I have a die with $N$ sides. It has a hidden transitive order which has to be uncovered. At the start of a trial, I throw the die until I get the highest-ranking side. I then throw the die until I get the next highest-ranking side, and repeat until side $1$, the lowest-ranking, remains. After one trial, I therefore have some confidence in a ranking $N > N - 1 > N - 2 >\ \cdots\ > 2 > 1$.
I represent each side as a Beta distribution with a uniform prior, which can be updated as the trial proceeds. I can have as many trials as are necessary to uncover the ranking.
Is there an elegant way to model this?