Large language models are increasingly deployed in multi-agent systems for strategic tasks, yet how design choices such as role-based personas and payoff visibility affect behavior remains poorly understood. We investigate whether LLM agents function as payoff-sensitive strategic actors or as identity-driven role followers. Using a 2x2 factorial experiment (persona presence x payoff visibility) with four models (Qwen-7B/32B, Llama-8B, Mistral-7B), we test 53 environmental policy scenarios in four-agent strategic games. We find that personas suppress payoff-aligned behavior: with personas present, all models achieve near-zero Nash equilibrium in Tragedy-dominant scenarios despite complete payoff information. Nearly every equilibrium reached is Green Transition. Removing personas and providing explicit payoffs are both near-necessary for payoff-aligned behavior, enabling only Qwen models to reach 65--90\% equilibrium rates. Our results reveal three behavioral profiles: Qwen adapts to framing, Mistral is disrupted without finding Tragedy equilibrium, and Llama remains near-invariant. We show that the same binary design choice can shift equilibrium attainment by up to 90 percentage points, establishing that representational choices are not implementation details but governance decisions.

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