Communities can sustainably manage shared resources (commons) through self-governance and cooperative norms, a central finding of Ostrom's theory of self-governance. However, real-world commons (e.g., fisheries, forests, and irrigation systems) are often governed under asymmetric power structures, where certain individuals or institutions possess disproportionate control over resource extraction and collective outcomes. As Large Language Models (LLMs) are increasingly explored as agents in synthetic governance simulations, understanding how LLM societies behave under asymmetric power structures is becoming increasingly important, yet existing evaluations largely ignore such asymmetries. We introduce Sovereignty over the Commons Simulation (SovSim), a generative multi-agent simulation framework that incorporates an agent with asymmetric power (boss or king) into a society of symmetric agents (workers or peasants), where all agents extract from a shared resource, collectively determining its sustainability over time. Across eleven state-of-the-art models, we find that introducing asymmetric power leads to severe breakdowns in cooperation and sustainability, with up to an 87.3% degradation in survival rate relative to symmetric settings.