When Does Combining Language Models Help? A Co-Failure Ceiling on Routing, Voting, and Mixture-of-Agents Across 67 Frontier Models

Recent advances in language models have enabled collaborative settings in which multiple models leverage one another's capabilities, iteratively improving, transforming, and extending each other's outputs. Each agent sees what the others assert before it answers, so peer opinion competes with the model's own parametric knowledge, and a wrong majority can overturn an answer the model would otherwise get right. We measure that displacement in 23 open-weight models, 19 conditions, and three datasets, yielding more than a million graded responses. A unanimous wrong majority reverses 22.8% of a model's correct MMLU answers, 54.8% on GPQA and 71.0% on SimpleQA, and 84-89% of the reversed answers match the peers' answers. Existing mitigations aim to increase Resistance, the rate at which a model keeps its correct answer under this pressure, which is only half of what a collaborating agent needs. We pair it with Receptivity, the rate at which a model adopts a correct peer answer after initially answering incorrectly. We score six methods on both axes, four drawn from prior work and two of our own. Each gains Resistance only by losing Receptivity, and their means fall on a single Resistance-Receptivity frontier with $R^2$ between 0.80 and 0.90. Reflection, the strongest published method, gains 7.9 points of MMLU Resistance and gives up 15.3 of Receptivity. Reasoning is the one exception. On GPQA and SimpleQA it trades like the rest, but on the MMLU subjects whose answers a model can derive for itself it raises Resistance by 7.2 points and Receptivity by 9.6 at once, the only intervention we find that improves both.

Selecting the correct answer from a pool of candidate reasoning chains is the engine of test-time scaling, yet the standard selectors each carry a cost: self-consistency inherits the errors of the single model it resamples, and trained reward models need labeled data and transfer poorly off-distribution. We study a third signal, free at inference time: cross-model consensus, the degree to which independently trained models, each solving the problem once, agree on a final answer. We treat the panel as an LLM-jury, in which the structure of agreement, not any model's score of another, is the verification signal. Across seven benchmarks it selects correct answers better than self-consistency and far better than a model scoring its own candidates: on competition math it closes the entire gap to an oracle selector, while self-scoring closes almost none. The mechanism is error decorrelation: independently trained models err differently, so their wrong answers scatter while the correct one accumulates agreement. We make this precise with a parameter-free law, derived in closed form, that predicts consensus accuracy from three measured panel statistics to a mean absolute error of $0.03$ and exposes the method's ceiling: a shared-error floor where models share a misconception, near zero on math but non-trivial on science. Against four trained verifiers spanning discriminative, outcome, and generative reward models, the free LLM-jury matches the strongest inside their math training domain and is the top selector outside it. Cross-model consensus is thus a verifier we can characterize in advance: a law that says when to trust it, and a floor that marks where it cannot.

Efficient routing across multiple LLMs enables cost-quality tradeoffs by directing queries to the cheapest capable model. Prior work attributes routing headroom to an "unsolvability ceiling", queries no model in the pool can solve. We present a large-scale study of multi-tier LLM routing with 206,000 query-model pairs across six benchmarks (MMLU, MedQA, HumanEval, MBPP, Alpaca, ShareGPT) using the Gemma 4 and Llama 3.1 families. Evaluating with both LLM-as-a-judge and exact-match metrics, we show that a substantial portion of reported unsolvability stems from evaluation artifacts: (i) systematic judge biases favoring verbosity over correctness, (ii) truncation under fixed generation budgets, and (iii) output format mismatches. Through dual-judge validation and exact-match grounding, we reduce measured unsolvability across tasks. We introduce a decomposition framework attributing failures to these artifacts, revealing consistent patterns across domains and model families. These artifacts also distort router training signals: standard routers collapse to majority-class prediction (~79% smallest-tier optimal), confirmed via random-feature and shuffled-label controls, incurring a 13-17 percentage point opportunity cost. We provide actionable recommendations including dual-judge validation, exact-match anchoring, and cost-sensitive objectives. Our findings suggest existing routing headroom estimates are substantially inflated, underscoring the need for reliable evaluation protocols in multi-LLM systems.