Research suggests that large-scale human cooperation is driven by shared narratives that encode common beliefs and values. This study explores whether such narratives can similarly nudge LLM agents toward collaboration. Therefore, we let LLM agents play a (networked) finitely repeated public goods game after being primed with different stories. Our experiments address four questions: (1) How do narratives influence negotiation behavior? (2) What differs when agents share the same story versus different ones? (3) What happens when the agent numbers grow? (4) Are agents resilient against self-serving participants? We find that story-based priming significantly affects collaboration. Common stories improve collaboration and benefit all participants, while different story priming reverses this effect, favoring self-interested agents. These patterns persist across network sizes and structures. These findings have implications for multi-agent coordination and AI alignment. Code is available at github.com/storyagents25/story-agents.