What are some examples of (non-total-order) relations that happen to be tournaments?

Question

I would like to see some (non-contrived) examples of relations that show up elsewhere in mathematics that happen to be tournaments (antisymmetric, connex, non-reflexive) without being linear orders.

Answer 1

If you plot the results of a round-robin tournament as follow, you'll get a very natural tournament that isn't a linear order (if there are more than 2 participants). We consider a tournament of a game where no draw can occur, such as Go.

Say there are $n > 2$ participants $p_1,\dots, p_n$. Take $V := \{p_1, \dots, p_n\}$. Now let $E := \{(p_i, p_j) \ \big| \ p_i \textrm{ has beaten } p_j\}$. The graph $G = (V, E)$ is a tournament (whence the name, btw).

Answer 2

Let $p$ be a prime of the form $4k+3$, and let $\mathbb{Z}_p$ be the integers modulo $p$. We can construct a directed graph on $\mathbb{Z}_p$ by drawing an edge $x \rightarrow y$ if $y-x$ is a non-zero quadratic residue modulo $p$. Because $-1$ is a quadratic non-residue, exactly one of $y-x$ and $x-y$ is a residue, so the resulting directed graph is a tournament.

This is sometimes referred to as the "Paley tournament" or "Paley digraph".