Sorting Out Single-Crossing Preferences

We extends the single-crossing property of preferences to tree graphs, and show that it is equivalent to intermediate preferences and order restriction, also extended to tree graphs. Moreover, to facilitate the use of single-crossing in network games, we develop algorithms to answer the following questions. Given a preference profile, can we construct a tree graph that supports single-crossing, if it is? Given a set of alternatives, can we generate single-crossing preference profiles with associated tree graphs?