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.
Large language model-based multi-agent systems (LLM-MAS) are designed through roles, debate protocols, and aggregation rules. These choices create implicit social expectations: agents may be expected to trust, challenge, defer to, or collaborate with peers. We study the effects of making inter-agent relation semantics explicit. We use a minimal signed-network formulation of relational priors and inject natural-language renderings into agent system prompts while holding the task protocol fixed. Across a commons-governance simulation and multi-agent debate, relational priors primarily act as convergence pressure: increasing relational positivity tends to make agents coordinate or agree more readily. This pressure can help when utility rewards behavioral alignment, as in sustainable resource governance and subjective consensus. It does not, however, reliably improve accuracy. In objective QA debates, higher positivity can increase agreement even when correctness-conditioned agreement does not improve and may decline in some settings. Effects vary by model backbone, relation type, and topology; explicit neutrality is not equivalent to omitting relational framing. We argue that relational priors should not be a default add-on for LLM-MAS. Their safer use is diagnostic and task-specific: compare against a no-prior baseline, monitor correctness-conditioned metrics when truth matters, and omit the relational layer when validation does not justify it.