar​\(ϵi\)=Var​\(ϵj\)\\mathrm\{Var\}\(\\epsilon\_\{i\}\)=\\mathrm\{Var\}\(\\epsilon\_\{j\}\)and simplifiesCorr​\(ϵi,ϵj\)=Cov​\(ϵi,ϵj\)/Var​\(ϵi\)​Var​\(ϵj\)\\mathrm\{Corr\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)=\\mathrm\{Cov\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)/\\sqrt\{\\mathrm\{Var\}\(\\epsilon\_\{i\}\)\\mathrm\{Var\}\(\\epsilon\_\{j\}\)\}toCov​\(ϵi,ϵj\)/Var​\(ϵj\)\\mathrm\{Cov\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)/\\mathrm\{Var\}\(\\epsilon\_\{j\}\):

Corr​\(ϵi,ϵj\)=ρ2​σpub2ρ2​σpub2\+\(1−ρ\)2​σpriv2\.\\mathrm\{Corr\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)=\\frac\{\\rho^\{2\}\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\}\{\\rho^\{2\}\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\+\(1\{\-\}\\rho\)^\{2\}\\sigma\_\{\\mathrm\{priv\}\}^\{2\}\}\.\(11\)In the homogeneous baseline whereρ=1\\rho=1\(all agents receive the same input\):Corr​\(ϵi,ϵj\)=1\\mathrm\{Corr\}\(\\epsilon\_\{i\},\\epsilon\_\{j\}\)=1\(perfectly correlated errors\)\. Asρ\\rhodecreases, correlation decreases monotonically, reaching 0 whenρ=0\\rho=0\. At our defaultρ=0\.5\\rho=0\.5with equal\-variance components,Corr=0\.25/\(0\.25\+0\.25\)=0\.5\\mathrm\{Corr\}=0\.25/\(0\.25\+0\.25\)=0\.5, compared toCorr=1\\mathrm\{Corr\}=1atρ=1\\rho=1, representing a50% reductionin inter\-agent error correlation\. ∎

## Appendix DProof of Corollary[3\.1\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1.Thmcorollary1)

Define thecommunicabilitybetween two agents as the mutual information between their forecast errors:

C​\(ρ\):=I​\(ϵi;ϵj\)\.C\(\\rho\)\\;:=\\;I\(\\epsilon\_\{i\};\\,\\epsilon\_\{j\}\)\.\(12\)Whenρ=0\\rho=0, the private componentsϵipriv\\epsilon\_\{i\}^\{\\mathrm\{priv\}\}andϵjpriv\\epsilon\_\{j\}^\{\\mathrm\{priv\}\}are independent by assumption \(Proposition[3\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1)\), soC​\(0\)=0C\(0\)=0\. Forρ\>0\\rho\>0, agents share the common componentϵpub\\epsilon^\{\\mathrm\{pub\}\}, soϵi\\epsilon\_\{i\}andϵj\\epsilon\_\{j\}are dependent andC​\(ρ\)\>0C\(\\rho\)\>0\. By the Data Processing Inequality\(Shin,[2026](https://arxiv.org/html/2607.01661#bib.bib36)\),C​\(ρ\)C\(\\rho\)upper\-bounds the information any function ofϵi\\epsilon\_\{i\}\(including agentii’s rationale\) can provide to agentjj: atρ=0\\rho=0, deliberation cannot transfer information between agents\.

#### Ensemble MSE decomposition\.

Our objective is to findρ∗=arg⁡minρ∈\[0,1\]⁡L​\(ρ\)\\rho^\{\*\}=\\arg\\min\_\{\\rho\\in\[0,1\]\}L\(\\rho\)\. TheJJ\-agent ensemble MSE decomposes as:

