Source-linked AI summary
A Theory of Post-hoc Debate Judgement
Xiang Yin, Adam Dejl, Antonio Rago, Lihu Chen, Francesca Toni
TL;DR
Post-hoc debate judges often rely on LLMs, but their formal properties are poorly understood. This paper develops and tests a theory of debate judgement, finding that argumentation semantics matches LLM judges in accuracy while better satisfying formal properties.
Problem
Post-hoc debate judges often rely on LLMs, yet their formal properties remain poorly understood despite their importance in high-stakes settings.
Method
The paper defines formal properties of debate judgement and evaluates LLM-based and computational-argumentation judges across claim-verification debate scenarios.
Results
The two judge families achieve similar accuracy across scenarios, while semantics-as-a-judge better satisfies the proposed formal properties.
Takeaways & Limitations
The study indicates argumentation semantics as a candidate for principled judges in debate-driven AI.
Takeaways & Limitations
Contestability can enable manipulation of judgements, so practical use may require restricting contestation to authorised users.
Abstract
from arXiv · showhide
Debates have recently emerged as a useful methodology for agentic AI to improve performance as well as to aid explainability and user engagement. For example, LLM-empowered agents may debate internally (with themselves) and/or externally (with other agents). In many settings where debates are used, debates' outcomes and resulting outputs are determined post-hoc by external judges, often LLMs. In this paper we develop and test a novel theory of debate judgement applicable to all settings where agents engage in debates by providing pros and cons for their opinions therein. Specifically, we identify a number of formal properties that debate judgement may be required to satisfy in general, as concerns reproducibility, robustness, groundedness and explainability. Then, we explore their satisfaction formally and/or experimentally, for claim verification settings, for two specific alternative debate judgement methods: variants of the LLMs as a judge idea and formal semantics drawn from computational argumentation. We show that the two methods give similar accuracy performances but the former may lack formal guarantees that the latter brings. Overall, our study indicates argumentation semantics as an ideal candidate for principled judges in debate-driven AI.
1 Introduction
The paper proposes a general theory of post-hoc debate judgement, identifying formal requirements for judging debates and evaluating two alternative judgement methods in claim verification. The methods achieve similar accuracy across scenarios, while formal computational-argumentation semantics provide better satisfaction of the proposed properties.
- Motivation: Debates support agentic AI performance and may occur internally among an agent’s reasoning processes or externally among multiple agents.They have been applied to tasks including reasoning enhancement, claim verification, AI safety, and fairness.
- Motivation: Post-hoc debate outcomes are often determined by LLM judges, whose outputs may be difficult to explain or contest.These judges can perform well but inherit limitations of the underlying LLMs.
- Theory: The paper develops a general theory identifying formal debate-judgement properties concerning reproducibility and robustness.Reproducibility includes deterministic behaviour and independence from agents’ permutation, while robustness concerns small variations in opinions and the judgement method.
- Evaluation: The study experimentally evaluates LLM-as-a-judge variants and formal semantics from computational argumentation in claim-verification settings.It considers internal debates with isolated opinion generation and external turn-taking debates that refine opinions.
- Findings: Similar accuracy performances are obtained across all tested scenarios, but computational-argumentation semantics show better satisfaction of the proposed properties.Judges may receive raw agent opinions or aggregated opinions before producing the final output.
2 Related Work
Existing debate research has largely prioritized improving debate generation over principled debate judgement. Related work therefore focuses on how debate outcomes should be judged, including approaches that aggregate debate-agent outputs through voting.
- Related Work: Existing debate-based research mainly improves debate generation while paying little attention to how debates should be judged.The paper treats debate judgement as a principled object of study.
- Non-argumentative Debate Judgement: Related work distinguishes non-argumentative debate judgement approaches from other forms of debate judgement.The supplied passage introduces this classification but does not provide the full list of forms.
- Non-argumentative Debate Judgement: Voting methods aggregate the outputs of debate agents, including majority voting.The passage identifies voting as one of the main forms adopted by existing approaches.
3 Background
The paper models debates with Quantitative Bipolar Argumentation Frameworks, which represent arguments, attacks, supports, and base beliefs. It evaluates dialectical strength using DF-QuAD semantics and highlights its balance and monotonicity properties for debate judgement.
