Quantifying Sycophancy as Deviations from Bayesian Rationality in LLMs

Sycophancy, or overly agreeable or flattering behavior, is a documented issue in large language models (LLMs), and is critical to understand in the context of human/AI collaboration. Prior works typically quantify sycophancy by measuring shifts in behavior or impacts on accuracy, but neither metric characterizes shifts in rationality, and accuracy measures can only be used in scenarios with a known ground truth. In this work, we utilize a Bayesian framework to quantify sycophancy as deviations from rational behavior when presented with user perspectives, thus distinguishing between rational and irrational updates based on the introduction of user perspectives. In comparison to other methods, this approach allows us to characterize excessive behavioral shifts, even for tasks that involve inherent uncertainty or do not have a ground truth. We study sycophancy for 3 different tasks, a combination of open-source and closed LLMs, and two different methods for probing sycophancy. We also experiment with multiple methods for eliciting probability judgments from LLMs. We hypothesize that probing LLMs for sycophancy will cause deviations in LLMs' predicted posteriors that will lead to increased Bayesian error. Our findings indicate that: 1) LLMs are not Bayesian rational, 2) probing for sycophancy results in significant increases to the predicted posterior in favor of the steered outcome, 3) sycophancy sometimes results in increased Bayesian error, and in a small number of cases actually decreases error, and 4) changes in Bayesian error due to sycophancy are not strongly correlated in Brier score, suggesting that studying the impact of sycophancy on ground truth alone does not fully capture errors in reasoning due to sycophancy.