We study LLM policy synthesis: using a large language model to iteratively generate programmatic agent policies for multi-agent environments. Rather than training neural policies via reinforcement learning, our framework prompts an LLM to produce Python policy functions, evaluates them in self-play, and refines them using performance feedback across iterations. We investigate feedback engineering (the design of what evaluation information is shown to the LLM during refinement) comparing sparse feedback (scalar reward only) against dense feedback (reward plus social metrics: efficiency, equality, sustainability, peace). Across two canonical Sequential Social Dilemmas (Gathering and Cleanup) and two frontier LLMs (Claude Sonnet 4.6, Gemini 3.1 Pro), dense feedback consistently matches or exceeds sparse feedback on all metrics. The advantage is largest in the Cleanup public goods game, where providing social metrics helps the LLM calibrate the costly cleaning-harvesting tradeoff. Rather than triggering over-optimization of fairness, social metrics serve as a coordination signal that guides the LLM toward more effective cooperative strategies, including territory partitioning, adaptive role assignment, and the avoidance of wasteful aggression. We further perform an adversarial experiment to determine whether LLMs can reward hack these environments. We characterize five attack classes and discuss mitigations, highlighting an inherent tension in LLM policy synthesis between expressiveness and safety. Code at https://github.com/vicgalle/llm-policies-social-dilemmas.

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