L​\(ρ\)=σ2​\(ρ\)J​\[1\+\(J−1\)​r​\(ρ\)\]⏟variance term​V​\(ρ\)\+D​\(ρ\)2⏟bias term,L\(\\rho\)\\;=\\;\\underbrace\{\\frac\{\\sigma^\{2\}\(\\rho\)\}\{J\}\\bigl\[1\+\(J\{\-\}1\)\\,r\(\\rho\)\\bigr\]\}\_\{\\text\{variance term \}V\(\\rho\)\}\\;\+\\;\\underbrace\{D\(\\rho\)^\{2\}\}\_\{\\text\{bias term\}\},\(13\)whereσ2​\(ρ\)=ρ2​σpub2\+\(1−ρ\)2​σpriv2\\sigma^\{2\}\(\\rho\)=\\rho^\{2\}\\sigma\_\{\\mathrm\{pub\}\}^\{2\}\+\(1\{\-\}\\rho\)^\{2\}\\sigma\_\{\\mathrm\{priv\}\}^\{2\}is the individual agent variance andr​\(ρ\)r\(\\rho\)is the inter\-agent correlation from Proposition[3\.1](https://arxiv.org/html/2607.01661#S3.Thmproposition1)\. We modelD​\(ρ\)D\(\\rho\)as a continuous, non\-increasing function ofC​\(ρ\)C\(\\rho\), withD​\(0\)=D0D\(0\)=D\_\{0\}\(single\-agent bias, since deliberation cannot transfer information whenC​\(0\)=0C\(0\)=0\) andD​\(ρ\)<D0D\(\\rho\)<D\_\{0\}for allρ\>0\\rho\>0\(any positive communicability enables strictly more effective deliberation than the no\-communication baseline\)\. Continuity ofDDfollows from the continuity ofC​\(ρ\)C\(\\rho\)as a function of the mixing weightρ\\rho\.

#### Boundary analysis\.

Atρ=1\\rho=1:r​\(1\)=1r\(1\)=1, soV​\(1\)=σpub2/J⋅J=σpub2V\(1\)=\\sigma\_\{\\mathrm\{pub\}\}^\{2\}/J\\cdot J=\\sigma\_\{\\mathrm\{pub\}\}^\{2\}, equal to single\-agent variance with no ensemble gain\. For anyρ=1−ε\\rho=1\-\\varepsilonwithε\>0\\varepsilon\>0:r​\(1−ε\)<1r\(1\{\-\}\\varepsilon\)<1, soV​\(1−ε\)<V​\(1\)V\(1\{\-\}\\varepsilon\)<V\(1\), whileD​\(1−ε\)≈D​\(1\)D\(1\{\-\}\\varepsilon\)\\approx D\(1\)by continuity\. ThereforeL​\(1−ε\)<L​\(1\)L\(1\{\-\}\\varepsilon\)<L\(1\), andρ=1\\rho=1is not a minimizer\.

Atρ=0\\rho=0:σ2​\(0\)=σpriv2\\sigma^\{2\}\(0\)=\\sigma\_\{\\mathrm\{priv\}\}^\{2\},r​\(0\)=0r\(0\)=0, andC​\(0\)=0C\(0\)=0, soV​\(0\)=σpriv2/JV\(0\)=\\sigma\_\{\\mathrm\{priv\}\}^\{2\}/JandD​\(0\)=D0D\(0\)=D\_\{0\}\. We show that moving slightly into the interior strictly decreases both terms\. Differentiatingσ2​\(ρ\)\\sigma^\{2\}\(\\rho\)andr​\(ρ\)r\(\\rho\):

d​σ2d​ρ\|ρ=0\\displaystyle\\left\.\\frac\{d\\sigma^\{2\}\}\{d\\rho\}\\right\|\_\{\\rho=0\}=−2​σpriv2<0,\\displaystyle=\-2\\sigma\_\{\\mathrm\{priv\}\}^\{2\}<0,d​rd​ρ\|ρ=0\\displaystyle\\left\.\\frac\{dr\}\{d\\rho\}\\right\|\_\{\\rho=0\}=0,\\displaystyle=0,\(14\)where the second identity follows becauseρ2​σpub2\\rho^\{2\}\\sigma\_\{\\mathrm\{pub\}\}^\{2\}in the numerator ofrrvanishes to second order atρ=0\\rho=0\. Therefore

