Fix an odd natural number $k$. Suppose we have $k$ total orders on the same (finite) set $X$. Define a tournament on the vertex set $X$ by putting a directed edge $x\rightarrow y$ if a majority of the total orders compare $x > y$.
What tournaments can be obtained this way? Of course, if $k = 1$, only linearly ordered tournaments are possible. I am most interested in the case of small $k$. For example, is there an excluded-substructure characterization of these tournaments?
What if we make the problem harder and ask whether a given directed graph $G$ can be extended to a tournament $T$ such that $T$ can be obtained in this way? Again, if $k = 1$, there are various simple characterizations, such as all digraphs that contain no directed cycles.
What can be said about the computational problem of determining the smallest $k$ that can represent a given tournament or digraph?
I assume, perhaps naively, that this problem already occurs in the literature, perhaps in the theory of voting/social choice, so I would be happy with references instead of solutions if that's easier.