  We consider $n$-sided dice whose face values lie between $1$ and $n$ and
whose faces sum to $n(n+1)/2$. For two dice $A$ and $B$, define $A \succ B$ if
it is more likely for $A$ to show a higher face than $B$. Suppose $k$ such dice
$A_1,\dots,A_k$ are randomly selected. We conjecture that the probability of
ties goes to 0 as $n$ grows. We conjecture and provide some supporting evidence
that---contrary to intuition---each of the $2^{k \choose 2}$ assignments of
$\succ$ or $\prec$ to each pair is equally likely asymptotically. For a
specific example, suppose we randomly select $k$ dice $A_1,\dots,A_k$ and
observe that $A_1 \succ A_2 \succ \ldots \succ A_k$. Then our conjecture
asserts that the outcomes $A_k \succ A_1$ and $A_1 \prec A_k$ both have
probability approaching $1/2$ as $n \rightarrow \infty$.
