Is there a word which means that ownership rotates or moves around a known set of options? For example, the Olympics are hosted by a different nation each time, so they are [word I am looking for].
The only thing I can think of is something derived from "round-robin", which is a way of rotating ownership of tasks for scheduling in Computer Science, but I don't think that is a universally understood term.
I'm not sure if round-robin is a good fit. As used in tournaments, it simply means that each participant plays against every other participant in turn. It might repeat a couple of times but it doesn't go on ad infinitum.
As used in computer science, round-robin does not reflect how Olympics host nations or cities are scheduled -- some nations/cities have hosted the Olympics more than once, and many have never hosted them.
A better example might be the Presidency of the European Union, which rotates amongst the member states according to a very specific formula:
The presidency of the Council of the European Union, also known as the presidency of the European Union (EU), is taken in turn by each of the 28 countries of the European Union, according to a rotation system for a predetermined period of six months. The order of rotation is determined unanimously by the Council of the EU, based on the principle of alternating between "major" and "minor" member states. The presidency change takes place on January 1st and July 1st each year.
The best word to describe that is rotation -- so the answer was in your question :-)
Round-robin is not limited to computer science and is actually used in the Olympics, e.g. in curling. It is indeed used for any kind of tournament that is, well, round-robin. Chess, poker, you name it. So it is safe to use it.
That being said, the key is, as you say, "around a known set of options", which is not true of countries, and even less so of individual cities, hosting the Olympics. There is no upper limit on either.
A circus (Collins, sense 4 & AHD sense 4a):
-4. (General Sporting Terms) a travelling group of professional sportsmen: a cricket circus.
-4. Informal Something suggestive of a circus ...
But the most commonly-used sense, and the pejorative polysemes:
-7. noisy or rowdy behaviour
-8. a person or group of people whose behaviour is wild, disorganized, or (esp unintentionally) comic
add an off-flavour.
At university, groups of chemistry students visited permanently-set-up experiment stations on a weekly basis, in rotation - this was known as a 'circus'.
You may call this process "rotational hosting". This process was exercised in the history but I do not have an example of contemporary usage.
Sources:
The 1987 Copa América was 33rd edition of the Copa América, CONMEBOL's national team competition. It was the first Copa América under the new rotational hosting system.
History of the American Automatic Control Council:
For several years prior to the emergence of the American Control Conference, JACC attendee surveys were taken and analyzed, old ideas about rotational hosting, inconsistent image and conflicts were removed from new operating principles
