Most social choice rules assume access to full rankings, while current alignment practice -- despite aiming for diversity -- typically treats voters as anonymous and comparisons as independent, effectively extracting only about one bit per voter. Motivated by this gap, we study social choice under an extreme communication budget in the linear social choice model, where each voter's utility is the inner product between a latent voter type and the embedding of the context and candidate. The candidate and voter spaces may be very large or even infinite. Our core idea is to model the electorate as an unknown distribution over voter types and to recover its moments as informative summary statistics for candidate selection. We show that one pairwise comparison per voter already suffices to select a candidate that maximizes social welfare, but this elicitation cannot identify the second moment and therefore cannot support objectives that account for inequality. We prove that two pairwise comparisons per voter, or alternatively a single graded comparison, identify the second moment; moreover, these richer queries suffice to identify all moments, and hence the entire voter-type distribution. These results enable principled solutions to a range of social choice objectives including inequality-aware welfare criteria such as taking into account the spread of voter utilities and choosing a representative subset.

翻译：大多数社会选择规则假设能够获取完整的排序信息，而当前的对齐实践——尽管旨在实现多样性——通常将选民视为匿名个体，并将比较视为独立事件，实际上每位选民仅提取约一个比特的信息。受这一差距的启发，我们在线性社会选择模型中研究了极端通信预算下的社会选择问题。在该模型中，每位选民的效用是潜在选民类型与语境及候选人嵌入向量的内积。候选人与选民空间可能非常庞大甚至无限。我们的核心理念是将全体选民建模为选民类型上的未知分布，并通过恢复其矩作为信息性统计摘要以辅助候选人选择。我们证明，每位选民只需一次成对比较即可选出使社会福利最大化的候选人，但这种 elicitation 方法无法识别二阶矩，因此无法支持考虑不平等性的目标。我们进一步证明，每位选民进行两次成对比较（或等效的单次分级比较）即可识别二阶矩；并且，这些更丰富的查询足以识别所有矩，从而完整还原选民类型分布。这些结果可为基础性的社会选择目标（包括考虑不平等性的福利准则，如兼顾选民效用分布并选择代表性子集）提供原则性解决方案。