Large language models (LLMs) increasingly operate in multi-agent systems where they collaborate by responding to others' choices, creating endogenous feedback loops with vulnerability to social influence, herding behavior and runaway cascades. Following the design of the classic Music Lab experiment, we tested this vulnerability in large populations of LLM agents that recommended unfamiliar songs, with or without exposure to cumulative recommendation counts. Exposure led to runaway cascades that generated extreme and unpredictable collective recommendations increasingly divergent from a song's intrinsic appeal. Compared to humans, LLM populations were generally far more susceptible to herding behavior, and unlike humans, LLM cascades were less able to self-correct. This vulnerability has implications for multi-agent collaboration in recommender systems, automated evaluation, complex problem-solving, and unauthorized conspiracies.

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