Recent evidence suggests Large Language Models (LLMs) display Theory of Mind (ToM) abilities. Most ToM experiments place participants in a spectatorial role, wherein they predict and interpret other agents' behavior. However, human ToM also contributes to dynamically planning action and strategically intervening on others' mental states. We present MindGames: a novel `planning theory of mind' (PToM) task which requires agents to infer an interlocutor's beliefs and desires to persuade them to alter their behavior. Unlike previous evaluations, we explicitly evaluate use cases of ToM. We find that humans significantly outperform o1-preview (an LLM) at our PToM task (11% higher; $p=0.006$). We hypothesize this is because humans have an implicit causal model of other agents (e.g., they know, as our task requires, to ask about people's preferences). In contrast, o1-preview outperforms humans in a baseline condition which requires a similar amount of planning but minimal mental state inferences (e.g., o1-preview is better than humans at planning when already given someone's preferences). These results suggest a significant gap between human-like social reasoning and LLM abilities.