Prompting is a standard tool for customizing language-model chatbots, enabling them to take on a specific "persona". An implicit assumption in the use of prompts is that they will be stable, so the chatbot will continue to generate text according to the stipulated persona for the duration of a conversation. We propose a quantitative benchmark to test this assumption, evaluating persona stability via self-chats between two personalized chatbots. Testing popular models like LLaMA2-chat-70B, we reveal a significant persona drift within eight rounds of conversations. An empirical and theoretical analysis of this phenomenon suggests the transformer attention mechanism plays a role, due to attention decay over long exchanges. To combat attention decay and persona drift, we propose a lightweight method called split-softmax, which compares favorably against two strong baselines. View paper on