Most election systems are "discrete" in the sense that each candidate is either elected or not elected. One could think of an election system that is "fractional", for example: if there are two candidates for presidency, one wins 60% and one wins 40% of the votes, then the first serves for 60% of the time and then rotates with the second, who serves for 40% of the time (to make it more practical, there may be a threshold of, say, 25%). It is similar to the idea of proportional representation, but applicable not only to parliaments but also to single-person offices such as presidency.
Is there a common term for such an election system?
I found the term "probabilistic social choice", which is somewhat similar, but the fractions are interpreted as probabilities, so in the above example, after the elections there is a lottery in which the first candidate is elected with probability 60%, otherwise the second is elected.