Intransitive Dice

We consider $n$-sided dice whose face values lie between $1$ and $n$ and whose faces sum to $n(n+1)\mathrm{/}2$. 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 $2^{\left(\genfrac{}{}{0px}{}{k}{2}\right)}$ 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 $A_1\succ A_2\succ \dots \succ A_k$. Then our conjecture asserts that the outcomes $A_k\succ A_1$ and $A_1\prec A_k$ both have probability approaching $1\mathrm{/}2$ as $n\to \mathrm{\infty }$.