d​Vd​ρ\|ρ=0\\displaystyle\\left\.\\frac\{dV\}\{d\\rho\}\\right\|\_\{\\rho=0\}=1J\[\(σ2\)′\(0\)\(1\+\(J−1\)r\(0\)\)\\displaystyle=\\frac\{1\}\{J\}\\Big\[\\,\(\\sigma^\{2\}\)^\{\\prime\}\(0\)\\,\\big\(1\+\(J\{\-\}1\)r\(0\)\\big\)\(15\)\+σ2\(0\)\(J−1\)r′\(0\)\]\\displaystyle\\qquad\\quad\+\\sigma^\{2\}\(0\)\\,\(J\{\-\}1\)\\,r^\{\\prime\}\(0\)\\,\\Big\]=−2​σpriv2J<0\.\\displaystyle=\-\\,\\frac\{2\\sigma\_\{\\mathrm\{priv\}\}^\{2\}\}\{J\}\\;<0\.By continuity, there existsεV\>0\\varepsilon\_\{V\}\>0such thatV​\(ε\)<V​\(0\)V\(\\varepsilon\)<V\(0\)for allε∈\(0,εV\)\\varepsilon\\in\(0,\\varepsilon\_\{V\}\)\. Simultaneously, for anyρ\>0\\rho\>0the agents share the common componentϵpub\\epsilon^\{\\mathrm\{pub\}\}, soC​\(ρ\)\>0C\(\\rho\)\>0and by the strict monotonicity assumptionD​\(ρ\)<D0=D​\(0\)D\(\\rho\)<D\_\{0\}=D\(0\), givingD​\(ε\)2<D​\(0\)2D\(\\varepsilon\)^\{2\}<D\(0\)^\{2\}for everyε\>0\\varepsilon\>0\. Combining,L​\(ε\)=V​\(ε\)\+D​\(ε\)2<V​\(0\)\+D​\(0\)2=L​\(0\)L\(\\varepsilon\)=V\(\\varepsilon\)\+D\(\\varepsilon\)^\{2\}<V\(0\)\+D\(0\)^\{2\}=L\(0\)for sufficiently smallε\>0\\varepsilon\>0, soρ=0\\rho=0is not a minimizer\.

#### Existence of interior optimum\.

Since neither boundary is a minimizer, andL​\(ρ\)L\(\\rho\)is continuous on the compact interval\[0,1\]\[0,1\], the Extreme Value Theorem guarantees that the minimum is attained at someρ∗∈\(0,1\)\\rho^\{\*\}\\in\(0,1\)\. ∎

## Appendix EExperimental Details

### E\.1Baseline Descriptions

#### ➠Single\-Agent Methods\.

- •Zero\-shot\(Kojimaet al\.,[2022](https://arxiv.org/html/2607.01661#bib.bib47)\): The LLM predicts without any retrieved evidence, relying solely on internal knowledge\.
- •Sequential Bayesian\(Shiet al\.,[2023](https://arxiv.org/html/2607.01661#bib.bib45)\): Evidence items are processed one at a time byK=5K\{=\}5sequential agents, each updating the previous agent’s probability estimate\.
- •Chain\-of\-Thought\(Weiet al\.,[2022](https://arxiv.org/html/2607.01661#bib.bib46)\): One agent receives all evidence and produces a forecast with chain\-of\-thought reasoning\.
- •Self\-Consistency\(Wanget al\.,[2022](https://arxiv.org/html/2607.01661#bib.bib20)\): The same CoT prompt is sampled 5 times at temperature 0\.7\. Then, predictions are averaged\.
- •Halawi et al\.\(Halawiet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib3)\): A 7\-step structured scratchpad prompt \(rephrase, pro/con arguments, aggregate, calibration check\) is run 3 times\. Results are combined via trimmed mean that downweights the prediction furthest from the median by 50%\.
- •Superforecaster\(Kargeret al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib5)\): A structured reasoning prompt that instructs the model to decompose the question, estimate base rates, update with evidence, and perform a calibration check\.

