Companies, including market rivals, have long collaborated on the development of open source software (OSS), resulting in a tangle of co-operation and competition known as "open source co-opetition". While prior work investigates open source co-opetition in OSS projects that are hosted by vendor-neutral foundations, we have a limited understanding thereof in OSS projects that are hosted and governed by one company. Given their prevalence, it is timely to investigate open source co-opetition in such contexts. Towards this end, we conduct a mixed-methods analysis of three company-hosted OSS projects in the artificial intelligence (AI) industry: Meta's PyTorch (prior to its donation to the Linux Foundation), Google's TensorFlow, and Hugging Face's Transformers. We contribute three key findings. First, while the projects exhibit similar code authorship patterns between host and external companies (80%/20% of commits), collaborations are structured differently (e.g., decentralised vs. hub-and-spoke networks). Second, host and external companies engage in strategic, non-strategic, and contractual collaborations, with varying incentives and collaboration practices. Some of the observed collaborations are specific to the AI industry (e.g., hardware-software optimizations or AI model integrations), while others are typical of the broader software industry (e.g., bug fixing or task outsourcing). Third, single-vendor governance creates a power imbalance that influences open source co-opetition practices and possibilities, from the host company's singular decision-making power (e.g., the risk of license change) to their community involvement strategy (e.g., from over-control to over-delegation). We conclude with recommendations for future research.

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