Let's suppose we have x people voting, and there are n proposals. People vote by ranking the proposal. We know that if the proposal can be ordered, so that each person has a favorite proposal, and then as you move away from the favorite proposal they like the most, their appreciation for the proposals go down. So if the proposals are A, B, C, D E, and are ordered in lexicographic order. Then someone that likes B will always like C more than D, and D more than E. So it will either be
B>A>C>D>E or
B>C>A>D>E or
B>C>D>A>E or
B>C>D>E>A
And if everybody shares the same order, then there is a condorcet winner, which will be the choice of the median voter (see median voter theorem).
Now, what if the order is not on a line but on a cycle.
For example suppose we are deciding the time to have a meeting. Meeting that we are going to have each day, always at the same time. Each of us has their favorite time, and as you move away from that time each of us will like that time less.
Then:
1) Does it still exist always a Condorcet Winner?
2) Is it still true that the median voter is Condorcet Winner?
3) And how do you calculate the median voter in a cycle? For example if I have a set of times all around the clock, what is the "median time"?
4) And if there is no Condorcet Winner, can someone come up with a counter example?
Thanks to anyone who can chip in something.
Thank you! There is plenty of material to dig my teeth in.
Did you actually wrote all this to answer my question, or was it a question you have worked before in your career?
Best Regards,
Pietro