Social choice theory assumes full preference rankings. In practice, asking each voter to rank all candidates is expensive — cognitively and computationally. The question is: how little can you ask?
Kong et al. study a linear social choice model where voter preferences are inner products between voter types and candidate embeddings. They prove that a single pairwise comparison per voter — “do you prefer A or B?” — suffices to select the candidate maximizing social welfare. One binary question per person identifies the best option for the group.
But one comparison cannot recover the second moment of the voter distribution. This matters because inequality-aware objectives — selecting a representative subset, ensuring minority preferences are weighted — require knowing not just the average preference but how preferences are spread. The mean tells you what the group wants. The variance tells you who is being ignored.
Two pairwise comparisons per voter, or a single graded comparison (“how much do you prefer A over B?”), recover the second moment and with it the full voter distribution. The jump from one question to two is qualitative, not merely quantitative: it unlocks an entirely different class of social objectives.
The through-claim is about the information geometry of preference elicitation. The first comparison captures the first moment — the direction of collective preference. The second captures the second moment — the dispersion around that direction. No number of additional first-moment measurements can substitute for the second. Welfare maximization and fairness maximization require different statistics, and the statistics require different instruments. The question you ask determines not just the precision of the answer but which questions become answerable at all.