- Quantitative Bipolar Argumentation Frameworks: A QBAF is a tuple containing a finite argument set, disjoint attack and support relations, and a base score function τ assigning beliefs in [0, 1].The attack and support relations are binary relations over the argument set.
- Gradual Semantics: Dialectical strength is evaluated by gradual semantics σ, with the paper focusing on DF-QuAD because of its broad applicability.DF-QuAD combines an argument’s base score with aggregated attacker and supporter strengths through an influence function.
- Illustration: 0.518 is the computed DF-QuAD strength of the illustrated topic argument, exceeding the neutral value 0.5 and therefore supporting the claim.The three pro and three con arguments have no attackers or supporters, so their strengths equal their displayed base scores.
- Desirable Properties: DF-QuAD satisfies balance, whereby equally strong support and attack cancel, and monotonicity, whereby stronger or additional support or attack changes topic strength in the expected direction.Increasing supporter strength or adding a supporter cannot decrease topic strength; increasing attacker strength or adding an attacker cannot increase it.
4 Abstract Framework
This section defines an abstract framework for post-hoc debate judgement and introduces formal properties for evaluating how judges transform agents’ stances and explanations into final judgements. The properties address reproducibility, robustness, groundedness, agreement, and responsiveness to supportive evidence, while allowing judgements to include abstention or uncertainty.
- Abstract framework: A judgement method maps an input claim and an ordered profile of agent opinions—each containing a stance plus pro and con explanations—to a final judgement.The final judgement may differ from agents’ opinions and may explicitly express uncertainty or abstention when evidence is insufficient.
- Evaluation framework: The framework evaluates judgement methods using abstract distance measures over opinion profiles, judgement stances, and judgement methods, alongside downstream metrics such as accuracy.Distance choices depend on the representation, with discrete metrics suitable for categorical outputs and semantic measures for textual content.
- Formal properties: The proposed properties require deterministic outputs, independence from opinion ordering, robustness to profile and judge changes, and selection of a stance proposed by at least one agent.Permutation independence promotes content-based assessment rather than identity-based bias, while judge robustness limits sensitivity to minor implementation changes.
- Formal properties: Judge unanimity requires adopting a stance shared by all agents, but both unanimity and non-hallucination can reproduce collective mistakes or systematic bias.Unanimity is weaker than non-hallucination because it applies only to unanimous profiles.
- Formal properties: Contestability requires more supportive opinion profiles to produce at least as supportive judgements, enabling users to challenge erroneous outputs while also creating a risk of manipulation.The paper notes that restricting contestation to authorised users may mitigate this risk.
5 Evaluation Setting
The evaluation tests post-hoc debate judgement across single-turn and multi-turn agent debates, varying opinion generation, judge input, and explanation-sharing conditions. It compares LLM-as-a-judge with semantics-as-a-judge while measuring accuracy and formal-property satisfaction.
- Evaluation scenarios: The evaluation uses three LLM-driven agents with real-valued stances in [0, 1] to generate claim-veracity opinions and pros-and-cons explanations.The prompts and underlying LLMs differ across agents, and the number of pros and cons varies by scenario.
- Evaluation scenarios: The single-turn scenario has m = 1 with independent opinions, each explained by one pro and one con.Agents generate their opinions once and independently.
- Evaluation scenarios: The multi-turn scenario sets m = 3, allowing agents to refine or completely change opinions with any number of pros and cons.It includes private explanations that agents adjust using others’ explanations and shared explanations collaboratively constructed while stances remain private.
- Experimental variants: Opinion generation varies whether stances precede explanations, follow explanations, or are omitted, labeled prior, post, and none respectively.The none variant is included to assess the usefulness of explanations without a generated stance score.
- Judges and measurement: Judges are either gpt-4o-based LLM-as-a-judge or DF-QuAD-based semantics-as-a-judge, receiving combined or separate agent explanations.Semantics-as-a-judge maps claims, pros, and cons to topic, supporting, and attacking arguments in QBAFs.
- Judges and measurement: Judgement accuracy and earlier properties are measured, with stance scores of 0.5 or higher classified as positive and lower scores as negative.Each judge uses information from the debate’s last time step.
