TITLE: Generalized nontransitive dice
QUESTION [22 upvotes]: Let $X_1, \ldots, X_n$ be a collection of random variables. Consider the directed graph with vertex set $\{ 1, 2, \ldots, n \}$ where there is a directed edge $i \to j$ if $\mathbb{P}(X_i > X_j) > \frac{1}{2}$. 
Question 1: What directed graphs can arise in this way? Certainly they must be simple and have no loops. Is that the only restriction? Alternatively, $\mathbb{P}(X_i > X_j) > \frac{1}{2}$ defines an irreflexive antisymmetric relation on $\{ 1, 2, ... n \}$. Which such relations arise in this way? 
Question 2: Does the answer change if we require the $X_i$ to all be defined on a finite sample space? 
Question 3: What if we require the $X_i$ to be independent? 
It is known (see nontransitive dice) that this graph can have directed cycles even if the $X_i$ are independent; in particular, the corresponding relation need not be transitive.

REPLY [3 votes]: This question is answered fully by NovaDenizen in this answer to a similar question. It turns out that arbitrary graphs can be achieved with independent random variables and a finite sample space. The construction is as follows:

I'm just going to describe how you'd build it up an edge at a time.
Say your graph has $n$ nodes.  Start out with $n$ identical 2-sided dice. Each having 0 on both sides will work fine.
Pick an arbitrary edge in the graph, specifying that $D_a$ should beat $D_b$.  All the dice currently have $k$ sides, and satisfy the requirements for previously added edges, and unconnected dice are even*.  Find $m$, the maximum of all the faces on the dice.  To $D_a$, add two faces with $m+3$.  To $D_b$, add two faces of $m+2$.  To all other dice, add two faces, one of $m+1$ and one of $m+4$.  Repeat this procedure for all edges in the graph, winding up with $2e+2$-sided dice.

*I assume "even" here means that neither beats the other with probability $>1/2$.