Approximate Single-Peakedness in Large Elections

Single-peaked preferences are a natural way to avoid paradoxes and impossibility theorems in social choice and have recently been involved in the study of various computational aspects of social choice. Since strict single-peakedness is hard to achieve in practice, approximate single-peakedness appears more appropriate and is gaining popularity. In this paper, we study approximate single-peakedness of large, randomly-generated profiles. We focused on characterizing the asymptotically optimal social axis, which is asymptotically consistent with most agents' preferences generated from a statistical model. We characterize all asymptotically optimal social axes under the Mallows model for two case: the case where the dispersion parameter $\varphi$ is close to 0, and the case where $\varphi$ is close to 1. We also design an algorithm to help characterize all asymptotically optimal social axes for all $\varphi$ when the number of alternative is no more than 10. These results help us understand the structure of approximate single-peakedness in large elections.