#### ➠Multi\-Agent Methods \(Homogeneous Input\)\.

- •Standard Debate\(Duet al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib37)\): 3 agents all receive the full evidence corpus, deliberate over 2 rounds sharing predictions and rationales, with final predictions aggregated by mean\.
- •MoA\(Wanget al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib21)\): 3 proposer agents independently forecast with full evidence\. The 4th aggregator agent receives all proposals with rationales and produces a final synthesized prediction\.
- •Crowd Ensemble\(Schoeneggeret al\.,[2024](https://arxiv.org/html/2607.01661#bib.bib19)\): 5 independent agents each receive all evidence and predict\. Then, the median prediction is used as the final forecast\.
- •AIA Forecaster\(Aluret al\.,[2025](https://arxiv.org/html/2607.01661#bib.bib44)\): 10 independent agents forecast with full evidence, and a supervisor agent identifies disagreements among their reasoning traces and produces a final prediction with a confidence label\. If confidence is “high”, the supervisor’s prediction is used; otherwise, the system falls back to the mean of the 10 agents\.

### E\.2Ablation Settings

Each ablation modifies one dimension of the default InfoDelphi configuration \(J=3J\{=\}3,R=2R\{=\}2,ρ=0\.5\\rho\{=\}0\.5, BM25 routing, confidence\-weighted aggregation, rationale sharing enabled\):

- •Rounds:R∈\{1,2,3\}R\\in\\\{1,2,3\\\}\. AtR=1R\{=\}1, agents predict once with no inter\-agent communication \(equivalent to independent forecasting\)\. AtR=3R\{=\}3, agents undergo two rounds of rationale exchange before the final prediction\.
- •Public ratio:ρ∈\{0\.3,0\.5,1\.0\}\\rho\\in\\\{0\.3,0\.5,1\.0\\\}\. Atρ=1\.0\\rho\{=\}1\.0, all evidence is shared \(no private subsets\), reducing to standard debate\.
- •Rationale sharing: When disabled, the history summary contains only\(pk,ℓk\)\(p\_\{k\},\\ell\_\{k\}\)pairs without rationale text\.
- •Aggregation: Uniform weighting replaces confidence\-weighted pooling with a simple mean of agent probabilities\.
- •Evidence routing: Random allocation shuffles evidence items with a fixed seed and assigns them round\-robin, without BM25 relevance ranking\.

### E\.3Model Details

Table[6](https://arxiv.org/html/2607.01661#A5.T6)summarizes the models used in our cross\-model experiments\. All models’ knowledge cutoffs predate our temporal filter \(resolution date≥\\geq2025\-09\-01\), ensuring no data leakage from training\.

Table 6:Model knowledge cutoff dates\. All cutoffs precede thePolyGymtemporal filter \(September 2025\), preventing prior knowledge contamination\.
### E\.4Computational Cost

Table[7](https://arxiv.org/html/2607.01661#A5.T7)reports the estimated API cost for a single full run \(375 questions, two settings\) per model\.

Table 7:Estimated cost per single run \(375 questions×\\times9 API calls each\)\. Pricing based on published per\-token rates at time of experiments\.

## Appendix FCase Study: Deliberation in Action

We present a complete example showing how InfoDelphi’s information asymmetry and rationale sharing enable self\-correction\. This is the same question used in Figure[1](https://arxiv.org/html/2607.01661#S1.F1)\.

