TITLE: Majority vote of total orders
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: For $k=3$, the following paper shows an example of a non-3-majority tournament with 8 vertices. 
http://www2.isye.gatech.edu/~ctovey/publications/papers/voting__19_may_08.pdf
A few years ago, I checked that every tournament with 7 vertices (even the Paley tournament) is 3-majority by using a computer.