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It is supported by our ability to form mental models of ourselves and others. Its ubiquity in human affairs entails that LLMs have seen innumerable examples of it in their training data and therefore may have learned to mimic it, but whether they have actually learned causal models that they can deploy in arbitrary settings is unclear. We therefore develop a novel experimental paradigm that requires that subjects form representations of the mental states of themselves and others and act on them strategically rather than merely describe them. We test a wide range of leading open and closed source LLMs released since 2024, as well as human subjects, on this paradigm. 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