#### Question\.

“Over $1\.8B committed to the MegaETH public sale?” \(Ground truth: No\.\)

#### Round 1 \(Independent predictions\)\.

Each agent receives the shared public evidence plus its own private subset:

- •Agent 1\(p=0\.86p=0\.86\): “The strongest evidence points to total commitments exceeding $1\.8B\. The official sale dashboard shows $1\.39B committed at one point, and multiple reports describe massive oversubscription…”
- •Agent 2\(p=0\.12p=0\.12\): “The strongest evidence indicates total committed amounts were around $1\.39B at the dashboard and over $1B in related coverage, but not over $1\.8B\. A Polymarket market page explicitly shows the outcome resolved No…”
- •Agent 3\(p=0\.03p=0\.03\): “The strongest direct source reports MegaETH’s total committed amount at $1,390,735,255, which is below $1\.8B\. Multiple other sources describe the sale as around $450M committed…”

Agent 1’s private evidence emphasized “massive oversubscription” narratives, while Agents 2 and 3 held private documents with the exact dashboard figure \($1\.39B\) and market resolution data\.

#### Round 2 \(After rationale sharing\)\.

After observing each other’s rationales:

- •Agent 1\(0\.86→0\.110\.86\\to 0\.11\): “The best direct evidence is the Sonar dashboard showing total committed amount of $1,390,735,255, which is below $1\.8B…”\(Self\-corrected after seeing Agents 2&3’s citation of the exact figure\.\)
- •Agent 2\(0\.12→0\.080\.12\\to 0\.08\): “The Polymarket resolution and dashboard figure consistently place total commitments below $1\.8B\. Agent 1’s revision further corroborates this\.”\(Maintained position with increased confidence\.\)
- •Agent 3\(0\.03→0\.080\.03\\to 0\.08\): “The $1\.39B dashboard figure is unambiguous\. Slight upward adjustment reflects residual uncertainty in interpreting ‘committed’ vs\. final sale volume\.”\(Slight adjustment, remained confident\.\)

#### Outcome\.

Final CW\-aggregated prediction:p^=0\.06\\hat\{p\}=0\.06\(Brier = 0\.004\)\. The single\-agent baseline with all evidence predictedp=0\.95p=0\.95\(Brier = 0\.9\), failing because the dominant narrative of “oversubscription” overwhelmed the specific dashboard figure buried among 20\+ documents\.

#### Key insight\.

Information asymmetry ensured that the critical evidence \(exact $1\.39B figure\) reached at least some agents as private, high\-salience information\. Rationale sharing then propagated this finding to Agent 1, who recognized the superior specificity of the dashboard data and self\-corrected\.

## Appendix GPrompt Templates

We list the exact prompts used in InfoDelphi\. All agents receive the same system prompt and structured user prompt; the only difference across agents is their assigned evidence subset\.

#### System Prompt

> You are a forecaster for binary \(Yes/No\) prediction markets\. Use only the evidence provided\. Prefer high\-credibility sources when they conflict\. Reflect uncertainty with probabilities near 0\.5\. You must respond with valid JSON only, no other text before or after\.

#### User Prompt \(Round 1\)

> Question: \{question\} Evidence: \[1\] doc\_id=doc\_001 \| source=\.\.\. \| title=\.\.\. \| content: \.\.\. \[2\] doc\_id=doc\_002 \| source=\.\.\. \| title=\.\.\. \| content: \.\.\. \.\.\. Respond with exactly one JSON object, no other text\. Schema: \{"p\_yes": <0\-1\>, "label": "YES"\|"NO", "rationale": "\.\.\.", "evidence\_used": \["doc\_id", \.\.\.\]\}

#### User Prompt \(Round 2\)

In round 2, the prompt includes a history summary of all other agents’ round\-1 outputs appended after the evidence:

> Question: \{question\} Evidence: \[1\] doc\_id=doc\_001 \| \.\.\. \.\.\. Previous round summary \(other agents\): Agent 1: p\_yes=0\.850 \(YES\) \-\-\- The evidence suggests\.\.\. Agent 2: p\_yes=0\.120 \(NO\) \-\-\- Based on the pricing data\.\.\. Agent 3: p\_yes=0\.630 \(YES\) \-\-\- Multiple sources indicate\.\.\. Round mean p\_yes=0\.533\. Consider whether you agree or disagree with the above\. Respond with exactly one JSON object, no other text\. Schema: \{"p\_yes": <0\-1\>, "label": "YES"\|"NO", "rationale": "\.\.\.", "evidence\_used": \["doc\_id", \.\.\.\]\}

#### Output Format

Each agent returns a JSON object:

> \{"p\_yes": 0\.85, "label": "YES", "rationale": "The strongest evidence from doc\_003 indicates\.\.\.", "evidence\_used": \["doc\_001", "doc\_003"\]\}

## Appendix HQualitative Analysis

We present two representative successes and two failures of InfoDelphi, illustrating the mechanisms through which information asymmetry and deliberation help or fail\.