6 Experiments
Experiments on claim verification compare LLM-based and semantics-based judges across debate settings and evaluate formal properties including determinism, permutation independence, robustness, non-hallucination, unanimity, and contestability. The judges achieve comparable accuracy, while semantics-based judging provides stronger formal-behavior guarantees in several tested properties.
- Experimental setup: The evaluation uses 500 approximately balanced True/False claims and three agents—GPT-4o-mini, Llama-3.3-70B-Instruct, and Qwen/Qwen3.5-9B—to generate debate stances.The single-turn dataset contains 1500 agent-level opinions; three debate time steps yield 4500 stances per scenario.
- Accuracy: Average accuracies exceed 70% for both judge families, with semantics-as-a-judge reaching 76.00% overall accuracy versus 75.00% for the best LLM-as-a-judge result.The highest True accuracy is 67.07% for semantics-as-a-judge, while the highest False accuracy is 94.02% for LLM-as-a-judge.
- Accuracy: In multi-turn debates, semantics-as-a-judge with private explanations and prior stance scores achieves 75.40%, compared with 74.80% for the best LLM-as-a-judge variant.Both configurations underperform the best single-turn configurations, suggesting that extended debate does not improve performance on this task.
- Judgement properties: Semantics-as-a-judge is deterministic because QBAF semantics uses fixed analytical functions, whereas LLM-as-a-judge accuracies vary between 73.20% and 73.40% across repeated runs.The LLM judge is also sensitive to the order in which agents’ opinions are presented.
- Judgement properties: Both judging methods change by at most 0.40% under the tested parameter perturbations, while QBAF-based judging preserves unanimous agreement at 99.39% versus 93.67% for LLMs and satisfies contestability in all 500 cases.Contestability is tested by increasing all pro argument base scores by 0.1, capped at 1.0.
7 Conclusions
The paper defines formal properties for post-hoc debate judgement, applies them to claim verification across debate settings, and experimentally studies their satisfaction under LLM and argumentation-semantics judges.
- The paper defines a novel set of formal properties for post-hoc debate judgement in the abstract.
- The theory is instantiated for claim verification across scenarios involving internal debates or direct agent interactions.
- The proposed properties are explored experimentally with LLM judges and argumentation semantics.
Supplementary Material for “A Theory of Post-hoc Debate Judgement” · A Computing Environment
The supplementary experiments used separate local computing environments for single-turn and multi-turn settings, with computations run on CPUs or remote API providers. The two setups differed in operating system, hardware, Python version, and supporting software libraries.
- A Computing Environment: Single-turn experiments ran on a Windows 11 laptop with a 14-core Intel Core Ultra 5 225H CPU, 32 GB RAM, and a Python-based software stack.The stack included Python 3.11.9, OpenAI 1.40.6, NumPy 2.4.0, pandas 2.3.3, scikit-learn 1.8.0, and Matplotlib 3.10.8.
- A Computing Environment: Multi-turn experiments ran on customized Ubuntu 22.04.5 LTS with an 8-core AMD Ryzen 7 pro 3700 CPU, 16 GB RAM, and a separate Python-based software stack.The stack included Python 3.12.10, OpenAI 2.44.0, Pydantic 2.13.4, and Tenacity 9.1.4.
B Experimental Results · B.1 Additional Results for Determinism and Permutation Independence
The experimental-results section includes additional analyses of the LLM-as-a-judge method, focusing on determinism across temperature settings and permutation independence across agent orders.
- B.1 Additional Results for Determinism and Permutation Independence: The section reports determinism results for the LLM-as-a-judge method under different temperature settings.These results are presented in Table 7.
- B.1 Additional Results for Determinism and Permutation Independence: Table 7 specifically evaluates how temperature settings relate to determinism in LLM-as-a-judge outcomes.The supplied passage identifies the method and experimental variable but provides no numerical values.
- B Experimental Results: The determinism analysis is an additional experimental result within the broader experimental-results section.No further setup or findings are specified in the supplied passage.
- B Experimental Results: Together, Tables 7 and 8 address reproducibility-related properties of debate judgement through temperature and agent-order analyses.This synthesis follows the two table captions; the passages do not state the observed outcomes.
