Statistical Scouting Finds Debate-Safe but Not Debate-Useful Cases: A Matched-Ceiling Study of Open-Weight LLM Reasoning Protocols

TL;DR

This study evaluates the effectiveness of debate protocols in open-weight LLM reasoning, revealing structural limitations in predicting debate safety and usefulness, and highlighting the challenge of recovering debate headroom from cheap signals.

Contribution

It provides a matched-ceiling analysis of debate protocols, showing vote entropy predicts debate safety but not necessity, and demonstrates the difficulty of recovering debate benefits from cheap signals.

Findings

Vote entropy predicts debate safety but not necessity.

High vote entropy reduces debate backfire.

Most debate-helpful examples occur with unanimous but wrong votes.

Abstract

When should a language model answer directly, sample and vote, or engage in multi-agent debate? Recent work shows voting often explains much of the gain attributed to debate, while selective-debate systems activate deliberation only on uncertain examples. We ask: under a matched ceiling on generated tokens (960 per example), how much per-example routing headroom exists, and how much is recoverable from cheap pre-deliberation signals? We evaluate greedy decoding, three-sample voting, and a two-agent critique-revise debate on MuSiQue and GSM8K using Llama 3.1 8B Instruct and Ministral 3 8B Instruct. On MuSiQue, an oracle selecting the correct protocol per example gains +14.0 and +13.7 pp over the best fixed one. The best fixed protocol is model- and dataset-dependent: each (model, dataset) cell has a different winner. This headroom is hard to recover from cheap ex-ante signals. A vote-entropy threshold is the only controller that directionally beats the best fixed protocol on both models (+1.3 and +1.7 pp), though individual paired-bootstrap CIs include zero. A joint analysis (meta-analysis +1.6 pp, p=0.125; Bayesian P(both>0)=0.59) is directionally consistent but not significant. Learned controllers (LR, GBT) do not outperform the threshold. The key finding is structural: vote entropy predicts where debate is safe, not where debate is needed. High entropy sharply reduces debate backfire, but 66% of debate-helpful examples (31/47) occur when voting is unanimous but wrong. A single-prompt self-critique probe on Llama flips the answer in 127/127 unanimous cases, yielding zero mutual information with the debate-helpful label; we cannot rule out a prompt-compliance artifact, but either interpretation disqualifies the probe as a router. Recovering the remaining headroom requires behavioral probes that avoid format-compliance confounds at the 8B scale.