Voting criteria under restrictricted expression

The Questions

• Assume we have an election between n candidates.
• Let S be the set of voting systems that allow voters to express both full ordering and indifference between candidates (e.g. Ranked Pairs, the Schulze method, Range voting, etc). If the voting system requires preferential ballots, generate them from ratings ballots with range 1..n without loss of generality.
• Each system s in S satisfies a set of criteria C_s(n).
• If we reduce the range of the ratings ballots to 1..t where t < n, for what values of s and t can we still claim C_s(n) = C_s(t)?
• What is C_s(n) – C_s(2) (aka approval ballots) for each value of s?

I don’t know the answer yet, but I think it’s worth asking.

Relevance of the questions

Election between 20+ candidates have the potential to confuse voters, more so with overly expressive ballots. If we can reduce perceived complexity without fundamentally restricting options (e.g. having voters rank each candidate out of 5 stars), then the ballots might be easier to understand. In that context, the questions ask if voting systems maintain their properties after such a preference compression.

Update

While I haven’t figured out how to solve this problem in the general case yet, I did give it a try with Ranked Pairs and the following criteria: majority, monotonicity, Condorcet winner, Condorcet loser, clone independence, and local independence of irrelevant alternatives. Ranked Pairs seems to satisfy these properties even when reduced to approval ballots. Neat.