✓\\checkmarkS1: Minority Evidence Overrides a Confident MajorityQ:Will the Monad public sale commitments be between $1\.8B and $2B? Ground truth:NOInfoDelphip^\\hat\{p\}:0\.01Brier:0\.0001Mechanism\.In Round 1, A0and A2predicted YES with high confidence, citing reports of a “record\-breaking” sale and large oversubscription\. A1, whose private evidence included Coinbase’s authoritative token\-sale page, correctly identified that the actual committed amount was 276M USDC, which is far below the $1\.8B threshold, and thus predicted NO \(0\.01\)\. After deliberation, A1’s rationale exposed that the majority had confused the token’s fully diluted valuation \($2\.5B\) with actual USDC commitments; both A0and A2immediately corrected to 0\.01\. This case illustrates the core benefit of information asymmetry: a single agent holding an authoritative private source can overturn a confidently wrong majority through rationale sharing\.

✓\\checkmarkS2: Deliberation Resolves a Shared MisinterpretationQ:Will Gemini 3\.0 be released on November 28, 2025? Ground truth:NOInfoDelphip^\\hat\{p\}:0\.03Mechanism\.All three agents in Round 1 had access to evidence stating that Gemini 3\.0 launched on November 18, 2025\. All three misinterpreted the question as asking whether Gemini 3\.0 would be releasedbyNovember 28 \(a deadline interpretation\), rather thanonNovember 28 specifically\. Under this reading, a November 18 launch satisfies the condition, leading to high YES probabilities despite rationales that correctly noted the release occurred on November 18\. After deliberation, agents read each other’s rationales and converged on the correct interpretation: the question asks about a specific date, and a November 18 release does not satisfy it\. All predictions corrected to 0\.03\. The case demonstrates that sharingreasoning, not just numeric estimates, is what enables deliberation to resolve shared misinterpretations, consistent with our theoretical analysis in Section[3\.2](https://arxiv.org/html/2607.01661#S3.SS2)\.

×\\timesF1: Evidence Gap Beyond RetrievalQ:Will Ethereum reach $4,600 between September 29 and October 5? Ground truth:YESInfoDelphip^\\hat\{p\}:0\.02Mechanism\.The retrieved evidence consistently showed Ethereum’s highest intraday price during the window as $4,591\.44 on October 3, which is $8\.56 below the threshold\. All agents independently concluded NO in Round 1, and deliberation only reinforced this shared view\. In reality, Ethereum briefly crossed $4,600 at a moment not captured in the retrieved documents\. No amount of rationale sharing can compensate for evidence that was never retrieved: the system’s upper bound is the quality of its retrieval\. This case highlights a fundamental limitation of RAG\-based forecasting and motivates future work on richer evidence retrieval\.

×\\timesF2: Correlated Evidence Produces an Overconfident Echo ChamberQ:Israel strikes Iran by November 30? Ground truth:NOInfoDelphip^\\hat\{p\}:0\.996Mechanism\.All agents’ private evidence contained multiple independent news reports describing Israeli strikes on Iran on June 13, 2025 \(Operation Rising Lion\)\. Every agent predicted YES with 0\.98 confidence in Round 1\. Deliberation made matters worse: agents reinforced each other’s certainty, pushing predictions to 0\.995–0\.998 in Round 2\. The market resolved NO, likely due to the question’s resolution criteria differing from the agents’ interpretation\. This case illustrates that when all agents’ private evidence is drawn from the same correlated news corpus, information asymmetry provides no diversification benefit and deliberation degenerates into an echo chamber\.