How Much Do You Trust A Dog? Quantifying Hint-Following in Reasoning Language Models

Summary

Large language models (LLMs) often exhibit sycophancy, altering their answers to align with user suggestions even on objective factual and reasoning tasks. Yet the quantitative form of this influence--- and whether it is calibrated to source reliability--- remains poorly understood. In this work, we investigate this behavior through hint-induced answer shifts on MMLU questions, evaluating prompts of the form ``[Source] thinks the answer is (X)''. We systematically vary the source to which the hint is attributed (a professor, a friend, or a dog) across three open-source model families, Qwen, Gemma, and SmolLM. We show that the change induced by the hint follows a predictable structure well-characterized by an exponential tilt: the hint induces an approximately constant mean shift in the hinted answer's log-odds, with question--target susceptibility largely shared across sources. To explore how this pattern can arise, we introduce a formal decomposition of the hint response and illustrate in controlled synthetic experiments how KL-regularized reward-based training reshapes both hinted and unhinted answer distributions. Finally, we evaluate whether this shift is calibrated to source reliability, finding that the log-odds shift persists even when the hint-giver is declared inaccurate or accompanied by explicit warnings not to trust the source, highlighting the vulnerability of LLMs to adversarial steering.