Reinforcement Learning from Human Feedback (RLHF) has become the predominant approach for aligning language models (LMs) to be more helpful and less harmful. At its core, RLHF uses a margin-based loss for preference optimization, which specifies the ideal LM behavior only in terms of the difference between preferred and dispreferred responses. This under-specification of ideal behavior for each response individually leads to two unintended consequences as the margin increases: (1) The probability of dispreferred (e.g., unsafe) responses may increase, resulting in potential safety alignment failures. (2) When the probability of dispreferred responses is reduced, this often coincides with a decrease in the probability of preferred responses, even when these responses are ideal. In this paper, we identify the fundamental issue: margin-based preference optimization loss under-specifies ideal LM behaviors. We derive key conditions under which the probabilities of both preferred and dispreferred responses increase or decrease together. These conditions occur when the inner products between the gradients of the log-probabilities of preferred and dispreferred responses are large. We theoretically analyze when such inner products are large and empirically validate our findings. Our framework also reveals important differences in the training dynamics of various preference optimization algorithms and suggests new directions for developing better algorithms for language model alignment.