Rubinstein, Ariel, and Uzi Segal. “On the Likelihood of Cyclic Comparisons”. Boston College Working Papers in Economics 773, 2011.
We investigate the procedure of "random sampling" where the alternatives are random variables. When comparing any two alternatives, the decision maker samples each of the alternatives once and ranks them according to the comparison between the two realizations. Our main result is that when applied to three alternatives, the procedure yields a cycle with a probability bounded above by 8/27. Bounds are also obtained for other related procedures.