Large language models (LLMs) are now being used in a wide variety of contexts, including as creativity support tools (CSTs) intended to help their users come up with new ideas. But do LLMs actually support user creativity? We hypothesized that the use of an LLM as a CST might make the LLM's users feel more creative, and even broaden the range of ideas suggested by each individual user, but also homogenize the ideas suggested by different users. We conducted a 36-participant comparative user study and found, in accordance with the homogenization hypothesis, that different users tended to produce less semantically distinct ideas with ChatGPT than with an alternative CST. Additionally, ChatGPT users generated a greater number of more detailed ideas, but felt less responsible for the ideas they generated. We discuss potential implications of these findings for users, designers, and developers of LLM-based CSTs.

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