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As LLMs are increasingly used to simulate and augment collective decision-making, it is critical to examine how they align with human social reasoning. The key novel contribution of this paper is a method and study of alignment for collective outcomes (as opposed to the body of work on individual behavior alignment). We adapt a classical social psychology task, Lost at Sea to study how identity cues affect group leader election in a large-scale human experiment (N=748); we also simulate the participants with the Gemini, GPT, and Claude Large Language Models (LLMs). This reveals a critical insight: the tension between alignment for simulation vs alignment for an idealized outcome. Some models mirror people, where others mask our collective biases. Moreover, when identity cues are hidden, contrary to our human study, some models use identity to compensate for male-associated dialogue resulting in more gender biased outcomes. These results highlight that understanding when LLMs mirror or mask human behavior is critical to advancing socially-aligned AI.