- B.1 Additional Results for Determinism and Permutation Independence: The section also reports permutation-independence results for the LLM-as-a-judge method under different agent orders.These results are presented in Table 8.
- B.1 Additional Results for Determinism and Permutation Independence: Table 8 examines whether agent ordering is associated with permutation independence in LLM-as-a-judge results.The supplied passage identifies agent orders as the experimental condition but gives no numerical values.
B.2 Additional Results for QE Semantics
Quadratic Energy Semantics (QE) computes argument strengths from base scores and direct supporters and attackers, evaluating acyclic QBAFs in topological order. Its energy adjustment decreases strength when energy is non-positive, increases it when positive, and leaves unattacked arguments at their base scores.
- Quadratic Energy Semantics: QE determines each argument’s strength from its base score and the strengths of its direct supporters and attackers.For acyclic QBAFs, all required predecessor strengths are available during evaluation.
- Quadratic Energy Semantics: For an acyclic QBAF, QE computes final strengths in a single pass by evaluating arguments in topological order.Evaluation begins with arguments that have no incoming relations.
- Quadratic Energy Semantics: Non-positive energy decreases an argument’s strength relative to its base score, whereas positive energy increases it.The final strength is adjusted by energy-dependent terms in the QE equations.
- Quadratic Energy Semantics: Arguments with no supporters or attackers have zero energy and retain their base score.For such arguments, E_α = 0 and σ(α) = τ(α).
C Proofs
The paper formally proves that QBAF-based judges using DF-QuAD or QE gradual semantics satisfy determinism and permutation independence.
- C Proofs: DF-QuAD and QE judges satisfy determinism because fixed QBAF constructions and deterministic strength computations yield unique argument strengths.For a fixed opinion profile, the arguments, relations, base scores, aggregation, and influence functions are fixed.
- C Proofs: DF-QuAD and QE judges satisfy permutation independence because reordering opinions preserves the constructed QBAF and resulting argument strengths.Permuting opinions leaves arguments, support/attack relations, and base scores unchanged.
D Prompts
The prompts define a staged debate pipeline that elicits stance scores, generates pro/con arguments, scores argument validity, and produces posterior judgments. Separate judge prompts vary evidence visibility, agent identity, and whether prior or posterior stance scores are available.
- Prompt 1: Prior Stance Score: Prior stance prompts ask agents to score claim truthfulness from 0.0 to 1.0 without access to the gold label.The scale maps 0.0 to likely false, 0.5 to balanced or uncertain evidence, and 1.0 to likely true.
- Prompt 2: Pro/Con Argument Generation: Argument-generation prompts require short claim-focused reasons supporting and attacking the claim, including weak steelman arguments when the requested side conflicts with likely truth.The two scenarios differ in output format: one requests exactly one argument, while the other permits multiple pros and cons.
- Prompt 3: Argument Base Score: Base-score prompts ask an analyst to estimate each argument’s validity and relevance to the claim on a calibrated 0%–100% likelihood scale.The assessment should consider support or refutation, correctness, accuracy, and truthfulness, with moderate probabilities allowed under uncertainty.
- Prompt 4: Posterior Stance Score: Posterior stance prompts provide each agent’s pro and con arguments plus their base scores before requesting a final truthfulness score.The posterior score uses the same 0.0, 0.5, and 1.0 interpretation as the prior score.
- Prompt 1: LLM Judge (Named Agents): Named-agent judge prompts expose agent identities, argument grouping, optional stance lines, and self-assessed base scores while requiring a 0.0–1.0 truth score.The judge is instructed to consider both pro and con arguments and their base scores.
- Prompt 2: Judge Evidence Block (Anonymised): Anonymised evidence prompts hide agent identifiers and the original grouping of stance scores, pro arguments, and con arguments, while preserving anonymous evidence items.The evidence is separated into support and attack arguments with corresponding base scores.
- Prompt 3: Judge Instruction Line Variants: Judge instruction variants control whether stance predictions are hidden or stance scores are available, and whether evidence retains named-agent grouping or is anonymised.Variants include no stance, prior or posterior stance information, and named or anonymised agents.