Puzzle: Suppose each voter's "vote" is an integer numerical score from 0 to 99 for each candidate, for example a legal vote in a 4-candidate election might be (99, 54, 0, 99). Suppose the "final score" for a candidate is the average of all his scores that remain after you discard the top X% and the bottom X% of them. If X=50, this is just the median score, and if X=0 then this is just the usual kind of range voting. These are both important special cases. But more generally any fixed X with 0≤X≤50 can be considered. One of the advantages of ordinary range voting, which appears to be sacrificed in this more general kind of voting, is the fact that with range voting only a small amount of information may be sent by each precinct to central tabulating, summarizing all the ballots (no matter how many) in that precinct. The entire set of ballots need not be sent.
Solution (a): for each candidate, send the 100 counts of 0-votes, 1-votes, 2-votes,... 99-votes, for that candidate to central tabulating. This is 100C numbers worth of information, where C is the number of candidates, regardless of the number of voters in the precincts (there could be a million voters and we don't care), and it suffices.
Solution (b): The main objection to "trimmed mean" range-voting variants where you discard X% of the outliers then take the average (median-range is the special case X=50, but one could also consider X=10, etc) is precisely that this discarding happens. For example,
Later note: However, Balinski & Laraki (after this puzzle was written) advanced a version of median-based range voting with a crucial "tie breaking" second stage. Their tie-breaking stage (when invoked) causes that 10% not to be totally discarded... they still can have some effect.
Solution (c): Massive ties would often happen with highest-median-score voting. In the French Approval-Voting study if voters gave zero scores to those they disapproved (or if they merely did so 80% of the time at random), then there would have been a 16-way tie for first place with all contenders getting an exactly-equal median score value of zero.
Solution b was pointed out by Jan Kok.