Preferences Single-Peaked on a Tree: Sampling and Tree Recognition

In voting theory, impossibility results and computational hardness results are often circumvented by recognising that voters' preferences are not arbitrary, but lie within a restricted domain. Uncovering the structure of the underlying domain often provides useful insights about the nature of the alternative space, and may be helpful in identifying a collective choice. Preferences single-peaked on a tree are an example of a relatively broad domain that nonetheless exhibits several desirable properties. We consider the setting where voters' preferences are independently sampled from rankings that are single-peaked on a given tree, and study the problem of reliably identifying the tree that generated the observed votes. We test our algorithm empirically; to this end, we develop an algorithm to uniformly sample preferences that are single-peaked on a given tree.