Did you know that the concept of utility was originally used to describe preferences over choices, not states of the world? Might be worth reading up on. One important idea is that you can write preferences entirely in a language of what decisions you make, not how much you value different amounts of X. This is the language that circular preferences can be written in. Pop quiz: if your utility was the angle of a wheel (so, if it turns twice, your utility is 4 pi), could you prove that writing utility in terms of the physical state of the wheel breaks down, but that you can still always have preferences over choices?
Edit: for reasons given in the comments, I don't think the question of what circular preferences actually do is well defined, so this an answer to a wrong question.
If I like Y more than X, at an exchange rate of 0.9Y for 1X, and I like Z more than Y, at an exchange rate of 0.9Z for 1Y, and I like X more than Z, at an exchange rate of 0.9X for 1Z, you might think that given 1X and the ability to trade X for Y at an exchange rate of 0.95Y for 1X, and Y for Z at an exchange rate of 0.95Z for 1Y, and Z for X at an exchange rate of 0.95X for 1Z, I would trade in a circle until I had nothing left.
But actually, if I knew that I had circular preferences, and I knew that if I had 0.95Y I would trade it for (0.95^2)Z, which I would trade for (0.95^3)X, then actually I'd be trading 1X for (0.95^3)X, which I'm obviously not going to do.
Similarly, if the exchange rates are all 1:1, but each trade costs 1 penny, and I care about 1 penny much much less than any of 1X, 1Y, or 1Z, and I trade my X for Y, I know I'm actually going to end up with X - 3 cents, so I won't make the trade.
Unless I can set a Schelling fence, in which case I will end up trading once.
So if instead of being given X, I have a 1/3 chance of each of X, Y, and Z, I would hope I wouldn't set a Schelling fence, because then my 1/3 chance of each thing becomes a 1/3 chance of each thing minus the trading penalty. So maybe I'd want to be bad at precommitments, or would I precommit not to precommit?