There's a brilliant, modern solution to this problem, and if the failed US democracy that I was born into can be fixed, I think this will be one of the cornerstones:

https://en.wikipedia.org/wiki/Instant-runoff_voting

It's often called RCV (Ranked Choice Voting) because that's what it was dubbed in Maine, the only US state so far to approve a plan to implement it.

It rolls the runoff(s) into one vote, and leaves the drudgery to computers (could be done by schoolchildren manually, though, or whatever).